Tonelli-Shanks algorithm

Tonelli-Shanks algorithm is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Tonelli–Shanks algorithm

Solve x² ≡ n (mod p)

In computational number theory, the Tonelli–Shanks algorithm is a technique for solving an equation of the form x² ≡ n (mod p), where p is an odd prime and x ,n Є Fp = {0, 1, ... p-1}. It is used in cryptography techniques.

To apply the algorithm we need the Legendre symbol.

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 is a ≡ 0

Algorithm pseudo-code copied from Wikipedia :

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. [Factors out powers of 2 from p-1] Define q -odd- and s such as p-1 = q * 2^s
  • if s = 1 , i.e p ≡ 3 (mod 4) , output the two solutions r ≡ +/- n^((p+1)/4) .
• Step 2. Select a non-square z such as (z | p) = -1 , and set c ≡ z^q .
• Step 3. Set r ≡ n ^((q+1)/2) , t ≡ n^q, m = s .
• Step 4. Loop.
  • if t ≡ 1 output r, p-r .
  • Otherwise find, by repeated squaring, the lowest i , 0 < i< m , such as t^(2^i) ≡ 1
  • Let b ≡ c^(2^(m-i-1)), and set r ≡ r*b, t ≡ t*b^2 , c ≡ b^2 and m = i.

Numerical Example

• n=10, p= 13. See Wikipedia

Task

Implement the above.

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

See also:

• Modular exponentiation
• Cipolla's algorithm

J

Implementation:

<lang J>leg=: dyad define

 x (y&|)@^ (y-1)%2

)

tosh=:dyad define

 assert. 1=x leg 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

)</lang>

Task examples:

<lang J> 10 tosh 13 7 6

  56 tosh 101

37 64

  8218 tosh 10007

9872 135

  8219 tosh 10007

|assertion failure: tosh | 1=x leg y

  331575 tosh 1000003

144161 855842

  665165880x tosh 1000000007x

475131702 524868305

  881398088036x tosh 1000000000039x

791399408049 208600591990

  34035243914635549601583369544560650254325084643201x tosh (10^50x) + 151

82563118828090362261378993957450213573687113690751 17436881171909637738621006042549786426312886309400</lang>