Tonelli-Shanks algorithm

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Tonelli-Shanks algorithm
You are encouraged to solve this task according to the task description, using any language you may know.
 This page uses content from Wikipedia. The original article was at Tonelli-Shanks algorithm. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

In computational number theory, the Tonelli–Shanks algorithm is a technique for solving for x in a congruence of the form:

x2 ≡ n (mod p)

where n is an integer which is a quadratic residue (mod p), p is an odd prime, and x,n ∈ Fp where Fp = {0, 1, ..., p - 1}.

It is used in cryptography techniques.

To apply the algorithm, we need the Legendre symbol:

The Legendre symbol (a | p) denotes the value of a(p-1)/2 (mod p).

• (a | p) ≡ 1    if a is a square (mod p)
• (a | p) ≡ -1    if a is not a square (mod p)
• (a | p) ≡ 0    if a ≡ 0 (mod p)

Algorithm pseudo-code

All ≡ are taken to mean (mod p) unless stated otherwise.

• Input: p an odd prime, and an integer n .
• Step 0: Check that n is indeed a square: (n | p) must be ≡ 1 .
• Step 1: By factoring out powers of 2 from p - 1, find q and s such that p - 1 = q2s with q odd .
• If p ≡ 3 (mod 4) (i.e. s = 1), output the two solutions r ≡ ± n(p+1)/4 .
• Step 2: Select a non-square z such that (z | p) ≡ -1 and set c ≡ zq .
• Step 3: Set r ≡ n(q+1)/2, t ≡ nq, m = s .
• Step 4: Loop the following:
• If t ≡ 1, output r and p - r .
• Otherwise find, by repeated squaring, the lowest i, 0 < i < m , such that t2i ≡ 1 .
• Let b ≡ c2(m - i - 1), and set r ≡ rb, t ≡ tb2, c ≡ b2 and m = i .

Implement the above algorithm.

Find solutions (if any) for

• n = 10 p = 13
• n = 56 p = 101
• n = 1030 p = 10009
• n = 1032 p = 10009
• n = 44402 p = 100049
Extra credit
• n = 665820697 p = 1000000009
• n = 881398088036 p = 1000000000039
• n = 41660815127637347468140745042827704103445750172002 p = 10^50 + 577

11l

Translation of: Python
```F legendre(a, p)
R pow(a, (p - 1) I/ 2, p)

F tonelli(n, p)
assert(legendre(n, p) == 1, ‘not a square (mod p)’)
V q = p - 1
V s = 0
L q % 2 == 0
q I/= 2
s++
I s == 1
R pow(n, (p + 1) I/ 4, p)
V z = 2
L
I p - 1 == legendre(z, p)
L.break
z++
V c = pow(z, q, p)
V r = pow(n, (q + 1) I/ 2, p)
V t = pow(n, q, p)
V m = s
V t2 = BigInt(0)
L (t - 1) % p != 0
t2 = (t * t) % p
V i = 1
L(ii) 1 .< m
I (t2 - 1) % p == 0
i = ii
L.break
t2 = (t2 * t2) % p
V b = pow(c, Int64(1 << (m - i - 1)), p)
r = (r * b) % p
c = (b * b) % p
t = (t * c) % p
m = i
R r

V ttest = [(BigInt(10), BigInt(13)), (BigInt(56), BigInt(101)), (BigInt(1030), BigInt(10009)), (BigInt(44402), BigInt(100049)),
(BigInt(665820697), BigInt(1000000009)), (BigInt(881398088036), BigInt(1000000000039)),
(BigInt(‘41660815127637347468140745042827704103445750172002’), BigInt(10) ^ 50 + 577)]
L(n, p) ttest
V r = tonelli(n, p)
assert((r * r - n) % p == 0)
print(‘n = #. p = #.’.format(n, p))
print("\t  roots : #. #.".format(r, p - r))```
Output:
```n = 10 p = 13
roots : 7 6
n = 56 p = 101
roots : 37 64
n = 1030 p = 10009
roots : 1632 8377
n = 44402 p = 100049
roots : 30468 69581
n = 665820697 p = 1000000009
roots : 378633312 621366697
n = 881398088036 p = 1000000000039
roots : 791399408049 208600591990
n = 41660815127637347468140745042827704103445750172002 p = 100000000000000000000000000000000000000000000000577
roots : 32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069
```

AArch64 Assembly

Works with: as version Raspberry Pi 3B version Buster 64 bits
or android 64 bits with application Termux
```/* ARM assembly AARCH64 Raspberry PI 3B or android 64 bits */
/* program tonshan64.s   */

/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"

/*******************************************/
/* Initialized data                        */
/*******************************************/
.data
szMessStartPgm:          .asciz "Program 64 bits start \n"
szMessEndPgm:            .asciz "Program normal end.\n"
szMessError:             .asciz "\033[31mError  !!!\n"
szMessErrGen:            .asciz "Error end program.\n"
szMessOverflow:          .asciz "Overflow function modulo.\n"
szMessNoSolution:        .asciz "No solution.\n"
szCarriageReturn:        .asciz "\n"

/* datas message display */
szMessEntry:             .asciz "Number : @ modulo : @ ==> "
szMessResult:            .asciz "Racine 1 : @ Racine 2 : @  \n"

/*******************************************/
/* UnInitialized data                      */
/*******************************************/
.bss
.align 4
sZoneConv:               .skip 24
/*******************************************/
/*  code section                           */
/*******************************************/
.text
.global main
main:                               // program start
ldr x0,qAdrszMessStartPgm       // display start message
bl affichageMess

mov x0,10
mov x1,13
bl displayEntry
bl computeTonSha
bl displayResult

mov x0,56
mov x1,101
bl displayEntry
bl computeTonSha
bl displayResult

mov x0,1030
mov x1,10009
bl displayEntry
bl computeTonSha
bl displayResult

mov x0,1032
mov x1,10009
bl displayEntry
bl computeTonSha
bcs 1f
bl displayResult
1:
ldr x0,[x4]
ldr x1,[x4]
bl displayEntry
bl computeTonSha
bl displayResult

ldr x0,qAdrszMessEndPgm         // display end message
bl affichageMess
b 100f
99:                                 // display error message
bl affichageMess
100:                                // standard end of the program
mov x0, #0                      // return code
mov x8, #EXIT                   // request to exit program
svc 0                           // perform system call

/******************************************************************/
/*     algorithm Tonelli–Shanks                                   */
/******************************************************************/
/* x0 contains number */
/* x1 contains modulus */
/* x0 return root 1 */
/* x1 return root 2 */
computeTonSha:
stp x10,lr,[sp,-16]!    // save  registres
stp x2,x3,[sp,-16]!     // save  registres
stp x4,x5,[sp,-16]!     // save  registres
stp x6,x7,[sp,-16]!     // save  registres
stp x8,x9,[sp,-16]!     // save  registres
stp x11,x12,[sp,-16]!   // save  registres
mov x9,x0               // save number
mov x10,x1              // save modulo p
mov x2,x10
sub x1,x2,1
lsr x1,x1,1
bl moduloPuR64
bcs 100f                // error ?
cmp x0,#1
bne 20f
sub x5,x10,1
mov x6,#1               // s
1:
lsr x5,x5,#1            // div by 2
tst x5,1                // even ?
cinc x6,x6,eq           // yes count
beq 1b                  // and loop
// x5 = q
cmp x6,#1               // s = 1 ?
bne 3f
add x1,x10,1            // compute root 1
lsr x1,x1,#2            // p + 1 / 4
mov x0,x9               // n
mov x2,x10              // p
bl moduloPuR64
bcs 100f                // error ?
neg x1,x0               // compute root 2 = - root 1
b 100f                  // and end
3:
mov x7,#3               // z
4:
mov x0,x7
mov x2,x10              //  p
sub x1,x2,1
lsr x1,x1,1             // power = p - 1 / 2
bl moduloPuR64
bcs 100f                // error ?
cmp x0,#1
cinc x7,x7,eq           // si égal à 1
cinc x7,x7,eq
beq 4b
cmp x0,0
cinc x7,x7,eq           // si egal à 0
cinc x7,x7,eq
beq 4b
mov x0,x7               // z
mov x1,x5               // q
mov x2,x10              // p
bl moduloPuR64
bcs 100f                // error ?
mov x12,x0              // c = z pow q mod p

add x1,x5,1             // = q +1
lsr x1,x1,1             // div 2
mov x0,x9               // n
mov x2,x10              // p
bl moduloPuR64
mov x4,x0               // r =  n puis (q+1)/2 mod p

mov x0,x9               // n
mov x1,x5               // = q
mov x2,x10              // p
bl moduloPuR64
bcs 100f                // error ?
mov x5,x0               // reuse r5 = t = n pow q mod p

8:                          // begin loop
cmp x5,1
beq 10f
mov x0,x5               // t
mov x1,x6               // m
mov x2,x10              // p
bl searchI              // search i for t puis 2 puis i = 1 mod p
cmp x0,-1               // not find -> no solution
beq 20f
mov x9,x0               // i
sub x8,x6,x0            // compute b
sub x8,x8,1             // m - i - 1
mov x1,1
lsl x1,x1,x8
mov x0,x12
mov x2,x10              // p
bl moduloPuR64
bcs 100f                // error ?
mov x7,x0               // b = c puis 2 puis 2 puis m-i-1  à verifier

mul x0,x7,x4            // r = r * b mod p
umulh x1,x7,x4
mov x2,x10
bl divisionReg128U
mov x4,x3               // r mod p
mul x0,x7,x7
umulh x1,x7,x7
mov x2,x10
bl divisionReg128U
mov x12,x3              // c mod p

mul x0,x5,x12
umulh x1,x5,x12
mov x2,x10
bl divisionReg128U
mov x5,x3               // t mod p

mov x6,x9               // m = i
b 8b
9:

10:
mov x0,x4               // x0 return root 1
sub x1,x10,x0           //  x1 return root 2
cmn x0,0                // carry à zero roots ok
b 100f
20:
bl affichageMess

mov x0,0
mov x1,0
cmp x0,0               // carry to 1 No solution
100:
ldp x11,x12,[sp],16
ldp x8,x9,[sp],16
ldp x6,x7,[sp],16
ldp x4,x5,[sp],16
ldp x2,x3,[sp],16
ldp x10,lr,[sp],16      // restaur des  2 registres
ret                     // retour adresse lr x30

/******************************************************************/
/*     search i                                               */
/******************************************************************/
// x0 contains t
// x1 contains maxi
// x2 contains modulo
searchI:
stp x1,lr,[sp,-16]!    // save  registres
stp x2,x3,[sp,-16]!    // save  registres
stp x4,x5,[sp,-16]!    // save  registres
stp x6,x7,[sp,-16]!    // save  registres
mov x4,x0              // t
mov x6,x1              // m
mov x3,1               // i
1:
mov x5,1
lsl x5,x5,x3           // compute 2 power i

mov x0,x4
mov x1,x5
bl moduloPuR64         // compute t pow 2 pow i mod p
cmp x0,1               // = 1 ?
beq 3f                 // yes it is ok
add x3,x3,1            // next i
cmp x3,x6
blt 1b                 // loop
mov x0,-1              // not find
b 100f
3:
mov x0,x3              // return i
100:
ldp x6,x7,[sp],16
ldp x4,x5,[sp],16
ldp x2,x3,[sp],16
ldp x1,lr,[sp],16         // restaur des  2 registres
ret                       // retour adresse lr x30
/******************************************************************/
/*     display numbers                                             */
/******************************************************************/
/* x0 contains number */
/* x1 contains modulo */
displayEntry:
stp x0,lr,[sp,-16]!        // save  registres
stp x1,x2,[sp,-16]!        // save  registres
mov x2,x1                  // root 2
ldr x1,qAdrsZoneConv       // convert root 1 in r0
bl conversion10S           // convert ascii string
bl strInsertAtCharInc      // and put in message
mov x3,x0
mov x0,x2                  // racine 2
bl conversion10S           // convert ascii string
mov x0,x3
bl strInsertAtCharInc      // and put in message
bl affichageMess
100:
ldp x1,x2,[sp],16
ldp x0,lr,[sp],16          // restaur des  2 registres
ret                        // retour adresse lr x30
/******************************************************************/
/*     display roots                                               */
/******************************************************************/
/* x0 contains root 1 */
/* x1 contains root 2 */
displayResult:
stp x1,lr,[sp,-16]!        // save  registres
stp x2,x3,[sp,-16]!        // save  registres
mov x2,x1                  // root 2
ldr x1,qAdrsZoneConv       // convert root 1 in r0
bl conversion10S           // convert ascii string
bl strInsertAtCharInc      // and put in message
mov x3,x0
mov x0,x2                  // racine 2
bl conversion10S           // convert ascii string
mov x0,x3
bl strInsertAtCharInc      // and put in message
bl affichageMess
100:
ldp x2,x3,[sp],16
ldp x1,lr,[sp],16          // restaur des  2 registres
ret                        // retour adresse lr x30
/**************************************************************/
/********************************************************/
/*   Calcul modulo de b puissance e modulo m  */
/*    Exemple 4 puissance 13 modulo 497 = 445         */
/********************************************************/
/* x0  nombre  */
/* x1 exposant */
/* x2 modulo   */
moduloPuR64:
stp x1,lr,[sp,-16]!        // save  registres
stp x3,x4,[sp,-16]!        // save  registres
stp x5,x6,[sp,-16]!        // save  registres
stp x7,x8,[sp,-16]!        // save  registres
stp x9,x10,[sp,-16]!        // save  registres
cbz x0,100f
cbz x1,100f
mov x8,x0
mov x7,x1
mov x6,1                   // resultat
udiv x4,x8,x2
msub x9,x4,x2,x8           // contient le reste
1:
tst x7,1
beq 2f
mul x4,x9,x6
umulh x5,x9,x6
mov x6,x4
mov x0,x6
mov x1,x5
bl divisionReg128U
cbnz x1,99f                // overflow
mov x6,x3
2:
mul x8,x9,x9
umulh x5,x9,x9
mov x0,x8
mov x1,x5
bl divisionReg128U
cbnz x1,99f                // overflow
mov x9,x3
lsr x7,x7,1
cbnz x7,1b
cmn x0,0                   // carry à zero pas d'erreur
mov x0,x6                  // result
b 100f
99:
bl  affichageMess
cmp x0,0                   // carry à un car erreur
mov x0,-1                  // code erreur

100:
ldp x9,x10,[sp],16          // restaur des  2 registres
ldp x7,x8,[sp],16          // restaur des  2 registres
ldp x5,x6,[sp],16          // restaur des  2 registres
ldp x3,x4,[sp],16          // restaur des  2 registres
ldp x1,lr,[sp],16          // restaur des  2 registres
ret                        // retour adresse lr x30
/***************************************************/
/*   division d un nombre de 128 bits par un nombre de 64 bits */
/***************************************************/
/* x0 contient partie basse dividende */
/* x1 contient partie haute dividente */
/* x2 contient le diviseur */
/* x0 retourne partie basse quotient */
/* x1 retourne partie haute quotient */
/* x3 retourne le reste */
divisionReg128U:
stp x6,lr,[sp,-16]!        // save  registres
stp x4,x5,[sp,-16]!        // save  registres
mov x5,#0                  // raz du reste R
mov x3,#128                // compteur de boucle
mov x4,#0                  // dernier bit
1:
lsl x5,x5,#1               // on decale le reste de 1
tst x1,1<<63               // test du bit le plus à gauche
lsl x1,x1,#1               // on decale la partie haute du quotient de 1
beq 2f
orr  x5,x5,#1              // et on le pousse dans le reste R
2:
tst x0,1<<63
lsl x0,x0,#1               // puis on decale la partie basse
beq 3f
orr x1,x1,#1               // et on pousse le bit de gauche dans la partie haute
3:
orr x0,x0,x4               // position du dernier bit du quotient
mov x4,#0                  // raz du bit
cmp x5,x2
blt 4f
sub x5,x5,x2                // on enleve le diviseur du reste
mov x4,#1                   // dernier bit à 1
4:
// et boucle
subs x3,x3,#1
bgt 1b
lsl x1,x1,#1               // on decale le quotient de 1
tst x0,1<<63
lsl x0,x0,#1              // puis on decale la partie basse
beq 5f
orr x1,x1,#1
5:
orr x0,x0,x4                  // position du dernier bit du quotient
mov x3,x5
100:
ldp x4,x5,[sp],16          // restaur des  2 registres
ldp x6,lr,[sp],16          // restaur des  2 registres
ret                        // retour adresse lr x30

/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"```
Output:
```Program 64 bits start
Number : +10 modulo : +13 ==> Racine 1 : +7 Racine 2 : +6
Number : +56 modulo : +101 ==> Racine 1 : +37 Racine 2 : +64
Number : +1030 modulo : +10009 ==> Racine 1 : +1632 Racine 2 : +8377
Number : +1032 modulo : +10009 ==> No solution.
Number : +44402 modulo : +100049 ==> Racine 1 : +30468 Racine 2 : +69581
Program normal end.
```

ARM Assembly

Works with: as version Raspberry Pi
or android 32 bits with application Termux
```/* ARM assembly Raspberry PI  or android 32 bits */
/* program tonshan.s   */

/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly
for the routine affichageMess conversion10
see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */
/************************************/
/* Constantes                       */
/************************************/
.include "../constantes.inc"

/*******************************************/
/* Initialized data                        */
/*******************************************/
.data
szMessStartPgm:          .asciz "Program 32 bits start \n"
szMessEndPgm:            .asciz "Program normal end.\n"
szMessError:             .asciz "\033[31mError  !!!\n"
szMessErrGen:            .asciz "Error end program.\n"
szMessOverflow:          .asciz "Overflow function modulo.\n"
szMessNoSolution:        .asciz "No solution.\n"
szCarriageReturn:        .asciz "\n"

/* datas message display */
szMessEntry:             .asciz "Number : @ modulo : @ ==> "
szMessResult:            .asciz "Racine 1 : @ Racine 2 : @  \n"

iNumberN:                .int 1030
iNumberP:                .int 10009

iNumberN1:               .int 1032
iNumberP1:               .int 10009

iNumberN2:               .int 44402
iNumberP2:               .int 100049

/*******************************************/
/* UnInitialized data                      */
/*******************************************/
.bss
.align 4
sZoneConv:               .skip 24
/*******************************************/
/*  code section                           */
/*******************************************/
.text
.global main
main:                               // program start
ldr r0,iAdrszMessStartPgm       // display start message
bl affichageMess

mov r0,#10
mov r1,#13
bl displayEntry                 // display entry number
bl computeTonSha                // compute roots
bl displayResult                // display roots

mov r0,#56
mov r1,#101
bl displayEntry
bl computeTonSha
bl displayResult

ldr r0,[r4]
ldr r1,[r4]
bl displayEntry
bl computeTonSha
bl displayResult

ldr r0,[r4]
ldr r1,[r4]
bl displayEntry
bl computeTonSha
bcs 1f
bl displayResult
1:
ldr r0,[r4]
ldr r1,[r4]
bl displayEntry
bl computeTonSha
bl displayResult

ldr r0,iAdrszMessEndPgm         // display end message
bl affichageMess
b 100f
99:                                 // display error message
bl affichageMess
100:                                // standard end of the program
mov r0, #0                      // return code
mov r7, #EXIT                   // request to exit program
svc 0                           // perform system call

/******************************************************************/
/*     algorithm Tonelli–Shanks                                   */
/******************************************************************/
/* r0 contains number */
/* r1 contains modulus */
/* r0 return root 1 */
/* r1 return root 2 */
computeTonSha:
push {r2-r12,lr}

mov r9,r0               // save number
mov r10,r1              // save modulo p
mov r2,r10
sub r1,r2,#1
lsr r1,r1,#1
bl moduloPuR32
cmp r0,#1
bne 20f
sub r5,r10,#1
mov r6,#1               // s
1:
lsr r5,r5,#1            // div by 2
tst r5,#1                // even ?
beq 1b                  // and loop
// r5 = q
cmp r6,#1               // s = 1 ?
bne 3f
add r1,r10,#1            // compute root 1
lsr r1,r1,#2            // p + 1 / 4
mov r0,r9               // n
mov r2,r10              // p
bl moduloPuR32
neg r1,r0               // compute root 2 = - root 1
b 100f                  // and end
3:
mov r7,#3               // z
4:
mov r0,r7
mov r2,r10              //  p
sub r1,r2,#1
lsr r1,r1,#1             // power = p - 1 / 2
bl moduloPuR32
cmp r0,#1
beq 4b
cmp r0,#0
beq 4b
mov r0,r7               // z
mov r1,r5               // q
mov r2,r10              // p
bl moduloPuR32
mov r12,r0              // c = z pow q mod p

add r1,r5,#1             // = q +1
lsr r1,r1,#1             // div 2
mov r0,r9               // n
mov r2,r10              // p
bl moduloPuR32
mov r4,r0               // r =  n puis (q+1)/2 mod p

mov r0,r9               // n
mov r1,r5               // = q
mov r2,r10              // p
bl moduloPuR32
mov r5,r0               // reuse r5 = t = n pow q mod p

8:                          // begin loop
cmp r5,#1
beq 10f
mov r0,r5               // t
mov r1,r6               // m
mov r2,r10              // p
bl searchI              // search i for t puis 2 puis i = 1 mod p
cmp r0,#-1               // not find -> no solution
beq 20f
mov r9,r0               // i
sub r8,r6,r0            // compute b
sub r8,r8,#1             // m - i - 1
mov r1,#1
lsl r1,r1,r8
mov r0,r12
mov r2,r10              // p
bl moduloPuR32
mov r7,r0               // b = c puis 2 puis 2 puis m-i-1  à verifier

umull r0,r1,r7,r4            // r = r * b mod p
mov r2,r10
bl division32R
mov r4,r2               // r mod p
umull r0,r1,r7,r7
mov r2,r10
bl division32R
mov r12,r2              // c mod p

umull r0,r1,r5,r12
mov r2,r10
bl division32R
mov r5,r2               // t mod p

mov r6,r9               // m = i
b 8b
9:

10:
mov r0,r4               // r0 return root 1
sub r1,r10,r0           //  r1 return root 2
cmn r0,#0               // carry à zero roots ok
b 100f
20:
bl affichageMess

mov r0,#0
mov r1,#0
cmp r0,#0               // carry to 1 No solution
100:
pop {r2-r12,lr}         // restaur registers
bx lr                   // return
/******************************************************************/
/*     search i                                               */
/******************************************************************/
// r0 contains t
// r1 contains maxi
// r2 contains modulo
// r0 return i
searchI:
push {r1-r6,lr}

mov r4,r0               // t
mov r6,r1               // m
mov r3,#1               // i
1:
mov r5,#1
lsl r5,r5,r3            // compute 2 power i

mov r0,r4
mov r1,r5
bl moduloPuR32          // compute t pow 2 pow i mod p
cmp r0,#1               // = 1 ?
beq 3f                  // yes it is ok
add r3,r3,#1            // next i
cmp r3,r6
blt 1b                  // loop
mov r0,#-1              // not find
b 100f
3:
mov r0,r3              // return i
100:
pop {r1-r6,lr}         // restaur registers
bx lr                  // return
/******************************************************************/
/*     display numbers                                             */
/******************************************************************/
/* r0 contains number */
/* r1 contains modulo */
displayEntry:
push {r0-r3,lr}
mov r2,r1                  // root 2
ldr r1,iAdrsZoneConv       // convert root 1 in r0
bl conversion10S           // convert ascii string
bl strInsertAtCharInc      // and put in message
mov r3,r0
mov r0,r2                  // racine 2
bl conversion10S           // convert ascii string
mov r0,r3
bl strInsertAtCharInc      // and put in message
bl affichageMess
100:
pop {r0-r3,lr}             // restaur registers
bx lr                      // return
/******************************************************************/
/*     display roots                                               */
/******************************************************************/
/* r0 contains root 1 */
/* r1 contains root 2 */
displayResult:
push {r1-r3,lr}
mov r2,r1                  // root 2
ldr r1,iAdrsZoneConv       // convert root 1 in r0
bl conversion10S           // convert ascii string
bl strInsertAtCharInc      // and put in message
mov r3,r0
mov r0,r2                  // racine 2
bl conversion10S           // convert ascii string
mov r0,r3
bl strInsertAtCharInc      // and put in message
bl affichageMess
100:

pop {r1-r3,lr}             // restaur registers
bx lr                      // return
/********************************************************/
/*   Calcul modulo de b puissance e modulo m  */
/*    Exemple 4 puissance 13 modulo 497 = 445         */
/*                                             */
/********************************************************/
/* r0  nombre  */
/* r1 exposant */
/* r2 modulo   */
/* r0 return result  */
moduloPuR32:
push {r1-r7,lr}    @ save registers
cmp r0,#0          @ verif <> zero
beq 90f
cmp r1,#0          @ verif <> zero
moveq r0,#0
beq 90f
cmp r2,#0          @ verif <> zero
moveq r0,#0
beq 90f            @
1:
mov r4,r2          @ save modulo
mov r5,r1          @ save exposant
mov r6,r0          @ save base
mov r3,#1          @ start result

mov r1,#0          @ division de r0,r1 par r2
bl division32R
mov r6,r2          @ base <- remainder
2:
tst r5,#1          @  exposant even or odd
beq 3f
umull r0,r1,r6,r3
mov r2,r4
bl division32R
mov r3,r2          @ result <- remainder
3:
umull r0,r1,r6,r6
mov r2,r4
bl division32R
mov r6,r2          @ base <- remainder

lsr r5,#1          @ left shift 1 bit
cmp r5,#0          @ end ?
bne 2b
mov r0,r3
90:
cmn r0,#0          @ no error
100:                   @ fin standard de la fonction
pop {r1-r7,lr}     @ restaur des registres
bx lr              @ retour de la fonction en utilisant lr

/***************************************************/
/*   division number 64 bits in 2 registers by number 32 bits */
/***************************************************/
/* r0 contains lower part dividende   */
/* r1 contains upper part dividende   */
/* r2 contains divisor   */
/* r0 return lower part quotient    */
/* r1 return upper part quotient    */
/* r2 return remainder               */
division32R:
push {r3-r9,lr}    @ save registers
mov r6,#0          @ init upper upper part remainder  !!
mov r7,r1          @ init upper part remainder with upper part dividende
mov r8,r0          @ init lower part remainder with lower part dividende
mov r9,#0          @ upper part quotient
mov r4,#0          @ lower part quotient
mov r5,#32         @ bits number
1:                     @ begin loop
lsl r6,#1          @ shift upper upper part remainder
lsls r7,#1         @ shift upper  part remainder
orrcs r6,#1
lsls r8,#1         @ shift lower  part remainder
orrcs r7,#1
lsls r4,#1         @ shift lower part quotient
lsl r9,#1          @ shift upper part quotient
orrcs r9,#1
@ divisor sustract  upper  part remainder
subs r7,r2
sbcs  r6,#0        @ and substract carry
bmi 2f             @ négative ?

@ positive or equal
orr r4,#1          @ 1 -> right bit quotient
b 3f
2:                     @ negative
orr r4,#0          @ 0 -> right bit quotient
adds r7,r2         @ and restaur remainder
3:
subs r5,#1         @ decrement bit size
bgt 1b             @ end ?
mov r0,r4          @ lower part quotient
mov r1,r9          @ upper part quotient
mov r2,r7          @ remainder
100:                   @ function end
pop {r3-r9,lr}     @ restaur registers
bx lr
/***************************************************/
/*      ROUTINES INCLUDE                           */
/***************************************************/
.include "../affichage.inc"```
Output:
```Program 32 bits start
Number :         +10 modulo :         +13 ==> Racine 1 :          +7 Racine 2 :          +6
Number :         +56 modulo :        +101 ==> Racine 1 :         +37 Racine 2 :         +64
Number :       +1030 modulo :      +10009 ==> Racine 1 :       +1632 Racine 2 :       +8377
Number :       +1032 modulo :      +10009 ==> No solution.
Number :      +44402 modulo :     +100049 ==> Racine 1 :      +30468 Racine 2 :      +69581
Program normal end.
```

C

Version 1

Translation of: C#
```#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>

uint64_t modpow(uint64_t a, uint64_t b, uint64_t n) {
uint64_t x = 1, y = a;
while (b > 0) {
if (b % 2 == 1) {
x = (x * y) % n; // multiplying with base
}
y = (y * y) % n; // squaring the base
b /= 2;
}
return x % n;
}

struct Solution {
uint64_t root1, root2;
bool exists;
};

struct Solution makeSolution(uint64_t root1, uint64_t root2, bool exists) {
struct Solution sol;
sol.root1 = root1;
sol.root2 = root2;
sol.exists = exists;
return sol;
}

struct Solution ts(uint64_t n, uint64_t p) {
uint64_t q = p - 1;
uint64_t ss = 0;
uint64_t z = 2;
uint64_t c, r, t, m;

if (modpow(n, (p - 1) / 2, p) != 1) {
return makeSolution(0, 0, false);
}

while ((q & 1) == 0) {
ss += 1;
q >>= 1;
}

if (ss == 1) {
uint64_t r1 = modpow(n, (p + 1) / 4, p);
return makeSolution(r1, p - r1, true);
}

while (modpow(z, (p - 1) / 2, p) != p - 1) {
z++;
}

c = modpow(z, q, p);
r = modpow(n, (q + 1) / 2, p);
t = modpow(n, q, p);
m = ss;

while (true) {
uint64_t i = 0, zz = t;
uint64_t b = c, e;
if (t == 1) {
return makeSolution(r, p - r, true);
}
while (zz != 1 && i < (m - 1)) {
zz = zz * zz % p;
i++;
}
e = m - i - 1;
while (e > 0) {
b = b * b % p;
e--;
}
r = r * b % p;
c = b * b % p;
t = t * c % p;
m = i;
}
}

void test(uint64_t n, uint64_t p) {
struct Solution sol = ts(n, p);
printf("n = %llu\n", n);
printf("p = %llu\n", p);
if (sol.exists) {
printf("root1 = %llu\n", sol.root1);
printf("root2 = %llu\n", sol.root2);
} else {
printf("No solution exists\n");
}
printf("\n");
}

int main() {
test(10, 13);
test(56, 101);
test(1030, 10009);
test(1032, 10009);
test(44402, 100049);

return 0;
}
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581```

Version 2

```// return (a * b) % mod, avoiding overflow errors while doing modular multiplication.
static unsigned multiplication_modulo(unsigned a, unsigned b, const unsigned mod) {
unsigned res = 0, tmp;
for (b %= mod; a; a & 1 ? b >= mod - res ? res -= mod : 0, res += b : 0, a >>= 1, (tmp = b) >= mod - b ? tmp -= mod : 0, b += tmp);
return res % mod;
}

// return (n ^ exp) % mod
static unsigned mod_pow(unsigned n, unsigned exp, const unsigned mod) {
unsigned res = 1;
for (n %= mod; exp; exp & 1 ? res = multiplication_modulo(res, n, mod) : 0, n = multiplication_modulo(n, n, mod), exp >>= 1);
return res;
}

static unsigned tonelli_shanks_1(const unsigned n, const unsigned mod) {
// return root such that (root * root) % mod congruent to n % mod.
// return 0 if no solution to the congruence exists.
// mod is assumed odd prime.
const unsigned a = n % mod;
unsigned res, b, c, d, e, f, g, h;
if (mod_pow(a, (mod - 1) >> 1, mod) != 1)
res = 0;
else
switch (mod & 7) {
case 3 : case 7 :
res = mod_pow(a, (mod + 1) >> 2, mod);
break;
case 5 :
res = mod_pow(a, (mod + 3) >> 3, mod);
if (multiplication_modulo(res, res, mod) != a){
b = mod_pow(2, (mod - 1) >> 2, mod);
res = multiplication_modulo(res, b, mod);
}
break;
default :
if (a == 1)
res = 1;
else {
for (c = mod - 1, d = 2; d < mod && mod_pow(d, c >> 1, mod) != c; ++d);
for (e = 0; !(c & 1); ++e, c >>= 1);
f = mod_pow(a, c, mod);
b = mod_pow(d, c, mod);
for (h = 0, g = 0; h < e; h++) {
d = mod_pow(b, g, mod);
d = multiplication_modulo(d, f, mod);
d = mod_pow(d, 1 << (e - 1 - h), mod);
if (d == mod - 1)
g += 1 << h;
}
f = mod_pow(a, (c + 1) >> 1, mod);
b = mod_pow(b, g >> 1, mod);
res = multiplication_modulo(f, b, mod);
}
}
return res;
}

// return root such that (root * root) % mod congruent to n % mod.
// return 0 (the default value of a) if no solution to the congruence exists.
static unsigned tonelli_shanks_2(unsigned n, const unsigned mod) {
unsigned a = 0, b = mod - 1, c, d = b, e = 0, f = 2, g;
if (mod_pow(n, b >> 1, mod) == 1) {
for (; !(d & 1); ++e, d >>= 1);
if (e == 1)
a = mod_pow(n, (mod + 1) >> 2, mod);
else {
for (; b != mod_pow(f, b >> 1, mod); ++f);
for (b = mod_pow(f, d, mod), a = mod_pow(n, (d + 1) >> 1, mod), c = mod_pow(n, d, mod), g = e; c != 1; g = d) {
for (d = 0, e = c, --g; e != 1 && d < g; ++d)
e = multiplication_modulo(e, e, mod);
for (f = b, n = g - d; n--;)
f = multiplication_modulo(f, f, mod);
a = multiplication_modulo(a, f, mod);
b = multiplication_modulo(f, f, mod);
c = multiplication_modulo(c, b, mod);
}
}
}
return a;
}

#include <assert.h>
int main() {
unsigned n, mod, root ; /* root_2 = mod - root */

n = 27875, mod = 26371, root = tonelli_shanks_1(n, mod);
assert(root == 14320); // 14320 * 14320  mod  26371 = 1504     and   1504 =    27875 mod 26371

n = 1111111111, mod = 1111111121, root = tonelli_shanks_1(n, mod);
assert(root == 88664850);

n = 5258, mod = 3851, root = tonelli_shanks_1(n, mod);
assert(root == 0); // no solution to the congruence exists.
}
```

A is assumed odd prime, the algorithm requires O(log A + r * r) multiplications modulo A, where r is the power of 2 dividing A − 1.

C#

Translation of: Java
```using System;
using System.Collections.Generic;
using System.Numerics;

namespace TonelliShanks {
class Solution {
private readonly BigInteger root1, root2;
private readonly bool exists;

public Solution(BigInteger root1, BigInteger root2, bool exists) {
this.root1 = root1;
this.root2 = root2;
this.exists = exists;
}

public BigInteger Root1() {
return root1;
}

public BigInteger Root2() {
return root2;
}

public bool Exists() {
return exists;
}
}

class Program {
static Solution Ts(BigInteger n, BigInteger p) {
if (BigInteger.ModPow(n, (p - 1) / 2, p) != 1) {
return new Solution(0, 0, false);
}

BigInteger q = p - 1;
BigInteger ss = 0;
while ((q & 1) == 0) {
ss = ss + 1;
q = q >> 1;
}

if (ss == 1) {
BigInteger r1 = BigInteger.ModPow(n, (p + 1) / 4, p);
return new Solution(r1, p - r1, true);
}

BigInteger z = 2;
while (BigInteger.ModPow(z, (p - 1) / 2, p) != p - 1) {
z = z + 1;
}
BigInteger c = BigInteger.ModPow(z, q, p);
BigInteger r = BigInteger.ModPow(n, (q + 1) / 2, p);
BigInteger t = BigInteger.ModPow(n, q, p);
BigInteger m = ss;

while (true) {
if (t == 1) {
return new Solution(r, p - r, true);
}
BigInteger i = 0;
BigInteger zz = t;
while (zz != 1 && i < (m - 1)) {
zz = zz * zz % p;
i = i + 1;
}
BigInteger b = c;
BigInteger e = m - i - 1;
while (e > 0) {
b = b * b % p;
e = e - 1;
}
r = r * b % p;
c = b * b % p;
t = t * c % p;
m = i;
}
}

static void Main(string[] args) {
List<Tuple<long, long>> pairs = new List<Tuple<long, long>>() {
new Tuple<long, long>(10, 13),
new Tuple<long, long>(56, 101),
new Tuple<long, long>(1030, 10009),
new Tuple<long, long>(1032, 10009),
new Tuple<long, long>(44402, 100049),
new Tuple<long, long>(665820697, 1000000009),
new Tuple<long, long>(881398088036, 1000000000039),
};

foreach (var pair in pairs) {
Solution sol = Ts(pair.Item1, pair.Item2);
Console.WriteLine("n = {0}", pair.Item1);
Console.WriteLine("p = {0}", pair.Item2);
if (sol.Exists()) {
Console.WriteLine("root1 = {0}", sol.Root1());
Console.WriteLine("root2 = {0}", sol.Root2());
} else {
Console.WriteLine("No solution exists");
}
Console.WriteLine();
}

BigInteger bn = BigInteger.Parse("41660815127637347468140745042827704103445750172002");
BigInteger bp = BigInteger.Pow(10, 50) + 577;
Solution bsol = Ts(bn, bp);
Console.WriteLine("n = {0}", bn);
Console.WriteLine("p = {0}", bp);
if (bsol.Exists()) {
Console.WriteLine("root1 = {0}", bsol.Root1());
Console.WriteLine("root2 = {0}", bsol.Root2());
} else {
Console.WriteLine("No solution exists");
}
}
}
}
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069```

C++

```#include <cstdint>
#include <iostream>
#include <vector>

struct Pair {
uint64_t n;
uint64_t p;
};

struct Solution {
uint64_t root1;
uint64_t root2;
bool is_square;
};

uint64_t multiply_modulus(uint64_t a, uint64_t b, const uint64_t& modulus) {
a %= modulus; b %= modulus;
if ( b < a ) {
uint64_t temp = a; a = b; b = temp;
}

uint64_t result = 0;
while ( a > 0 ) {
if ( a % 2 == 1 ) {
result = ( result + b ) % modulus;
};
b = ( b << 1 ) % modulus;
a >>= 1;
}
return result;
}

uint64_t power_modulus(uint64_t base, uint64_t exponent, const uint64_t& modulus) {
if ( modulus == 1 ) {
return 0;
}

base %= modulus;
uint64_t result = 1;
while ( exponent > 0 ) {
if ( ( exponent & 1 ) == 1 ) {
result = multiply_modulus(result, base, modulus);
}
base = multiply_modulus(base, base, modulus);
exponent >>= 1;
}
return result;
}

uint64_t legendre(const uint64_t& a, const uint64_t& p) {
return power_modulus(a, ( p - 1 ) / 2, p);
}

Solution tonelli_shanks(const uint64_t& n, const uint64_t& p) {
if ( legendre(n, p) != 1 ) {
return Solution(0, 0, false);
}

// Factor out powers of 2 from p - 1
uint64_t q = p - 1;
uint64_t s = 0;
while ( q % 2 == 0 ) {
q /= 2;
s += 1;
}

if ( s == 1 ) {
uint64_t result = power_modulus(n, ( p + 1 ) / 4, p);
return Solution(result, p - result, true);
}

// Find a non-square z such as ( z | p ) = -1
uint64_t z = 2;
while ( legendre(z, p) != p - 1 ) {
z += 1;
}

uint64_t c = power_modulus(z, q, p);
uint64_t t = power_modulus(n, q, p);
uint64_t m = s;
uint64_t result = power_modulus(n, ( q + 1 ) >> 1, p);

while ( t != 1 ) {
uint64_t i = 1;
z = multiply_modulus(t, t, p);
while ( z != 1 && i < m - 1 ) {
i += 1;
z = multiply_modulus(z, z, p);
}
uint64_t b = power_modulus(c, 1 << ( m - i - 1 ), p);
c = multiply_modulus(b, b, p);
t = multiply_modulus(t, c, p);
m = i;
result = multiply_modulus(result, b, p);
}
return Solution(result, p - result, true);
}

int main() {
const std::vector<Pair> tests = { Pair(10, 13), Pair(56, 101), Pair(1030, 1009), Pair(1032, 1009),
Pair(44402, 100049), Pair(665820697, 1000000009), Pair(881398088036, 1000000000039) };

for ( const Pair& test : tests ) {
Solution solution = tonelli_shanks(test.n, test.p);
std::cout << "n = " << test.n << ", p = " << test.p;
if ( solution.is_square == true ) {
std::cout << " has solutions: " << solution.root1 << " and " << solution.root2 << std::endl << std::endl;
} else {
std::cout << " has no solutions because n is not a square modulo p" << std::endl << std::endl;
}
}
}
```
Output:
```n = 10, p = 13 has solutions: 7 and 6

n = 56, p = 101 has solutions: 37 and 64

n = 1030, p = 1009 has solutions: 651 and 358

n = 1032, p = 1009 has no solutions because n is not a square modulo p

n = 44402, p = 100049 has solutions: 30468 and 69581

n = 665820697, p = 1000000009 has solutions: 378633312 and 621366697

n = 881398088036, p = 1000000000039 has solutions: 791399408049 and 208600591990
```

Clojure

```(defn find-first
" Finds first element of collection that satisifies predicate function pred "
[pred coll]
(first (filter pred coll)))

(defn modpow
" b^e mod m (using Java which solves some cases the pure clojure method has to be modified to tackle--i.e. with large b & e and
calculation simplications when gcd(b, m) == 1 and gcd(e, m) == 1) "
[b e m]
(.modPow (biginteger b) (biginteger e) (biginteger m)))

(defn legendre [a p]
(modpow a (quot (dec p) 2) p)
)

(defn tonelli [n p]
" Following Wikipedia https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm "
(assert (= (legendre n p) 1) "not a square (mod p)")
(loop [q (dec p)                                                  ; Step 1 in Wikipedia
s 0]
(if (zero? (rem q 2))
(recur (quot q 2) (inc s))
(if (= s 1)
(modpow n (quot (inc p) 4) p)
(let [z (find-first #(= (dec p) (legendre % p)) (range 2 p))] ; Step 2 in Wikipedia
(loop [
M s
c (modpow z q p)
t (modpow n q p)
R (modpow n (quot (inc q) 2) p)]
(if (= t 1)
R
(let [i (long (find-first #(= 1 (modpow t (bit-shift-left 1 %) p)) (range 1 M))) ; Step 3
b (modpow c (bit-shift-left 1 (- M i 1)) p)
M i
c (modpow b 2 p)
t (rem (* t c) p)
R (rem (* R b) p)]
(recur M c t R)
)
)
)
)
)
)
)
)

; Testing--using Python examples
(doseq [[n p]  [[10, 13], [56, 101], [1030, 10009], [44402, 100049],
[665820697, 1000000009], [881398088036, 1000000000039],
[41660815127637347468140745042827704103445750172002, 100000000000000000000000000000000000000000000000577]]
:let [r (tonelli n p)]]
(println (format "n: %5d p: %d \n\troots: %5d %5d" (biginteger n) (biginteger p) (biginteger r) (biginteger (- p r)))))
```
Output:

n: 10 p: 13 roots: 7 6 n: 56 p: 101 roots: 37 64 n: 1030 p: 10009 roots: 1632 8377 n: 44402 p: 100049 roots: 30468 69581 n: 665820697 p: 1000000009 roots: 378633312 621366697 n: 881398088036 p: 1000000000039 roots: 791399408049 208600591990 n: 41660815127637347468140745042827704103445750172002 p: 100000000000000000000000000000000000000000000000577 roots: 32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069

D

Translation of: Kotlin
```import std.bigint;
import std.stdio;
import std.typecons;

alias Pair = Tuple!(long, "n", long, "p");

enum BIGZERO = BigInt("0");
enum BIGONE = BigInt("1");
enum BIGTWO = BigInt("2");
enum BIGTEN = BigInt("10");

struct Solution {
BigInt root1, root2;
bool exists;
}

/// https://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method
BigInt modPow(BigInt b, BigInt e, BigInt n) {
if (n == 1) return BIGZERO;
BigInt result = 1;
b = b % n;
while (e > 0) {
if (e % 2 == 1) {
result = (result * b) % n;
}
e >>= 1;
b = (b*b) % n;
}
return result;
}

Solution ts(long n, long p) {
return ts(BigInt(n), BigInt(p));
}

Solution ts(BigInt n, BigInt p) {
auto powMod(BigInt a, BigInt e) {
return a.modPow(e, p);
}

auto ls(BigInt a) {
return powMod(a, (p-1)/2);
}

if (ls(n) != 1) return Solution(BIGZERO, BIGZERO, false);
auto q = p - 1;
auto ss = BIGZERO;
while ((q & 1) == 0) {
ss = ss + 1;
q = q >> 1;
}

if (ss == BIGONE) {
auto r1 = powMod(n, (p + 1) / 4);
return Solution(r1, p - r1, true);
}

auto z = BIGTWO;
while (ls(z) != p - 1) z = z + 1;
auto c = powMod(z, q);
auto r = powMod(n, (q + 1) / 2);
auto t = powMod(n, q);
auto m = ss;

while (true) {
if (t == 1) return Solution(r, p - r, true);
auto i = BIGZERO;
auto zz = t;
while (zz != 1 && i < m - 1) {
zz  = zz * zz % p;
i = i + 1;
}
auto b = c;
auto e = m - i - 1;
while (e > 0) {
b = b * b % p;
e = e - 1;
}
r = r * b % p;
c = b * b % p;
t = t * c % p;
m = i;
}
}

void main() {
auto pairs = [
Pair(             10L,                13L),
Pair(             56L,               101L),
Pair(          1_030L,            10_009L),
Pair(          1_032L,            10_009L),
Pair(         44_402L,           100_049L),
Pair(    665_820_697L,     1_000_000_009L),
Pair(881_398_088_036L, 1_000_000_000_039L),
];

foreach (pair; pairs) {
auto sol = ts(pair.n, pair.p);

writeln("n = ", pair.n);
writeln("p = ", pair.p);
if (sol.exists) {
writeln("root1 = ", sol.root1);
writeln("root2 = ", sol.root2);
}
else writeln("No solution exists");
writeln();
}

auto bn = BigInt("41660815127637347468140745042827704103445750172002");
auto bp = BIGTEN ^^ 50 + 577L;
auto sol = ts(bn, bp);
writeln("n = ", bn);
writeln("p = ", bp);
if (sol.exists) {
writeln("root1 = ", sol.root1);
writeln("root2 = ", sol.root2);
}
else writeln("No solution exists");
}
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069```

EchoLisp

```(require 'bigint)
;; test equality mod p
(define-syntax-rule (mod= a b p)
(zero?  (% (- a b) p)))
;; assign mod p
(define-syntax-rule (mod:≡ s v p)
(set! s (% v p)))

(define (Legendre a p)
(powmod a (/ (1- p) 2) p))

(define (Tonelli n p)
(unless (= 1 (Legendre n p)) (error "not a square (mod p)" (list n p)))
(define q (1- p))
(define s 0)
(while (even? q)
(/= q 2)
(++ s))
(if (= s 1) (powmod n (/ (1+ p) 4) p)
(begin
(define z
(for ((z (in-range 2 p)))
#:break (= (1- p)  (Legendre z p)) => z ))

(define c (powmod z q p))
(define r (powmod n (/ (1+ q) 2) p))
(define t (powmod n q p))
(define m s)
(define t2 0)
(while #t
#:break (mod= 1  t p) => r
(mod:≡ t2 (* t t) p)
(define i
(for ((i (in-range 1 m)))
#:break (mod= t2 1 p) => i
(mod:≡ t2 (* t2 t2) p)))
(define b (powmod c (expt 2 (- m i 1)) p))
(mod:≡ r (* r b) p)
(mod:≡ c (* b b) p)
(mod:≡ t (* t c) p)
(set! m i)))))
```
Output:
```(define ttest
`((10 13) (56 101) (1030 10009) (44402 100049)
(665820697 1000000009)
(881398088036  1000000000039)
(41660815127637347468140745042827704103445750172002  ,(+ 1e50 577))))

(for ((test ttest))
(define n (first test))
(define p (second test))
(define r (Tonelli n p))
(assert (mod= (* r r) n p))
(printf "n = %d p = %d" n p)
(printf "\t  roots : %d %d"  r (- p r))))

n = 10 p = 13
roots : 7 6
n = 56 p = 101
roots : 37 64
n = 1030 p = 10009
roots : 1632 8377
n = 44402 p = 100049
roots : 30468 69581
n = 665820697 p = 1000000009
roots : 378633312 621366697
n = 881398088036 p = 1000000000039
roots : 791399408049 208600591990
n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
roots : 32102985369940620849741983987300038903725266634508
67897014630059379150258016012699961096274733366069
(Tonelli 1032 10009)
❌ error: not a square (mod p) (1032 10009)
```

FreeBASIC

LongInt version

```' version 11-04-2017
' compile with: fbc -s console
' maximum for p is 17 digits to be on the save side

' TRUE/FALSE are built-in constants since FreeBASIC 1.04
' But we have to define them for older versions.
#Ifndef TRUE
#Define FALSE 0
#Define TRUE Not FALSE
#EndIf

Function mul_mod(a As ULongInt, b As ULongInt, modulus As ULongInt) As ULongInt
' returns a * b mod modulus
Dim As ULongInt x, y = a Mod modulus

While b > 0
If (b And 1) = 1 Then
x = (x + y) Mod modulus
End If
y = (y Shl 1) Mod modulus
b = b Shr 1
Wend

Return x

End Function

Function pow_mod(b As ULongInt, power As ULongInt, modulus As ULongInt) As ULongInt
' returns b ^ power mod modulus
Dim As ULongInt x = 1

While power > 0
If (power And 1) = 1 Then
' x = (x * b) Mod modulus
x = mul_mod(x, b, modulus)
End If
' b = (b * b) Mod modulus
b = mul_mod(b, b, modulus)
power = power Shr 1
Wend

Return x

End Function

Function Isprime(n As ULongInt, k As Long) As Long
' miller-rabin prime test
If n > 9223372036854775808ull Then ' limit 2^63, pow_mod/mul_mod can't handle bigger numbers
Print "number is to big, program will end"
Sleep
End
End If

' 2 is a prime, if n is smaller then 2 or n is even then n = composite
If n = 2 Then Return TRUE
If (n < 2) OrElse ((n And 1) = 0) Then Return FALSE

Dim As ULongInt a, x, n_one = n - 1, d = n_one
Dim As UInteger s

While (d And 1) = 0
d = d Shr 1
s = s + 1
Wend

While k > 0
k = k - 1
a = Int(Rnd * (n -2)) +2          ' 2 <= a < n
x = pow_mod(a, d, n)
If (x = 1) Or (x = n_one) Then Continue While
For r As Integer = 1 To s -1
x = pow_mod(x, 2, n)
If x = 1 Then Return FALSE
If x = n_one Then Continue While
Next
If x <> n_one Then Return FALSE
Wend
Return TRUE

End Function

Function legendre_symbol (a As LongInt, p As LongInt) As LongInt

Dim As LongInt x = pow_mod(a, ((p -1) \ 2), p)
If p -1 = x Then
Return x - p
Else
Return x
End If

End Function

' ------=< MAIN >=------

Dim As LongInt b, c, i, k, m, n, p, q, r, s, t, z

For k = 1 To 7
Print "Find solution for n ="; n; " and p =";p

If legendre_symbol(n, p) <> 1 Then
Print n;" is not a quadratic residue"
Print
Continue For
End If

If p = 2 OrElse Isprime(p, 15) = FALSE Then
Print p;" is not a odd prime"
Print
Continue For
End If

s = 0 : q = p -1
Do
s += 1
q \= 2
Loop Until (q And 1) = 1

If s = 1 And (p Mod 4) = 3 Then
r = pow_mod(n, ((p +1) \ 4), p)
Print "Solution found:"; r; " and"; p - r
Print
Continue For
End If

z = 1
Do
z += 1
Loop Until legendre_symbol(z, p) = -1
c = pow_mod(z, q, p)
r = pow_mod(n, (q +1) \ 2, p)
t = pow_mod(n, q, p)
m = s

Do
i = 0
If (t Mod p) = 1 Then
Print "Solution found:"; r; " and"; p - r
Print
Continue For
End If

Do
i += 1
If i >= m Then Continue For
Loop Until pow_mod(t, 2 ^ i, p) = 1
b = pow_mod(c, (2 ^ (m - i -1)), p)
r = mul_mod(r, b, p)
c = mul_mod(b, b, p)
t = mul_mod(t, c, p)' t = t * b ^ 2
m = i
Loop

Next

Data 10, 13, 56, 101, 1030, 10009, 1032, 10009, 44402, 100049
Data 665820697, 1000000009, 881398088036, 1000000000039

' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End```
Output:
```Find solution for n = 10 and p = 13
Solution found: 7 and 6

Find solution for n = 56 and p = 101
Solution found: 37 and 64

Find solution for n = 1030 and p = 10009
Solution found: 1632 and 8377

Find solution for n = 1032 and p = 10009
1032 is not a quadratic residue

Find solution for n = 44402 and p = 100049
Solution found: 30468 and 69581

Find solution for n = 665820697 and p = 1000000009
Solution found: 378633312 and 621366697

Find solution for n = 881398088036 and p = 1000000000039
Solution found: 791399408049 and 208600591990```

GMP version

Library: GMP
```' version 12-04-2017
' compile with: fbc -s console

#Include Once "gmp.bi"

Data "10", "13", "56", "101", "1030", "10009", "1032", "10009"
Data "44402", "100049", "665820697", "1000000009"
Data "881398088036", "1000000000039"
Data "41660815127637347468140745042827704103445750172002"   ' p = 10^50 + 577

' ------=< MAIN >=------

Dim As uLong k
Dim As ZString Ptr zstr
Dim As String n_str, p_str

Dim As Mpz_ptr b, c, i, m, n, p, q, r, s, t, z, tmp
b = Allocate(Len(__Mpz_struct)) : Mpz_init(b)
c = Allocate(Len(__Mpz_struct)) : Mpz_init(c)
i = Allocate(Len(__Mpz_struct)) : Mpz_init(i)
m = Allocate(Len(__Mpz_struct)) : Mpz_init(m)
n = Allocate(Len(__Mpz_struct)) : Mpz_init(n)
p = Allocate(Len(__Mpz_struct)) : Mpz_init(p)
q = Allocate(Len(__Mpz_struct)) : Mpz_init(q)
r = Allocate(Len(__Mpz_struct)) : Mpz_init(r)
s = Allocate(Len(__Mpz_struct)) : Mpz_init(s)
t = Allocate(Len(__Mpz_struct)) : Mpz_init(t)
z = Allocate(Len(__Mpz_struct)) : Mpz_init(z)
tmp = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp)

For k = 1 To 8
Mpz_set_str(n, n_str, 10)
If k < 8 Then
Mpz_set_str(p, p_str, 10)
Else
p_str = "10^50 + 577"
Mpz_set_str(p, "1" + String(50, "0"), 10)
End If

Print "Find solution for n = "; n_str; " and p = "; p_str

If Mpz_legendre(n, p) <> 1 Then
Print n_str; " is not a quadratic residue"
Print
Continue For
End If

If Mpz_tstbit(p, 0) = 0 OrElse Mpz_probab_prime_p(p, 20) = 0 Then
Print p_str; "is not a odd prime"
Print
Continue For
End If

Mpz_set_ui(s, 0) : Mpz_set(q, p) : Mpz_sub_ui(q, q, 1) ' q = p -1
Do
Mpz_fdiv_q_2exp(q, q, 1)
Loop Until Mpz_tstbit(q, 0) = 1

If Mpz_cmp_ui(s, 1) = 0 Then
If Mpz_tstbit(p, 1) = 1 Then
Mpz_fdiv_q_2exp(tmp, tmp, 2)         ' tmp = p +1 \ 4
Mpz_powm(r, n, tmp, p)
zstr = Mpz_get_str(0, 10, r)
Print "Solution found: "; *zstr;
Mpz_sub(r, p, r)
zstr = Mpz_get_str(0, 10, r)
Print " and "; *zstr
Print
Continue For
End If
End If

Mpz_set_ui(z, 1)
Do
Loop Until Mpz_legendre(z, p) = -1
Mpz_powm(c, z, q, p)
Mpz_fdiv_q_2exp(tmp, tmp, 1)
Mpz_powm(r, n, tmp, p)
Mpz_powm(t, n, q, p)
Mpz_set(m, s)

Do
Mpz_set_ui(i, 0)
Mpz_mod(tmp, t, p)
If Mpz_cmp_ui(tmp, 1) = 0 Then
zstr = Mpz_get_str(0, 10, r)
Print "Solution found: "; *zstr;
Mpz_sub(r, p, r)
zstr = Mpz_get_str(0, 10, r)
Print " and "; *zstr
Print
Continue For
End If

Mpz_set_ui(q, 1)
Do
If Mpz_cmp(i, m) >= 0 Then
Continue For
end if
Mpz_mul_ui(q, q, 2)                  ' q = 2^i
Mpz_powm(tmp, t, q, p)
Loop Until Mpz_cmp_ui(tmp, 1) = 0

Mpz_set_ui(q, 2)
Mpz_sub(tmp, m, i) : Mpz_sub_ui(tmp, tmp, 1) : Mpz_powm(tmp, q, tmp, p)
Mpz_powm(b, c, tmp, p)
Mpz_mul(r, r, b) : Mpz_mod(r, r, p)
Mpz_mul(tmp, b, b) : Mpz_mod(c, tmp, p)
Mpz_mul(tmp, t, c) : Mpz_mod(t, tmp, p)
Mpz_set(m, i)
Loop

Next

Mpz_clear(b) : Mpz_clear(c) : Mpz_clear(i) : Mpz_clear(m)
Mpz_clear(n) : Mpz_clear(p) : Mpz_clear(q) : Mpz_clear(r)
Mpz_clear(s) : Mpz_clear(t) : Mpz_clear(z) : Mpz_clear(tmp)

' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End```
Output:
```Find solution for n = 10 and p = 13
Solution found: 7 and 6

Find solution for n = 56 and p = 101
Solution found: 37 and 64

Find solution for n = 1030 and p = 10009
Solution found: 1632 and 8377

Find solution for n = 1032 and p = 10009
1032 is not a quadratic residue

Find solution for n = 44402 and p = 100049
Solution found: 30468 and 69581

Find solution for n = 665820697 and p = 1000000009
Solution found: 378633312 and 621366697

Find solution for n = 881398088036 and p = 1000000000039
Solution found: 791399408049 and 208600591990

Find solution for n = 41660815127637347468140745042827704103445750172002 and p = 10^50 + 577
Solution found: 32102985369940620849741983987300038903725266634508 and 67897014630059379150258016012699961096274733366069```

Go

int

Implementation following Wikipedia, using similar variable names, and using the int type for simplicity.

```package main

import "fmt"

// Arguments n, p as described in WP
// If Legendre symbol != 1, ok return is false.  Otherwise ok return is true,
// R1 is WP return value R and for convenience R2 is p-R1.
func ts(n, p int) (R1, R2 int, ok bool) {
// a^e mod p
powModP := func(a, e int) int {
s := 1
for ; e > 0; e-- {
s = s * a % p
}
return s
}
// Legendre symbol, returns 1, 0, or -1 mod p -- that's 1, 0, or p-1.
ls := func(a int) int {
return powModP(a, (p-1)/2)
}
// argument validation
if ls(n) != 1 {
return 0, 0, false
}
// WP step 1, factor out powers two.
// variables Q, S named as at WP.
Q := p - 1
S := 0
for Q&1 == 0 {
S++
Q >>= 1
}
// WP step 1, direct solution
if S == 1 {
R1 = powModP(n, (p+1)/4)
return R1, p - R1, true
}
// WP step 2, select z, assign c
z := 2
for ; ls(z) != p-1; z++ {
}
c := powModP(z, Q)
// WP step 3, assign R, t, M
R := powModP(n, (Q+1)/2)
t := powModP(n, Q)
M := S
// WP step 4, loop
for {
// WP step 4.1, termination condition
if t == 1 {
return R, p - R, true
}
// WP step 4.2, find lowest i...
i := 0
for z := t; z != 1 && i < M-1; {
z = z * z % p
i++
}
// WP step 4.3, using a variable b, assign new values of R, t, c, M
b := c
for e := M - i - 1; e > 0; e-- {
b = b * b % p
}
R = R * b % p
c = b * b % p // more convenient to compute c before t
t = t * c % p
M = i
}
}

func main() {
fmt.Println(ts(10, 13))
fmt.Println(ts(56, 101))
fmt.Println(ts(1030, 10009))
fmt.Println(ts(1032, 10009))
fmt.Println(ts(44402, 100049))
}
```
Output:
```7 6 true
37 64 true
1632 8377 true
0 0 false
30468 69581 true
```

big.Int

For the extra credit, we use big.Int from the math/big package of the Go standard library. While the method call syntax is not as easy on the eyes as operator syntax, the package provides modular exponentiation and even the Legendre symbol as the Jacobi function.

```package main

import (
"fmt"
"math/big"
)

func ts(n, p big.Int) (R1, R2 big.Int, ok bool) {
if big.Jacobi(&n, &p) != 1 {
return
}
var one, Q big.Int
one.SetInt64(1)
Q.Sub(&p, &one)
S := 0
for Q.Bit(0) == 0 {
S++
Q.Rsh(&Q, 1)
}
if S == 1 {
R1.Exp(&n, R1.Rsh(R1.Add(&p, &one), 2), &p)
R2.Sub(&p, &R1)
return R1, R2, true
}
var z, c big.Int
for z.SetInt64(2); big.Jacobi(&z, &p) != -1; z.Add(&z, &one) {
}
c.Exp(&z, &Q, &p)
var R, t big.Int
R.Exp(&n, R.Rsh(R.Add(&Q, &one), 1), &p)
t.Exp(&n, &Q, &p)
M := S
for {
if t.Cmp(&one) == 0 {
R2.Sub(&p, &R)
return R, R2, true
}
i := 0
// reuse z as a scratch variable
for z.Set(&t); z.Cmp(&one) != 0 && i < M-1; {
z.Mod(z.Mul(&z, &z), &p)
i++
}
// and instead of a new scratch variable b, continue using z
z.Set(&c)
for e := M - i - 1; e > 0; e-- {
z.Mod(z.Mul(&z, &z), &p)
}
R.Mod(R.Mul(&R, &z), &p)
c.Mod(c.Mul(&z, &z), &p)
t.Mod(t.Mul(&t, &c), &p)
M = i
}
}

func main() {
var n, p big.Int
n.SetInt64(665820697)
p.SetInt64(1000000009)
R1, R2, ok := ts(n, p)
fmt.Println(&R1, &R2, ok)

n.SetInt64(881398088036)
p.SetInt64(1000000000039)
R1, R2, ok = ts(n, p)
fmt.Println(&R1, &R2, ok)
n.SetString("41660815127637347468140745042827704103445750172002", 10)
p.SetString("100000000000000000000000000000000000000000000000577", 10)
R1, R2, ok = ts(n, p)
fmt.Println(&R1)
fmt.Println(&R2)
}
```
Output:
```378633312 621366697 true
791399408049 208600591990 true
32102985369940620849741983987300038903725266634508
67897014630059379150258016012699961096274733366069
```

Library

It gets better; the library has a ModSqrt function that uses Tonelli-Shanks internally. Output is same as above.

```package main

import (
"fmt"
"math/big"
)

func main() {
var n, p, R1, R2 big.Int
n.SetInt64(665820697)
p.SetInt64(1000000009)
R1.ModSqrt(&n, &p)
R2.Sub(&p, &R1)
fmt.Println(&R1, &R2)

n.SetInt64(881398088036)
p.SetInt64(1000000000039)
R1.ModSqrt(&n, &p)
R2.Sub(&p, &R1)
fmt.Println(&R1, &R2)

n.SetString("41660815127637347468140745042827704103445750172002", 10)
p.SetString("100000000000000000000000000000000000000000000000577", 10)
R1.ModSqrt(&n, &p)
R2.Sub(&p, &R1)
fmt.Println(&R1)
fmt.Println(&R2)
}
```

Translation of: Python
```import Data.List (genericTake, genericLength)
import Data.Bits (shiftR)

powMod :: Integer -> Integer -> Integer -> Integer
powMod m b e = go b e 1
where
go b e r
| e == 0 = r
| odd e  = go ((b*b) `mod` m) (e `div` 2) ((r*b) `mod` m)
| even e = go ((b*b) `mod` m) (e `div` 2) r

legendre :: Integer -> Integer -> Integer
legendre a p = powMod p a ((p - 1) `div` 2)

tonelli :: Integer -> Integer -> Maybe (Integer, Integer)
tonelli n p | legendre n p /= 1 = Nothing
tonelli n p =
let s = length \$ takeWhile even \$ iterate (`div` 2) (p-1)
q = shiftR (p-1) s
in if s == 1
then let r = powMod p n ((p+1) `div` 4)
in Just (r, p - r)
else let z = (2 +) . genericLength
\$ takeWhile (\i -> p - 1 /= legendre i p)
\$ [2..p-1]
in loop s
( powMod p z q )
( powMod p n \$ (q+1) `div` 2 )
( powMod p n q )
where
loop m c r t
| (t - 1) `mod` p == 0 = Just (r, p - r)
| otherwise =
let i = (1 +) . genericLength . genericTake (m - 2)
\$ takeWhile (\t2 -> (t2 - 1) `mod` p /= 0)
\$ iterate (\t2 -> (t2*t2) `mod` p)
\$ (t*t) `mod` p
b = powMod p c (2^(m - i - 1))
r' = (r*b)  `mod` p
c' = (b*b)  `mod` p
t' = (t*c') `mod` p
in loop i c' r' t'
```
```λ> tonelli 10 13
Just (7,6)
λ> tonelli 56 101
Just (37,64)
λ> tonelli 1030 10009
Just (1632,8377)
λ> tonelli 1032 10009
Nothing
λ> tonelli 44402 100049
Just (30468,69581)
λ> tonelli 665820697 1000000009
Just (378633312,621366697)
λ> tonelli 881398088036 1000000000039
Just (791399408049,208600591990)
λ> tonelli 41660815127637347468140745042827704103445750172002 \$ (10^50)+577
Just (32102985369940620849741983987300038903725266634508,67897014630059379150258016012699961096274733366069)```

J

Implementation:

```leg=: dyad define
x (y&|)@^ (y-1)%2
)

assert. 1=1 p: y [ 'y must be prime'
assert. 1=x leg y [ 'x must be square mod y'
pow=. y&|@^
if. 1=m=. {.1 q: y-1 do.
r=. x pow (y+1)%4
else.
z=. 1x while. 1>: z leg y do. z=.z+1 end.
c=. z pow q=. (y-1)%2^m
r=. x pow (q+1)%2
t=. x pow q
while. t~:1 do.
n=. t
i=. 0
whilst. 1~:n do.
n=. n pow 2
i=. i+1
end.
r=. y|r*b=. c pow 2^m-i+1
m=. i
t=. y|t*c=. b pow 2
end.
end.
y|(,-)r
)
```

```   10 tosh 13
7 6
56 tosh 101
37 64
1030 tosh 10009
1632 8377
1032 tosh 10009
|assertion failure: tosh
|   1=x leg y['x must be square mod y'
44402 tosh 100049
30468 69581
665820697x tosh 1000000009x
378633312 621366697
881398088036 tosh 1000000000039x
791399408049 208600591990
41660815127637347468140745042827704103445750172002x tosh (10^50x)+577
32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069
```

Java

Translation of: Kotlin
Works with: Java version 9
```import java.math.BigInteger;
import java.util.List;
import java.util.Map;
import java.util.function.BiFunction;
import java.util.function.Function;

public class TonelliShanks {
private static final BigInteger ZERO = BigInteger.ZERO;
private static final BigInteger ONE = BigInteger.ONE;
private static final BigInteger TEN = BigInteger.TEN;
private static final BigInteger TWO = BigInteger.valueOf(2);
private static final BigInteger FOUR = BigInteger.valueOf(4);

private static class Solution {
private BigInteger root1;
private BigInteger root2;
private boolean exists;

Solution(BigInteger root1, BigInteger root2, boolean exists) {
this.root1 = root1;
this.root2 = root2;
this.exists = exists;
}
}

private static Solution ts(Long n, Long p) {
return ts(BigInteger.valueOf(n), BigInteger.valueOf(p));
}

private static Solution ts(BigInteger n, BigInteger p) {
BiFunction<BigInteger, BigInteger, BigInteger> powModP = (BigInteger a, BigInteger e) -> a.modPow(e, p);
Function<BigInteger, BigInteger> ls = (BigInteger a) -> powModP.apply(a, p.subtract(ONE).divide(TWO));

if (!ls.apply(n).equals(ONE)) return new Solution(ZERO, ZERO, false);

BigInteger q = p.subtract(ONE);
BigInteger ss = ZERO;
while (q.and(ONE).equals(ZERO)) {
q = q.shiftRight(1);
}

if (ss.equals(ONE)) {
BigInteger r1 = powModP.apply(n, p.add(ONE).divide(FOUR));
return new Solution(r1, p.subtract(r1), true);
}

BigInteger z = TWO;
while (!ls.apply(z).equals(p.subtract(ONE))) z = z.add(ONE);
BigInteger c = powModP.apply(z, q);
BigInteger r = powModP.apply(n, q.add(ONE).divide(TWO));
BigInteger t = powModP.apply(n, q);
BigInteger m = ss;

while (true) {
if (t.equals(ONE)) return new Solution(r, p.subtract(r), true);
BigInteger i = ZERO;
BigInteger zz = t;
while (!zz.equals(BigInteger.ONE) && i.compareTo(m.subtract(ONE)) < 0) {
zz = zz.multiply(zz).mod(p);
}
BigInteger b = c;
BigInteger e = m.subtract(i).subtract(ONE);
while (e.compareTo(ZERO) > 0) {
b = b.multiply(b).mod(p);
e = e.subtract(ONE);
}
r = r.multiply(b).mod(p);
c = b.multiply(b).mod(p);
t = t.multiply(c).mod(p);
m = i;
}
}

public static void main(String[] args) {
List<Map.Entry<Long, Long>> pairs = List.of(
Map.entry(10L, 13L),
Map.entry(56L, 101L),
Map.entry(1030L, 10009L),
Map.entry(1032L, 10009L),
Map.entry(44402L, 100049L),
Map.entry(665820697L, 1000000009L),
Map.entry(881398088036L, 1000000000039L)
);

for (Map.Entry<Long, Long> pair : pairs) {
Solution sol = ts(pair.getKey(), pair.getValue());
System.out.printf("n = %s\n", pair.getKey());
System.out.printf("p = %s\n", pair.getValue());
if (sol.exists) {
System.out.printf("root1 = %s\n", sol.root1);
System.out.printf("root2 = %s\n", sol.root2);
} else {
System.out.println("No solution exists");
}
System.out.println();
}

BigInteger bn = new BigInteger("41660815127637347468140745042827704103445750172002");
BigInteger bp = TEN.pow(50).add(BigInteger.valueOf(577));
Solution sol = ts(bn, bp);
System.out.printf("n = %s\n", bn);
System.out.printf("p = %s\n", bp);
if (sol.exists) {
System.out.printf("root1 = %s\n", sol.root1);
System.out.printf("root2 = %s\n", sol.root2);
} else {
System.out.println("No solution exists");
}
}
}
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069```

jq

Works with gojq and fq, two Go implementations of jq

The Go implementations of jq provide indefinite-precision integer arithmetic.

See Modular_exponentiation for suitable jq definitions of `power/1` and `modPow/2` as used here.

```include "rc-modular-exponentiation";  # see remark above

# If \$j is 0, then an error condition is raised;
# otherwise, assuming infinite-precision integer arithmetic,
# if the input and \$j are integers, then the result will be an integer.
def idivide(\$j):
. as \$i
| (\$i % \$j) as \$mod
| (\$i - \$mod) / \$j ;

def Solution(a;b;c):
{"root1": a, "root2": b, "exists": c};

# pretty print a Solution
def pp:
if .exists
then "root1 = \(.root1)",
"root2 = \(.root2)"
else "No solution exists"
end;

# Tonelli-Shanks
def ts(\$n; \$p):
def powModP(\$a; \$e): \$a | modPow(\$e; \$p);

def ls(\$a): powModP(\$a; (\$p - 1) | idivide(2));

if ls(\$n) != 1 then Solution(0; 0; false)
else { q: (\$p - 1), ss: 0}
| until (.q % 2 != 0;
.ss += 1
| .q |= idivide(2) )
| if .ss == 1
then powModP(n; (\$p+1) | idivide(4)) as \$r1
| Solution(\$r1; \$p - \$r1; true)
else .z = 2
| until ( ls(.z) == (\$p - 1); .z += 1 )
| .c = powModP(.z; .q)
| .r = powModP(\$n; (.q+1) | idivide(2))
| .t = powModP(\$n; .q)
| .m = .ss
| until (.emit;
if .t == 1 then .emit = Solution(.r; \$p - .r; true)
else .i = 0
| .zz = .t
| until (.zz == 1 or .i >= (.m - 1);
.zz = (.zz * .zz) % p
| .i += 1 )
| .b = .c
| .e = .m - (1 + .i)
| until (.e <= 0;
.b = (.b * .b) % \$p
| .e += -1 )
| .r = (.r * .b) % \$p
| .c = (.b * .b) % \$p
| .t = (.t * .c) % \$p
| .m = .i
end )
| .emit
end
end;

def pairs: [
[10, 13], [56, 101], [1030, 10009], [1032, 10009], [44402, 100049],
[665820697, 1000000009], [881398088036, 1000000000039]
];

pairs[] as [\$n, \$p]
| ts(\$n; \$p) as \$sol
| "n     = \(\$n)",
"p     = \(\$p)",
(\$sol | pp),
"";

def bn: 41660815127637347468140745042827704103445750172002;
def bp: (10 | power(50)) + 577;
ts(bn; bp) as \$bsol
| "n     = \(bn)",
"p     = \(bp)",
( \$bsol | pp );

Output:

See Wren.

Julia

Works with: Julia version 0.6

Module:

```module TonelliShanks

legendre(a, p) = powermod(a, (p - 1) ÷ 2, p)

function solve(n::T, p::T) where T <: Union{Int, Int128, BigInt}
legendre(n, p) != 1 && throw(ArgumentError("\$n not a square (mod \$p)"))
local q::T = p - one(p)
local s::T = 0
while iszero(q % 2)
q ÷= 2
s += one(s)
end
if s == one(s)
r = powermod(n, (p + 1) >> 2, p)
return r, p - r
end
local z::T
for z in 2:(p - 1)
p - 1 == legendre(z, p) && break
end
local c::T = powermod(z, q, p)
local r::T = powermod(n, (q + 1) >> 1, p)
local t::T = powermod(n, q, p)
local m::T = s
local t2::T = zero(p)
while !iszero((t - 1) % p)
t2 = (t * t) % p
local i::T
for i in Base.OneTo(m)
iszero((t2 - 1) % p) && break
t2 = (t2 * t2) % p
end
b = powermod(c, 1 << (m - i - 1), p)
r = (r * b) % p
c = (b * b) % p
t = (t * c) % p
m = i
end
return r, p - r
end

end  # module TonelliShanks
```

Main:

```@show TonelliShanks.solve(10, 13)
@show TonelliShanks.solve(56, 101)
@show TonelliShanks.solve(1030, 10009)
@show TonelliShanks.solve(44402, 100049)
@show TonelliShanks.solve(665820697, 1000000009)
@show TonelliShanks.solve(881398088036, 1000000000039)
@show TonelliShanks.solve(41660815127637347468140745042827704103445750172002, big"10" ^ 50 + 577)
```
Output:
```TonelliShanks.solve(10, 13) = (7, 6)
TonelliShanks.solve(56, 101) = (37, 64)
TonelliShanks.solve(1030, 10009) = (1632, 8377)
TonelliShanks.solve(44402, 100049) = (30468, 69581)
TonelliShanks.solve(665820697, 1000000009) = (378633312, 621366697)
TonelliShanks.solve(881398088036, 1000000000039) = (791399408049, 208600591990)
TonelliShanks.solve(@big_str("41660815127637347468140745042827704103445750172002"), @big_str("10") ^ 50 + 577) = (32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069)```

Kotlin

Translation of: Go
```// version 1.1.3

import java.math.BigInteger

data class Solution(val root1: BigInteger, val root2: BigInteger, val exists: Boolean)

val bigZero = BigInteger.ZERO
val bigOne  = BigInteger.ONE
val bigTwo  = BigInteger.valueOf(2L)
val bigFour = BigInteger.valueOf(4L)
val bigTen  = BigInteger.TEN

fun ts(n: Long, p: Long) = ts(BigInteger.valueOf(n), BigInteger.valueOf(p))

fun ts(n: BigInteger, p: BigInteger): Solution {

fun powModP(a: BigInteger, e: BigInteger) = a.modPow(e, p)

fun ls(a: BigInteger) = powModP(a, (p - bigOne) / bigTwo)

if (ls(n) != bigOne) return Solution(bigZero, bigZero, false)
var q = p - bigOne
var ss = bigZero
while (q.and(bigOne) == bigZero) {
ss = ss + bigOne
q = q.shiftRight(1)
}

if (ss == bigOne) {
val r1 = powModP(n, (p + bigOne) / bigFour)
return Solution(r1, p - r1, true)
}

var z = bigTwo
while (ls(z) != p - bigOne) z = z + bigOne
var c = powModP(z, q)
var r = powModP(n, (q + bigOne) / bigTwo)
var t = powModP(n, q)
var m = ss

while (true) {
if (t == bigOne) return Solution(r, p - r, true)
var i = bigZero
var zz = t
while (zz != bigOne && i < m - bigOne) {
zz  = zz * zz % p
i = i + bigOne
}
var b = c
var e = m - i - bigOne
while (e > bigZero) {
b = b * b % p
e = e - bigOne
}
r = r * b % p
c = b * b % p
t = t * c % p
m = i
}
}

fun main(args: Array<String>) {
val pairs = listOf<Pair<Long, Long>>(
10L to 13L,
56L to 101L,
1030L to 10009L,
1032L to 10009L,
44402L to 100049L,
665820697L to 1000000009L,
881398088036L to 1000000000039L
)

for (pair in pairs) {
val (n, p) = pair
val (root1, root2, exists) = ts(n, p)
println("n = \$n")
println("p = \$p")
if (exists) {
println("root1 = \$root1")
println("root2 = \$root2")
}
else println("No solution exists")
println()
}

val bn = BigInteger("41660815127637347468140745042827704103445750172002")
val bp = bigTen.pow(50) + BigInteger.valueOf(577L)
val (broot1, broot2, bexists) = ts(bn, bp)
println("n = \$bn")
println("p = \$bp")
if (bexists) {
println("root1 = \$broot1")
println("root2 = \$broot2")
}
else println("No solution exists")
}
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069
```

Nim

Based algorithm pseudo-code, referencing python 3.

```proc pow*[T: SomeInteger](x, n, p: T): T =
var t = x mod p
var e = n
result = 1
while e > 0:
if (e and 1) == 1:
result = result * t mod p
t = t * t mod p
e = e shr 1

proc legendre*[T: SomeInteger](a, p: T): T = pow(a, (p-1) shr 1, p)

proc tonelliShanks*[T: SomeInteger](n, p: T): T =
# Check that n is indeed a square.
if legendre(n, p) != 1:
raise newException(ValueError, "Not a square")

# Factor out power of 2 from p-1.
var q = p - 1
var s = 0
while (q and 1) == 0:
s += 1
q = q shr 1

if s == 1:
return pow(n, (p+1) shr 2, p)

# Select a non-square z such as (z | p) = -1.
var z = 2
while legendre(z, p) != p - 1:
z += 1

var
c = pow(z, q, p)
t = pow(n, q, p)
m = s
result = pow(n, (q+1) shr 1, p)
while t != 1:
var
i = 1
z = t * t mod p
while z != 1 and i < m-1:
i += 1
z = z * z mod p

var b = pow(c, 1 shl (m-i-1), p)
c = b * b mod p
t = t * c mod p
m = i
result = result * b mod p

when isMainModule:
proc run(n, p: SomeInteger) =
try:
let r = tonelliShanks(n, p)
echo r, " ", p-r
except ValueError:
echo getCurrentExceptionMsg()

run(10, 13)
run(56, 101)
run(1030, 10009)
run(1032, 10009)
run(44402, 100049)
run(665820697, 1000000009)
```

output:

```7 6
37 64
1632 8377
Not a square
30468 69581
378633312 621366697```

OCaml

Translation of: Java
Library: zarith

An extra test case has been added for the `s = 1` branch.

```let tonelli n p =
let open Z in
let two = ~\$2 in
let pp = pred p in
let pph = pred p / two in
let pow_mod_p a e = powm a e p in
let legendre_p a = pow_mod_p a pph in

if legendre_p n <> one then None
else
let s = trailing_zeros pp in
if s = 1 then
let r = pow_mod_p n (succ p / ~\$4) in
Some (r, p - r)
else
let q = pp asr s in
let z =
let rec find_non_square z =
if legendre_p z = pp then z else find_non_square (succ z)
in
find_non_square two
in
let rec loop c r t m =
if t = one then (r, p - r)
else
let mp = pred m in
let rec find_i n i =
if n = one || i >= mp then i else find_i (n * n mod p) (succ i)
in
let rec exp_pow2 b e =
if e <= zero then b else exp_pow2 (b * b mod p) (pred e)
in
let i = find_i t zero in
let b = exp_pow2 c (mp - i) in
let c = b * b mod p in
loop c (r * b mod p) (t * c mod p) i
in
Some
(loop (pow_mod_p z q) (pow_mod_p n (succ q / two)) (pow_mod_p n q) ~\$s)

let () =
let open Z in
[
(~\$9, ~\$11);
(~\$10, ~\$13);
(~\$56, ~\$101);
(~\$1030, ~\$10009);
(~\$1032, ~\$10009);
(~\$44402, ~\$100049);
(~\$665820697, ~\$1000000009);
(~\$881398088036, ~\$1000000000039);
( of_string "41660815127637347468140745042827704103445750172002",
pow ~\$10 50 + ~\$577 );
]
|> List.iter (fun (n, p) ->
Printf.printf "n = %s\np = %s\n%!" (to_string n) (to_string p);
match tonelli n p with
| Some (r1, r2) ->
Printf.printf "root1 = %s\nroot2 = %s\n\n%!" (to_string r1)
(to_string r2)
| None -> print_endline "No solution exists\n")
```
Output:
```n = 9
p = 11
root1 = 3
root2 = 8

n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069

```

Perl

Translation of: Raku
Library: ntheory
```use bigint;
use ntheory qw(is_prime powmod kronecker);

sub tonelli_shanks {
my(\$n,\$p) = @_;
return if kronecker(\$n,\$p) <= 0;
my \$Q = \$p - 1;
my \$S = 0;
\$Q >>= 1 and \$S++ while 0 == \$Q%2;
return powmod(\$n,int((\$p+1)/4), \$p) if \$S == 1;

my \$c;
for \$n (2..\$p) {
next if kronecker(\$n,\$p) >= 0;
\$c = powmod(\$n, \$Q, \$p);
last;
}

my \$R = powmod(\$n, (\$Q+1) >> 1, \$p ); # ?
my \$t = powmod(\$n, \$Q, \$p );
while ((\$t-1) % \$p) {
my \$b;
my \$t2 = \$t**2 % \$p;
for (1 .. \$S) {
if (0 == (\$t2-1)%\$p) {
\$b = powmod(\$c, 1 << (\$S-1-\$_), \$p);
\$S = \$_;
last;
}
\$t2 = \$t2**2 % \$p;
}
\$R = (\$R * \$b) % \$p;
\$c = \$b**2 % \$p;
\$t = (\$t * \$c) % \$p;
}
\$R;
}

my @tests = (
(10, 13),
(56, 101),
(1030, 10009),
(1032, 10009),
(44402, 100049),
(665820697, 1000000009),
(881398088036, 1000000000039),
);

while (@tests) {
\$n = shift @tests;
\$p = shift @tests;
my \$t = tonelli_shanks(\$n, \$p);
if (!\$t or (\$t**2 - \$n) % \$p) {
printf "No solution for (%d, %d)\n", \$n, \$p;
} else {
printf "Roots of %d are (%d, %d) mod %d\n", \$n, \$t, \$p-\$t, \$p;
}
}
```
Output:
```Roots of 10 are (7, 6) mod 13
Roots of 56 are (37, 64) mod 101
Roots of 1030 are (1632, 8377) mod 10009
No solution for (1032, 10009)
Roots of 44402 are (30468, 69581) mod 100049
Roots of 665820697 are (378633312, 621366697) mod 1000000009
Roots of 881398088036 are (791399408049, 208600591990) mod 1000000000039```

Phix

Translation of: C#
Library: Phix/mpfr
```with javascript_semantics
include mpfr.e

function ts(string ns, ps)
mpz n = mpz_init(ns),
p = mpz_init(ps),
t = mpz_init(),
r = mpz_init(),
pm1 = mpz_init(),
pm2 = mpz_init()
mpz_sub_ui(pm1,p,1)                 -- pm1 = p-1
mpz_fdiv_q_2exp(pm2,pm1,1)          -- pm2 = pm1/2
mpz_powm(t,n,pm2,p)                 -- t = mod(n^pm2,p)
if mpz_cmp_si(t,1)!=0 then
return "No solution exists"
end if
mpz q = mpz_init_set(pm1)
integer ss = 0
while mpz_even(q) do
ss += 1
mpz_fdiv_q_2exp(q,q,1)          -- q/=2
end while
if ss=1 then
mpz_fdiv_q_2exp(t,t,2)
mpz_powm(r,n,t,p)               -- r = mod(n^((p+1)/4),p)
else
mpz z = mpz_init(2)
while true do
mpz_powm(t,z,pm2,p)         -- t = mod(z^pm2,p)
if mpz_cmp(t,pm1)=0 then exit end if
mpz_add_ui(z,z,1)           -- z+= 1
end while
mpz {b,c,zz} = mpz_inits(3)
mpz_powm(c,z,q,p)               -- c = mod(z^q,p)
mpz_fdiv_q_2exp(t,t,1)
mpz_powm(r,n,t,p)               -- r = mod(n^((q+1)/2),p)
mpz_powm(t,n,q,p)               -- t = mod(n^q,p)
integer m = ss
while mpz_cmp_si(t,1) do        -- t!=1
integer i = 0
mpz_set(zz,t)
while mpz_cmp_si(zz,1)!=0 and i<m-1 do
mpz_powm_ui(zz,zz,2,p)  -- zz = mod(zz^2,p)
i += 1
end while
mpz_set(b,c)
integer e = m-i-1
while e>0 do
mpz_powm_ui(b,b,2,p)    -- b = mod(b^2,p)
e -= 1
end while
mpz_mul(r,r,b)
mpz_mod(r,r,p)              -- r = mod(r*b,p)
mpz_powm_ui(c,b,2,p)        -- c = mod(b^2,p)
mpz_mul(t,t,c)
mpz_mod(t,t,p)              -- t = mod(t*c,p)
m = i
end while
end if
mpz_sub(p,p,r)
return mpz_get_str(r)&" and "&mpz_get_str(p)
end function

constant tests = {{"10","13"},
{"56","101"},
{"1030","10009"},
{"1032","10009"},
{"44402","100049"},
{"665820697","1000000009"},
{"881398088036","1000000000039"},
{"41660815127637347468140745042827704103445750172002",
sprintf("1%s577",repeat('0',47))}} -- 10^50+577

for i=1 to length(tests) do
string {p1,p2} = tests[i]
printf(1,"For n = %s and p = %s, %s\n",{p1,p2,ts(p1,p2)})
end for
```
Output:
```For n = 10 and p = 13, 7 and 6
For n = 56 and p = 101, 37 and 64
For n = 1030 and p = 10009, 1632 and 8377
For n = 1032 and p = 10009, No solution exists
For n = 44402 and p = 100049, 30468 and 69581
For n = 665820697 and p = 1000000009, 378633312 and 621366697
For n = 881398088036 and p = 1000000000039, 791399408049 and 208600591990
For n = 41660815127637347468140745042827704103445750172002 and p = 100000000000000000000000000000000000000000000000577,
32102985369940620849741983987300038903725266634508 and 67897014630059379150258016012699961096274733366069
```

PicoLisp

Translation of: Go
```# from @lib/rsa.l
(de **Mod (X Y N)
(let M 1
(loop
(when (bit? 1 Y)
(setq M (% (* M X) N)) )
(T (=0 (setq Y (>> 1 Y)))
M )
(setq X (% (* X X) N)) ) ) )
(de legendre (N P)
(**Mod N (/ (dec P) 2) P) )
(de ts (N P)
(and
(=1 (legendre N P))
(let
(Q (dec P)
S 0
Z 0
C 0
R 0
D 0
M 0
B 0
I 0 )
(until (bit? 1 Q)
(setq Q (>> 1 Q))
(inc 'S) )
(if (=1 S)
(list
(setq @@ (**Mod N (/ (inc P) 4) P))
(- P @@) )
(setq Z 2)
(until (= (legendre Z P) (dec P))
(inc 'Z) )
(setq
C (**Mod Z Q P)
R (**Mod N (/ (inc Q) 2) P)
D (**Mod N Q P)
M S )
(until (=1 D)
(zero I)
(for
(Z
D
(and (<> Z 1) (< I (dec M)))
(setq Z (% (* Z Z) P)) )
(inc 'I) )
(setq B C)
(for
(Z
(- M I 1)
(> Z 0) (dec Z) )
(setq B (% (* B B) P)) )
(setq
R (% (* R B) P)
C (% (* B B) P)
D (% (* D C) P)
M I ) )
(list R (- P R)) ) ) ) )

(println (ts 10 13))
(println (ts 56 101))
(println (ts 1030 10009))
(println (ts 1032 10009))
(println (ts 44402 100049))
(println (ts 665820697 1000000009))
(println (ts 881398088036 1000000000039))
(println (ts 41660815127637347468140745042827704103445750172002 (+ (** 10 50) 577)))```
Output:
```(7 6)
(37 64)
(1632 8377)
NIL
(30468 69581)
(378633312 621366697)
(791399408049 208600591990)
(32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069)
```

Powershell

Translation of: Python
Works with: Powershell version 7
```Function Invoke-ModuloExponentiation ([BigInt]\$Base, [BigInt]\$Exponent, \$Modulo) {
\$Result = 1
\$Base = \$Base % \$Modulo
If (\$Base -eq 0) {return 0}

While (\$Exponent -gt 0) {
If ((\$Exponent -band 1) -eq 1) {\$Result = (\$Result * \$Base) % \$Modulo}
\$Exponent = \$Exponent -shr 1
\$Base = (\$Base * \$Base) % \$Modulo
}
return (\$Result % \$Modulo)
}

Function Get-Legendre ([BigInt]\$Integer, [BigInt]\$Prime) {
return (Invoke-ModuloExponentiation -Base \$Integer -Exponent ((\$Prime - 1) / 2) -Modulo \$Prime)
}

Function Invoke-TonelliShanks ([BigInt]\$Integer, [BigInt]\$Prime) {
If ((Get-Legendre \$Integer \$Prime) -ne 1) {throw "\$Integer not a square (mod \$Prime)"}
[bigint]\$q = \$Prime - 1
\$s = 0
While ((\$q % 2) -eq 0) {
\$q = \$q / 2
\$s++
}
If (\$s -eq 1) {
return (Invoke-ModuloExponentiation \$Integer -Exponent ((\$Prime + 1) / 4) -Modulo \$Prime)
}
For (\$z = 2; [Bigint]::Compare(\$z, \$Prime) -lt 0; \$z++) {
If ([BigInt]::Compare((\$Prime - 1), (Get-Legendre \$z \$Prime)) -eq 0) {
break
}
}
\$c = Invoke-ModuloExponentiation -Base \$z -Exponent \$q -Modulo \$Prime
\$r = Invoke-ModuloExponentiation -Base \$Integer -Exponent ((\$q + 1) / 2) -Modulo \$Prime
\$t = Invoke-ModuloExponentiation -Base \$Integer -Exponent \$q -Modulo \$Prime
\$m = \$s
\$t2 = 0

While (((\$t - 1) % \$Prime) -ne 0) {
\$t2 = \$t * \$t % \$Prime
Foreach (\$i in (1..\$m)) {
If (((\$t2 -1) % \$Prime) -eq 0) {
break
}
\$t2 = Invoke-ModuloExponentiation -Base \$t2 -Exponent 2 -Modulo \$Prime
}
\$b = Invoke-ModuloExponentiation -Base \$c -Exponent ([Math]::Pow(2, (\$m - \$i - 1))) -Modulo \$Prime
\$r = (\$r * \$b) % \$Prime
\$c = (\$b * \$b) % \$Prime
\$t = (\$t * \$c) % \$Prime
\$m = \$i
}
return \$r
}

\$TonelliTests = @(
@{Integer = [BigInt]::Parse('10'); Prime = [BigInt]::Parse('13')},
@{Integer = [BigInt]::Parse('56'); Prime = [BigInt]::Parse('101')},
@{Integer = [BigInt]::Parse('1030'); Prime = [BigInt]::Parse('10009')},
@{Integer = [BigInt]::Parse('44402'); Prime = [BigInt]::Parse('100049')},
@{Integer = [BigInt]::Parse('665820697'); Prime = [BigInt]::Parse('1000000009')},
@{Integer = [BigInt]::Parse('881398088036'); Prime = [BigInt]::Parse('1000000000039')},
@{Integer = [BigInt]::Parse('41660815127637347468140745042827704103445750172002'); Prime = [BigInt]::Parse('100000000000000000000000000000000000000000000000577')}
)

\$TonelliTests | Foreach-Object {
\$Result = Invoke-TonelliShanks @_
[PSCustomObject]@{
n = \$_['Integer']
p = \$_['Prime']
Roots = @(\$Result, (\$_['Prime'] - \$Result))
}
} | Format-List
```
Output:
```n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}

n     : 41660815127637347468140745042827704103445750172002
p     : 100000000000000000000000000000000000000000000000577
Roots : {32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069}
```

Python

Translation of: EchoLisp
Works with: Python version 3
```def legendre(a, p):
return pow(a, (p - 1) // 2, p)

def tonelli(n, p):
assert legendre(n, p) == 1, "not a square (mod p)"
q = p - 1
s = 0
while q % 2 == 0:
q //= 2
s += 1
if s == 1:
return pow(n, (p + 1) // 4, p)
for z in range(2, p):
if p - 1 == legendre(z, p):
break
c = pow(z, q, p)
r = pow(n, (q + 1) // 2, p)
t = pow(n, q, p)
m = s
t2 = 0
while (t - 1) % p != 0:
t2 = (t * t) % p
for i in range(1, m):
if (t2 - 1) % p == 0:
break
t2 = (t2 * t2) % p
b = pow(c, 1 << (m - i - 1), p)
r = (r * b) % p
c = (b * b) % p
t = (t * c) % p
m = i
return r

if __name__ == '__main__':
ttest = [(10, 13), (56, 101), (1030, 10009), (44402, 100049),
(665820697, 1000000009), (881398088036, 1000000000039),
(41660815127637347468140745042827704103445750172002, 10**50 + 577)]
for n, p in ttest:
r = tonelli(n, p)
assert (r * r - n) % p == 0
print("n = %d p = %d" % (n, p))
print("\t  roots : %d %d" % (r, p - r))
```
Output:
```n = 10 p = 13
roots : 7 6
n = 56 p = 101
roots : 37 64
n = 1030 p = 10009
roots : 1632 8377
n = 44402 p = 100049
roots : 30468 69581
n = 665820697 p = 1000000009
roots : 378633312 621366697
n = 881398088036 p = 1000000000039
roots : 791399408049 208600591990
n = 41660815127637347468140745042827704103445750172002 p = 100000000000000000000000000000000000000000000000577
roots : 32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069
```

Racket

Translation of: EchoLisp
```#lang racket

(require math/number-theory)

(define (Legendre a p)
(modexpt a (quotient (sub1 p) 2)))

(define (Tonelli n p (err (λ (n p) (error "not a square (mod p)" (list n p)))))
(with-modulus p
(unless (= 1 (Legendre n p)) (err n p))

(define-values (q s)
(let even?-q-loop ((q (sub1 p)) (s 0))
(if (even? q)
(even?-q-loop (quotient q 2) (add1 s))
(values q s))))

(cond
[(= s 1)
(modexpt n (/ (add1 p) 4))]
[else
(define z (for/first ((z (in-range 2 p)) #:when (= (sub1 p) (Legendre z p))) z))
(let loop ((c (modexpt z q))
(r (modexpt n (quotient (add1 q) 2)))
(t (modexpt n q))
(m s))
(cond
[(mod= 1 t)
r]
[else
(define-values (t2 m′) (for/fold ((t2 (modsqr t)) (i 1))
((j (in-range 1 m)) #:final (mod= t2 1))
(values (modsqr t2) j)))
(define b (modexpt c (expt 2 (- m m′ 1))))
(define c′ (modsqr b))
(loop c′ (mod* r b) (mod* t c′) m′)]))])))

(module+ test
(require rackunit)

(define ttest
`((10 13)
(56 101)
(1030 10009)
(44402 100049)
(665820697 1000000009)
(881398088036  1000000000039)
(41660815127637347468140745042827704103445750172002
,(+ #e1e50 577))))

(for ((test ttest))
(define n (first test))
(define p (second test))
(define r (Tonelli n p))
(printf "n = ~a p = ~a~%  roots : ~a ~a~%" n p r (- p r))))

(check-exn exn:fail? (λ () (Tonelli 1032 1009))))
```
Output:
```n = 10 p = 13
roots : 7 6
n = 56 p = 101
roots : 37 64
n = 1030 p = 10009
roots : 1632 8377
n = 44402 p = 100049
roots : 30468 69581
n = 665820697 p = 1000000009
roots : 378633312 621366697
n = 881398088036 p = 1000000000039
roots : 791399408049 208600591990
n = 41660815127637347468140745042827704103445750172002 p = 100000000000000000000000000000000000000000000000577
roots : 32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069```

Raku

(formerly Perl 6)

Works with: Rakudo version 2018.04

Translation of the Wikipedia pseudocode, heavily influenced by Sidef and Python.

```#  Legendre operator (𝑛│𝑝)
sub infix:<│> (Int \𝑛, Int \𝑝 where 𝑝.is-prime && (𝑝 != 2)) {
given 𝑛.expmod( (𝑝-1) div 2, 𝑝 ) {
when 0  {  0 }
when 1  {  1 }
default { -1 }
}
}

sub tonelli-shanks ( \𝑛, \𝑝 where (𝑛│𝑝) > 0 ) {
my \$𝑄 = 𝑝 - 1;
my \$𝑆 = 0;
\$𝑄 +>= 1 and \$𝑆++ while \$𝑄 %% 2;
return 𝑛.expmod((𝑝+1) div 4, 𝑝) if \$𝑆 == 1;
my \$𝑐 = ((2..𝑝).first: (*│𝑝) < 0).expmod(\$𝑄, 𝑝);
my \$𝑅 = 𝑛.expmod( (\$𝑄+1) +> 1, 𝑝 );
my \$𝑡 = 𝑛.expmod( \$𝑄, 𝑝 );
while (\$𝑡-1) % 𝑝 {
my \$b;
my \$𝑡2 = \$𝑡² % 𝑝;
for 1 .. \$𝑆 {
if (\$𝑡2-1) %% 𝑝 {
\$b = \$𝑐.expmod(1 +< (\$𝑆-1-\$_), 𝑝);
\$𝑆 = \$_;
last;
}
\$𝑡2 = \$𝑡2² % 𝑝;
}
\$𝑅 = (\$𝑅 * \$b) % 𝑝;
\$𝑐 = \$b² % 𝑝;
\$𝑡 = (\$𝑡 * \$𝑐) % 𝑝;
}
\$𝑅;
}

my @tests = (
(10, 13),
(56, 101),
(1030, 10009),
(1032, 10009),
(44402, 100049),
(665820697, 1000000009),
(881398088036, 1000000000039),
(41660815127637347468140745042827704103445750172002,
100000000000000000000000000000000000000000000000577)
);

for @tests -> (\$n, \$p) {
try my \$t = tonelli-shanks(\$n, \$p);
say "No solution for ({\$n}, {\$p})." and next if !\$t or (\$t² - \$n) % \$p;
say "Roots of \$n are (\$t, {\$p-\$t}) mod \$p";
}
```
Output:
```Roots of 10 are (7, 6) mod 13
Roots of 56 are (37, 64) mod 101
Roots of 1030 are (1632, 8377) mod 10009
No solution for (1032, 10009).
Roots of 44402 are (30468, 69581) mod 100049
Roots of 665820697 are (378633312, 621366697) mod 1000000009
Roots of 881398088036 are (791399408049, 208600591990) mod 1000000000039
Roots of 41660815127637347468140745042827704103445750172002 are (32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069) mod 100000000000000000000000000000000000000000000000577```

REXX

Translation of: Python

The large numbers cannot reasonably be handled by the pow function shown here.

```/* REXX (required by some interpreters) */
Numeric Digits 1000000
ttest ='[(10, 13), (56, 101), (1030, 10009), (44402, 100049)]'
Do While pos('(',ttest)>0
Parse Var ttest '(' n ',' p ')' ttest
r = tonelli(n, p)
Say "n =" n "p =" p
Say "          roots :" r (p - r)
End
Exit

legendre: Procedure
Parse Arg a, p
return pow(a, (p - 1) % 2, p)

tonelli: Procedure
Parse Arg n, p
q = p - 1
s = 0
Do while q // 2 == 0
q = q % 2
s = s+1
End
if s == 1 Then
return pow(n, (p + 1) % 4, p)
Do z=2 To p
if p - 1 == legendre(z, p) Then
Leave
End
c = pow(z, q, p)
r = pow(n, (q + 1) / 2, p)
t = pow(n, q, p)
m = s
t2 = 0
Do while (t - 1) // p <> 0
t2 = (t * t) // p
Do i=1 To m
if (t2 - 1) // p == 0 Then
Leave
t2 = (t2 * t2) // p
End
y=2**(m - i - 1)
b = pow(c, y, p)
If b=10008 Then Trace ?R
r = (r * b) // p
c = (b * b) // p
t = (t * c) // p
m = i
End
return r
pow: Procedure
Parse Arg x,y,z
If y>0 Then
p=x**y
Else p=x
If z>'' Then
p=p//z
Return p
```
Output:
```n = 10 p =  13
roots : 7 6
n = 56 p =  101
roots : 37 64
n = 1030 p =  10009
roots : 1632 8377
n = 44402 p =  100049
roots : 30468 69581```

Sidef

Translation of: Python
```func tonelli(n, p) {
legendre(n, p) == 1 || die "not a square (mod p)"
var q = p-1
var s = valuation(q, 2)
s == 1 ? return(powmod(n, (p + 1) >> 2, p)) : (q >>= s)
var c = powmod(2 ..^ p -> first {|z| legendre(z, p) == -1}, q, p)
var r = powmod(n, (q + 1) >> 1, p)
var t = powmod(n, q, p)
var m = s
var t2 = 0
while (!p.divides(t - 1)) {
t2 = ((t * t) % p)
var b
for i in (1 ..^ m) {
if (p.divides(t2 - 1)) {
b = powmod(c, 1 << (m - i - 1), p)
m = i
break
}
t2 = ((t2 * t2) % p)
}

r = ((r * b) % p)
c = ((b * b) % p)
t = ((t * c) % p)
}
return r
}

var tests = [
[10, 13], [56, 101], [1030, 10009], [44402, 100049],
[665820697, 1000000009], [881398088036, 1000000000039],
[41660815127637347468140745042827704103445750172002, 10**50 + 577],
]

for n,p in tests {
var r = tonelli(n, p)
assert((r*r - n) % p == 0)
say "Roots of #{n} are (#{r}, #{p-r}) mod #{p}"
}
```
Output:
```Roots of 10 are (7, 6) mod 13
Roots of 56 are (37, 64) mod 101
Roots of 1030 are (1632, 8377) mod 10009
Roots of 44402 are (30468, 69581) mod 100049
Roots of 665820697 are (378633312, 621366697) mod 1000000009
Roots of 881398088036 are (791399408049, 208600591990) mod 1000000000039
Roots of 41660815127637347468140745042827704103445750172002 are (32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069) mod 100000000000000000000000000000000000000000000000577
```

Visual Basic .NET

Translation of: C#
```Imports System.Numerics

Module Module1

Class Solution
ReadOnly root1 As BigInteger
ReadOnly root2 As BigInteger
ReadOnly exists As Boolean

Sub New(r1 As BigInteger, r2 As BigInteger, e As Boolean)
root1 = r1
root2 = r2
exists = e
End Sub

Public Function GetRoot1() As BigInteger
Return root1
End Function

Public Function GetRoot2() As BigInteger
Return root2
End Function

Public Function GetExists() As Boolean
Return exists
End Function
End Class

Function Ts(n As BigInteger, p As BigInteger) As Solution
If BigInteger.ModPow(n, (p - 1) / 2, p) <> 1 Then
Return New Solution(0, 0, False)
End If

Dim q As BigInteger = p - 1
Dim ss = BigInteger.Zero
While (q Mod 2) = 0
ss += 1
q >>= 1
End While

If ss = 1 Then
Dim r1 = BigInteger.ModPow(n, (p + 1) / 4, p)
Return New Solution(r1, p - r1, True)
End If

Dim z As BigInteger = 2
While BigInteger.ModPow(z, (p - 1) / 2, p) <> p - 1
z += 1
End While
Dim c = BigInteger.ModPow(z, q, p)
Dim r = BigInteger.ModPow(n, (q + 1) / 2, p)
Dim t = BigInteger.ModPow(n, q, p)
Dim m = ss

Do
If t = 1 Then
Return New Solution(r, p - r, True)
End If
Dim i = BigInteger.Zero
Dim zz = t
While zz <> 1 AndAlso i < (m - 1)
zz = zz * zz Mod p
i += 1
End While
Dim b = c
Dim e = m - i - 1
While e > 0
b = b * b Mod p
e = e - 1
End While
r = r * b Mod p
c = b * b Mod p
t = t * c Mod p
m = i
Loop
End Function

Sub Main()
Dim pairs = New List(Of Tuple(Of Long, Long)) From {
New Tuple(Of Long, Long)(10, 13),
New Tuple(Of Long, Long)(56, 101),
New Tuple(Of Long, Long)(1030, 10009),
New Tuple(Of Long, Long)(1032, 10009),
New Tuple(Of Long, Long)(44402, 100049),
New Tuple(Of Long, Long)(665820697, 1000000009),
New Tuple(Of Long, Long)(881398088036, 1000000000039)
}

For Each pair In pairs
Dim sol = Ts(pair.Item1, pair.Item2)
Console.WriteLine("n = {0}", pair.Item1)
Console.WriteLine("p = {0}", pair.Item2)
If sol.GetExists() Then
Console.WriteLine("root1 = {0}", sol.GetRoot1())
Console.WriteLine("root2 = {0}", sol.GetRoot2())
Else
Console.WriteLine("No solution exists")
End If
Console.WriteLine()
Next

Dim bn = BigInteger.Parse("41660815127637347468140745042827704103445750172002")
Dim bp = BigInteger.Pow(10, 50) + 577
Dim bsol = Ts(bn, bp)
Console.WriteLine("n = {0}", bn)
Console.WriteLine("p = {0}", bp)
If bsol.GetExists() Then
Console.WriteLine("root1 = {0}", bsol.GetRoot1())
Console.WriteLine("root2 = {0}", bsol.GetRoot2())
Else
Console.WriteLine("No solution exists")
End If
End Sub

End Module
```
Output:
```n = 10
p = 13
root1 = 7
root2 = 6

n = 56
p = 101
root1 = 37
root2 = 64

n = 1030
p = 10009
root1 = 1632
root2 = 8377

n = 1032
p = 10009
No solution exists

n = 44402
p = 100049
root1 = 30468
root2 = 69581

n = 665820697
p = 1000000009
root1 = 378633312
root2 = 621366697

n = 881398088036
p = 1000000000039
root1 = 791399408049
root2 = 208600591990

n = 41660815127637347468140745042827704103445750172002
p = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069```

Wren

Translation of: Kotlin
Library: Wren-dynamic
Library: Wren-big
```import "./dynamic" for Tuple
import "./big" for BigInt

var Solution = Tuple.create("Solution", ["root1", "root2", "exists"])

var ts = Fn.new { |n, p|
if (n is Num) n = BigInt.new(n)
if (p is Num) p = BigInt.new(p)

var powModP = Fn.new { |a, e| a.modPow(e, p) }

var ls = Fn.new { |a| powModP.call(a, p.dec / BigInt.two) }

if (ls.call(n) != BigInt.one) return Solution.new(BigInt.zero, BigInt.zero, false)
var q = p.dec
var ss = BigInt.zero
while (q & BigInt.one == BigInt.zero) {
ss = ss.inc
q = q >> 1
}
if (ss == BigInt.one) {
var r1 = powModP.call(n, p.inc / BigInt.four)
return Solution.new(r1, p - r1, true)
}
var z = BigInt.two
while (ls.call(z) != p.dec) z = z.inc
var c = powModP.call(z, q)
var r = powModP.call(n, q.inc/BigInt.two)
var t = powModP.call(n, q)
var m = ss
while (true) {
if (t == BigInt.one) return Solution.new(r, p - r, true)
var i = BigInt.zero
var zz = t
while (zz != BigInt.one && i < m.dec) {
zz = zz * zz % p
i = i.inc
}
var b = c
var e = m - i.inc
while (e > BigInt.zero) {
b = b * b % p
e = e.dec
}
r = r * b % p
c = b * b % p
t = t * c % p
m = i
}
}

var pairs = [
[10, 13], [56, 101], [1030, 10009], [1032, 10009], [44402, 100049],
[665820697, 1000000009], [881398088036, 1000000000039]
]

for (pair in pairs) {
var n = pair[0]
var p = pair[1]
var sol = ts.call(n, p)
System.print("n     = %(n)")
System.print("p     = %(p)")
if (sol.exists) {
System.print("root1 = %(sol.root1)")
System.print("root2 = %(sol.root2)")
} else {
System.print("No solution exists")
}
System.print()
}

var bn = BigInt.new("41660815127637347468140745042827704103445750172002")
var bp = BigInt.ten.pow(50) + BigInt.new(577)
var bsol = ts.call(bn, bp)
System.print("n     = %(bn)")
System.print("p     = %(bp)")
if (bsol.exists) {
System.print("root1 = %(bsol.root1)")
System.print("root2 = %(bsol.root2)")
} else {
System.print("No solution exists")
}
```
Output:
```n     = 10
p     = 13
root1 = 7
root2 = 6

n     = 56
p     = 101
root1 = 37
root2 = 64

n     = 1030
p     = 10009
root1 = 1632
root2 = 8377

n     = 1032
p     = 10009
No solution exists

n     = 44402
p     = 100049
root1 = 30468
root2 = 69581

n     = 665820697
p     = 1000000009
root1 = 378633312
root2 = 621366697

n     = 881398088036
p     = 1000000000039
root1 = 791399408049
root2 = 208600591990

n     = 41660815127637347468140745042827704103445750172002
p     = 100000000000000000000000000000000000000000000000577
root1 = 32102985369940620849741983987300038903725266634508
root2 = 67897014630059379150258016012699961096274733366069
```

zkl

Translation of: EchoLisp
```var BN=Import("zklBigNum");
fcn modEq(a,b,p) { (a-b)%p==0 }
fcn legendre(a,p){ a.powm((p - 1)/2,p) }

fcn tonelli(n,p){ //(BigInt,Int|BigInt)
_assert_(legendre(n,p)==1, "not a square (mod p)"+vm.arglist);
q,s:=p-1,0;
while(q.isEven){ q/=2; s+=1; }
if(s==1) return(n.powm((p+1)/4,p));
z:=[BN(2)..p].filter1('wrap(z){ legendre(z,p)==(p-1) });
c,r,t,m,t2:=z.powm(q,p), n.powm((q+1)/2,p), n.powm(q,p), s, 0;
while(not modEq(t,1,p)){
t2=(t*t)%p;
i:=1; while(not modEq(t2,1,p)){ i+=1; t2=(t2*t2)%p; } // assert(i<m)
b:=c.powm(BN(1).shiftLeft(m-i-1), p);
r,c,t,m = (r*b)%p, (b*b)%p, (t*c)%p, i;
}
r
}```
```ttest:=T(T(10,13), T(56,101), T(1030,10009), T(44402,100049),
T(665820697,1000000009), T(881398088036,1000000000039),
T("41660815127637347468140745042827704103445750172002", BN(10).pow(50) + 577),
T(1032,10009) );
foreach n,p in (ttest){ n=BN(n);
r:=tonelli(n,p);
assert((r*r-n)%p == 0,"(r*r-n)%p == 0 : %s,%s,%s-->%s".fmt(r,n,p,(r*r-n)%p));
println("n=%d p=%d".fmt(n,p));
println("   roots: %d %d".fmt(r, p-r));
}```
Output:
```n=10 p=13
roots: 7 6
n=56 p=101
roots: 37 64
n=1030 p=10009
roots: 1632 8377
n=44402 p=100049
roots: 30468 69581
n=665820697 p=1000000009
roots: 378633312 621366697
n=881398088036 p=1000000000039
roots: 791399408049 208600591990
n=41660815127637347468140745042827704103445750172002 p=100000000000000000000000000000000000000000000000577
roots: 32102985369940620849741983987300038903725266634508 67897014630059379150258016012699961096274733366069
VM#1 caught this unhandled exception:
AssertionError : not a square (mod p)L(1032,10009)
Stack trace for VM#1 ():
bbb.assert addr:13  args(2) reg(0)
bbb.tonelli addr:29  args(2) reg(10) R
...
```