RSA code: Difference between revisions
| Line 1,057: | Line 1,057: | ||
e = 65537 |
e = 65537 |
||
d = 5617843187844953170308463622230283376298685 |
d = 5617843187844953170308463622230283376298685 |
||
print('public key ', n, e) |
|||
print('private key ', n, d, '\n') |
|||
message='Rosetta Code!' |
message='Rosetta Code!' |
||
| Line 1,084: | Line 1,081: | ||
{{output}} |
{{output}} |
||
<pre> |
<pre> |
||
public key 9516311845790656153499716760847001433441357 65537 |
|||
private key 9516311845790656153499716760847001433441357 5617843187844953170308463622230283376298685 |
|||
message Rosetta Code! |
message Rosetta Code! |
||
hex data b'526f736574746120436f646521' |
hex data b'526f736574746120436f646521' |
||
Revision as of 20:55, 26 January 2019
You are encouraged to solve this task according to the task description, using any language you may know.
Given an RSA key (n,e,d), construct a program to encrypt and decrypt plaintext messages strings.
Background
RSA code is used to encode secret messages. It is named after Ron Rivest, Adi Shamir, and Leonard Adleman who published it at MIT in 1977. The advantage of this type of encryption is that you can distribute the number “” and “” (which makes up the Public Key used for encryption) to everyone. The Private Key used for decryption “” is kept secret, so that only the recipient can read the encrypted plaintext.
The process by which this is done is that a message, for example “Hello World” is encoded as numbers (This could be encoding as ASCII or as a subset of characters ). This yields a string of numbers, generally referred to as "numerical plaintext", “”. For example, “Hello World” encoded with a=1,...,z=26 by hundreds would yield .
The plaintext must also be split into blocks so that the numerical plaintext is smaller than otherwise the decryption will fail.
The ciphertext, , is then computed by taking each block of , and computing
Similarly, to decode, one computes
To generate a key, one finds 2 (ideally large) primes and . the value “” is simply: . One must then choose an “” such that . That is to say, and are relatively prime to each other.
The decryption value is then found by solving
The security of the code is based on the secrecy of the Private Key (decryption exponent) “” and the difficulty in factoring “”. Research into RSA facilitated advances in factoring and a number of factoring challenges. Keys of 768 bits have been successfully factored. While factoring of keys of 1024 bits has not been demonstrated, NIST expected them to be factorable by 2010 and now recommends 2048 bit keys going forward (see Asymmetric algorithm key lengths or NIST 800-57 Pt 1 Revised Table 4: Recommended algorithms and minimum key sizes).
Summary of the task requirements:
- Encrypt and Decrypt a short message or two using RSA with a demonstration key.
- Implement RSA do not call a library.
- Encode and decode the message using any reversible method of your choice (ASCII or a=1,..,z=26 are equally fine).
- Either support blocking or give an error if the message would require blocking)
- Demonstrate that your solution could support real keys by using a non-trivial key that requires large integer support (built-in or libraries). There is no need to include library code but it must be referenced unless it is built into the language. The following keys will be meet this requirement;however, they are NOT long enough to be considered secure:
- n = 9516311845790656153499716760847001433441357
- e = 65537
- d = 5617843187844953170308463622230283376298685
- Messages can be hard-coded into the program, there is no need for elaborate input coding.
- Demonstrate that your implementation works by showing plaintext, intermediate results, encrypted text, and decrypted text.
ALGOL 68
The code below uses Algol 68 Genie which provides arbitrary precision arithmetic for LONG LONG modes.
<lang algol68> COMMENT
First cut. Doesn't yet do blocking and deblocking. Also, as encryption and decryption are identical operations but for the reciprocal exponents used, only one has been implemented below.
A later release will address these issues.
COMMENT
BEGIN
PR precision=1000 PR
MODE LLI = LONG LONG INT; CO For brevity CO
PROC mod power = (LLI base, exponent, modulus) LLI :
BEGIN
LLI result := 1, b := base, e := exponent;
IF exponent < 0
THEN
put (stand error, (("Negative exponent", exponent, newline)))
ELSE
WHILE e > 0 DO (ODD e | result := (result * b) MOD modulus); e OVERAB 2; b := (b * b) MOD modulus OD
FI;
result
END;
PROC modular inverse = (LLI a, m) LLI :
BEGIN
PROC extended gcd = (LLI x, y) []LLI :
BEGIN
LLI v := 1, a := 1, u := 0, b := 0, g := x, w := y; WHILE w>0 DO LLI q := g % w, t := a - q * u; a := u; u := t; t := b - q * v; b := v; v := t; t := g - q * w; g := w; w := t OD; a PLUSAB (a < 0 | u | 0); (a, b, g)
END;
[] LLI egcd = extended gcd (a, m);
(egcd[3] > 1 | 0 | egcd[1] MOD m)
END;
PROC number to string = (LLI number) STRING :
BEGIN
[] CHAR map = (blank + "ABCDEFGHIJKLMNOPQRSTUVWXYZ")[@0];
LLI local number := number;
INT length := SHORTEN SHORTEN ENTIER long long log(number) + 1;
(ODD length | length PLUSAB 1);
[length % 2] CHAR text;
FOR i FROM length % 2 BY -1 TO 1
DO
INT index = SHORTEN SHORTEN (local number MOD 100); text[i] := (index > 26 | "?" | map[index]); local number := local number % 100
OD;
text
END;
CO The parameters of a particular RSA cryptosystem CO
LLI p = 3490529510847650949147849619903898133417764638493387843990820577; LLI q = 32769132993266709549961988190834461413177642967992942539798288533; LLI n = p * q; LLI phi n = (p-1) * (q-1); LLI e = 9007; LLI d = modular inverse (e, phi n);
CO A ciphertext CO
LLI cipher text = 96869613754622061477140922254355882905759991124574319874695120930816298225145708356931476622883989628013391990551829945157815154;
CO Print out the corresponding plain text CO
print (number to string (mod power (ciphertext, d, n)))
END </lang>
- Output:
THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE
C
<lang C>
- include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <gmp.h>
int main(void) {
mpz_t n, d, e, pt, ct;
mpz_init(pt); mpz_init(ct); mpz_init_set_str(n, "9516311845790656153499716760847001433441357", 10); mpz_init_set_str(e, "65537", 10); mpz_init_set_str(d, "5617843187844953170308463622230283376298685", 10);
const char *plaintext = "Rossetta Code"; mpz_import(pt, strlen(plaintext), 1, 1, 0, 0, plaintext);
if (mpz_cmp(pt, n) > 0)
abort();
mpz_powm(ct, pt, e, n);
gmp_printf("Encoded: %Zd\n", ct);
mpz_powm(pt, ct, d, n);
gmp_printf("Decoded: %Zd\n", pt);
char buffer[64];
mpz_export(buffer, NULL, 1, 1, 0, 0, pt);
printf("As String: %s\n", buffer);
mpz_clears(pt, ct, n, e, d, NULL); return 0;
} </lang>
- Output:
Encoded: 5278143020249600501803788468419399384934220 Decoded: 6531201733672758787904906421349 As String: Rossetta Code
C#
<lang csharp>using System; using System.Numerics; using System.Text;
class Program {
static void Main(string[] args)
{
BigInteger n = BigInteger.Parse("9516311845790656153499716760847001433441357");
BigInteger e = 65537;
BigInteger d = BigInteger.Parse("5617843187844953170308463622230283376298685");
const string plaintextstring = "Hello, Rosetta!";
byte[] plaintext = ASCIIEncoding.ASCII.GetBytes(plaintextstring);
BigInteger pt = new BigInteger(plaintext);
if (pt > n)
throw new Exception();
BigInteger ct = BigInteger.ModPow(pt, e, n);
Console.WriteLine("Encoded: " + ct);
BigInteger dc = BigInteger.ModPow(ct, d, n);
Console.WriteLine("Decoded: " + dc);
string decoded = ASCIIEncoding.ASCII.GetString(dc.ToByteArray());
Console.WriteLine("As ASCII: " + decoded);
}
}</lang>
- Output:
Encoded: 8545729659809274764853392532557102329563535 Decoded: 173322416552962951144796590453843272 As ASCII: Hello, Rosetta!
Common Lisp
In this example, the functions encode-string and decode-string are responsible for converting the string to an integer (which can be encoded) and back. They are not very important to the RSA algorithm, which happens in encode-rsa, decode-rsa, and mod-exp.
The string is encoded as follows: each character is converted into 2 digits based on ASCII value (subtracting 32, so that SPACE=00, and so on.) To decode we simply read every 2 digits from the given integer in order, adding 32 and converting back into characters.
<lang lisp>(defparameter *n* 9516311845790656153499716760847001433441357) (defparameter *e* 65537) (defparameter *d* 5617843187844953170308463622230283376298685)
- magic
(defun encode-string (message)
(parse-integer (reduce #'(lambda (x y) (concatenate 'string x y))
(loop for c across message collect (format nil "~2,'0d" (- (char-code c) 32))))))
- sorcery
(defun decode-string (message) (coerce (loop for (a b) on
(loop for char across (write-to-string message) collect char) by #'cddr collect (code-char (+ (parse-integer (coerce (list a b) 'string)) 32))) 'string))
- ACTUAL RSA ALGORITHM STARTS HERE ;;
- fast modular exponentiation
- runs in O(log exponent)
- acc is initially 1 and contains the result by the end
(defun mod-exp (base exponent modulus acc)
(if (= exponent 0) acc (mod-exp (mod (* base base) modulus) (ash exponent -1) modulus
(if (= (mod exponent 2) 1) (mod (* acc base) modulus) acc))))
- to encode a message, we first convert it to its integer form.
- then, we raise it to the *e* power, modulo *n*
(defun encode-rsa (message)
(mod-exp (encode-string message) *e* *n* 1))
- to decode a message, we raise it to *d* power, modulo *n*
- and then convert it back into a string
(defun decode-rsa (message)
(decode-string (mod-exp message *d* *n* 1)))
</lang> Interpreter output (the star * represents the interpreter prompt):
* (load "rsa.lisp") T * (encode-rsa "Rosetta Code") 4330737636866106722999010287941987299297557 * (decode-rsa 4330737636866106722999010287941987299297557) "Rosetta Code"
D
This used the D module of the Modular Exponentiation Task.
<lang d>void main() {
import std.stdio, std.bigint, std.algorithm, std.string, std.range,
modular_exponentiation;
immutable txt = "Rosetta Code";
writeln("Plain text: ", txt);
// A key set big enough to hold 16 bytes of plain text in
// a single block (to simplify the example) and also big enough
// to demonstrate efficiency of modular exponentiation.
immutable BigInt n = "2463574872878749457479".BigInt *
"3862806018422572001483".BigInt;
immutable BigInt e = 2 ^^ 16 + 1;
immutable BigInt d = "5617843187844953170308463622230283376298685";
// Convert plain text to a number.
immutable txtN = reduce!q{ (a << 8) | uint(b) }(0.BigInt, txt);
if (txtN >= n)
return writeln("Plain text message too long.");
writeln("Plain text as a number: ", txtN);
// Encode a single number.
immutable enc = txtN.powMod(e, n);
writeln("Encoded: ", enc);
// Decode a single number.
auto dec = enc.powMod(d, n);
writeln("Decoded: ", dec);
// Convert number to text.
char[] decTxt;
for (; dec; dec >>= 8)
decTxt ~= (dec & 0xff).toInt;
writeln("Decoded number as text: ", decTxt.retro);
}</lang>
- Output:
Plain text: Rosetta Code Plain text as a number: 25512506514985639724585018469 Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 Decoded number as text: Rosetta Code
F#
<lang fsharp> //Nigel Galloway February 12th., 2018 let RSA n g l = bigint.ModPow(l,n,g) let encrypt = RSA 65537I 9516311845790656153499716760847001433441357I let m_in = System.Text.Encoding.ASCII.GetBytes "The magic words are SQUEAMISH OSSIFRAGE"|>Array.chunkBySize 16|>Array.map(Array.fold(fun n g ->(n*256I)+(bigint(int g))) 0I) let n = Array.map encrypt m_in let decrypt = RSA 5617843187844953170308463622230283376298685I 9516311845790656153499716760847001433441357I let g = Array.map decrypt n let m_out = Array.collect(fun n->Array.unfold(fun n->if n>0I then Some(byte(int (n%256I)),n/256I) else None) n|>Array.rev) g|>System.Text.Encoding.ASCII.GetString printfn "'The magic words are SQUEAMISH OSSIFRAGE' as numbers -> %A\nEncrypted -> %A\nDecrypted -> %A\nAs text -> %A" m_in n g m_out </lang>
- Output:
'The magic words are SQUEAMISH OSSIFRAGE' as numbers -> [|112197201611743344895286521511035564832; 129529088517466560781735691575334293331; 23442989443532613|] Encrypted -> [|3493129515654757560886946157927565680562316; 5582490186309277335090560762784439391588703; 8700785834706594190338047528968122486721264|] Decrypted -> [|112197201611743344895286521511035564832; 129529088517466560781735691575334293331; 23442989443532613|] As text -> "The magic words are SQUEAMISH OSSIFRAGE"
FreeBASIC
<lang freebasic>' version 17-01-2017 ' compile with: fbc -s console
- Include Once "gmp.bi"
Dim As Mpz_ptr e, d, n, pt, ct
e = Allocate(Len(__mpz_struct)) d = Allocate(Len(__mpz_struct)) n = Allocate(Len(__mpz_struct)) pt = Allocate(Len(__mpz_struct)) : mpz_init(pt) ct = Allocate(Len(__mpz_struct)) : mpz_init(ct)
mpz_init_set_str(e, "65537", 10) mpz_init_set_str(d, "5617843187844953170308463622230283376298685", 10) mpz_init_set_str(n, "9516311845790656153499716760847001433441357", 10)
Dim As ZString Ptr plaintext : plaintext = Allocate(1000) Dim As ZString Ptr text : text = Allocate(1000)
- plaintext = "Rosetta Code"
mpz_import(pt, Len(*plaintext), 1, 1, 0, 0, plaintext)
If mpz_cmp(pt, n) > 0 Then GoTo clean_up
mpz_powm(ct, pt, e, n) gmp_printf(!" Encoded: %Zd\n", ct)
mpz_powm(pt, ct, d, n) gmp_printf(!" Decoded: %Zd\n", pt)
mpz_export(text, NULL, 1, 1, 0, 0, pt) Print "As string: "; *text
clean_up: DeAllocate(plaintext) : DeAllocate(text) mpz_clear(e) : mpz_clear(d) : mpz_clear(n) mpz_clear(pt) : mpz_clear(ct)
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 As string: Rosetta Code
Go
Note: see the crypto/rsa package included with Go for a full implementation. <lang go>package main
import (
"fmt" "math/big"
)
func main() {
var n, e, d, bb, ptn, etn, dtn big.Int
pt := "Rosetta Code"
fmt.Println("Plain text: ", pt)
// a key set big enough to hold 16 bytes of plain text in
// a single block (to simplify the example) and also big enough
// to demonstrate efficiency of modular exponentiation.
n.SetString("9516311845790656153499716760847001433441357", 10)
e.SetString("65537", 10)
d.SetString("5617843187844953170308463622230283376298685", 10)
// convert plain text to a number
for _, b := range []byte(pt) {
ptn.Or(ptn.Lsh(&ptn, 8), bb.SetInt64(int64(b)))
}
if ptn.Cmp(&n) >= 0 {
fmt.Println("Plain text message too long")
return
}
fmt.Println("Plain text as a number:", &ptn)
// encode a single number
etn.Exp(&ptn, &e, &n)
fmt.Println("Encoded: ", &etn)
// decode a single number
dtn.Exp(&etn, &d, &n)
fmt.Println("Decoded: ", &dtn)
// convert number to text
var db [16]byte
dx := 16
bff := big.NewInt(0xff)
for dtn.BitLen() > 0 {
dx--
db[dx] = byte(bb.And(&dtn, bff).Int64())
dtn.Rsh(&dtn, 8)
}
fmt.Println("Decoded number as text:", string(db[dx:]))
}</lang> Output:
Plain text: Rosetta Code Plain text as a number: 25512506514985639724585018469 Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 Decoded number as text: Rosetta Code
Haskell
<lang Haskell>module RSAMaker
where
import Data.Char ( chr )
encode :: String -> [Integer] encode s = map (toInteger . fromEnum ) s
rsa_encode :: Integer -> Integer -> [Integer] -> [Integer] rsa_encode n e numbers = map (\num -> mod ( num ^ e ) n ) numbers
rsa_decode :: Integer -> Integer -> [Integer] -> [Integer] rsa_decode d n ciphers = map (\c -> mod ( c ^ d ) n ) ciphers
decode :: [Integer] -> String decode encoded = map ( chr . fromInteger ) encoded
divisors :: Integer -> [Integer] divisors n = [m | m <- [1..n] , mod n m == 0 ]
isPrime :: Integer -> Bool isPrime n = divisors n == [1,n]
totient :: Integer -> Integer -> Integer totient prime1 prime2 = (prime1 - 1 ) * ( prime2 - 1 )
myE :: Integer -> Integer myE tot = head [n | n <- [2..tot - 1] , gcd n tot == 1]
myD :: Integer -> Integer -> Integer -> Integer myD e n phi = head [d | d <- [1..n] , mod ( d * e ) phi == 1]
main = do
putStrLn "Enter a test text!"
text <- getLine
let primes = take 90 $ filter isPrime [1..]
p1 = last primes
p2 = last $ init primes
tot = totient p1 p2
e = myE tot
n = p1 * p2
rsa_encoded = rsa_encode n e $ encode text
d = myD e n tot
encrypted = concatMap show rsa_encoded
decrypted = decode $ rsa_decode d n rsa_encoded
putStrLn ("Encrypted: " ++ encrypted )
putStrLn ("And now decrypted: " ++ decrypted )</lang>
- Output:
Enter a test text! Rosettacode Encrypted: 65646265111107071564791028551028551458331139502651145035156479 And now decrypted: Rosettacode
Icon and Unicon
Please read talk pages.
<lang Icon>procedure main() # rsa demonstration
n := 9516311845790656153499716760847001433441357
e := 65537
d := 5617843187844953170308463622230283376298685
b := 2^integer(log(n,2)) # for blocking
write("RSA Demo using\n n = ",n,"\n e = ",e,"\n d = ",d,"\n b = ",b)
every m := !["Rosetta Code", "Hello Word!",
"This message is too long.", repl("x",*decode(n+1))] do {
write("\nMessage = ",image(m))
write( "Encoded = ",m := encode(m))
if m := rsa(m,e,n) then { # unblocked
write( "Encrypt = ",m)
write( "Decrypt = ",m := rsa(m,d,n))
}
else { # blocked
every put(C := [], rsa(!block(m,b),e,n))
writes("Encrypt = ") ; every writes(!C," ") ; write()
every put(P := [], rsa(!C,d,n))
writes("Decrypt = ") ; every writes(!P," ") ; write()
write("Unblocked = ",m := unblock(P,b))
}
write( "Decoded = ",image(decode(m)))
}
end
procedure mod_power(base, exponent, modulus) # fast modular exponentation
result := 1
while exponent > 0 do {
if exponent % 2 = 1 then
result := (result * base) % modulus
exponent /:= 2
base := base ^ 2 % modulus
}
return result
end
procedure rsa(text,e,n) # return rsa encryption of numerically encoded message; fail if text < n return mod_power(text,e,text < n) end
procedure encode(text) # numerically encode ascii text as int
every (message := 0) := ord(!text) + 256 * message return message
end
procedure decode(message) # numerically decode int to ascii text
text := ""
while text ||:= char((0 < message) % 256) do
message /:= 256
return reverse(text)
end
procedure block(m,b) # break lg int into blocks of size b
M := []
while push(M, x := (0 < m) % b) do
m /:= b
return M
end
procedure unblock(M,b) # reassemble blocks of size b into lg int
every (m := 0) := !M + b * m return m
end</lang>
Output:
RSA Demo using n = 9516311845790656153499716760847001433441357 e = 65537 d = 5617843187844953170308463622230283376298685 b = 5575186299632655785383929568162090376495104 Message = "Rosetta Code" Encoded = 25512506514985639724585018469 Encrypt = 916709442744356653386978770799029131264344 Decrypt = 25512506514985639724585018469 Decoded = "Rosetta Code" Message = "Hello Word!" Encoded = 87521618088882533792113697 Encrypt = 1798900477268307339588642263628429901019383 Decrypt = 87521618088882533792113697 Decoded = "Hello Word!" Message = "This message is too long." Encoded = 529836718428469753460978059376661024804668788418205881100078 Encrypt = 3376966937987363040878203966915676619521252 7002174816151673360605669161609885530980579 Decrypt = 95034800624219541 4481988526688939374478063610382714873472814 Unblocked = 529836718428469753460978059376661024804668788418205881100078 Decoded = "This message is too long." Message = "xxxxxxxxxxxxxxxxxx" Encoded = 10494468328720293243075632128305111296931960 Encrypt = 1 829820657892505002815717051746917810425013 Decrypt = 1 4919282029087637457691702560143020920436856 Unblocked = 10494468328720293243075632128305111296931960 Decoded = "xxxxxxxxxxxxxxxxxx"
J
Note, for an implementation with blocking (and a much smaller key) see [1]
<lang j> N=: 9516311845790656153499716760847001433441357x
E=: 65537x D=: 5617843187844953170308463622230283376298685x ] text=: 'Rosetta Code'
Rosetta Code
] num=: 256x #. a.i.text
25512506514985639724585018469
num >: N NB. check if blocking is necessary (0 means no)
0
] enc=: N&|@^&E num
916709442744356653386978770799029131264344
] dec=: N&|@^&D enc
25512506514985639724585018469
] final=: a. {~ 256x #.inv dec
Rosetta Code</lang>
Note: as indicated at http://www.jsoftware.com/help/dictionary/special.htm, N&|@^ does not bother with creating the exponential intermediate result.
Java
<lang java> public static void main(String[] args) {
/*
This is probably not the best method...or even the most optimized way...however it works since n and d are too big to be ints or longs
This was also only tested with 'Rosetta Code' and 'Hello World'
It's also pretty limited on plainText size (anything bigger than the above will fail)
*/
BigInteger n = new BigInteger("9516311845790656153499716760847001433441357");
BigInteger e = new BigInteger("65537");
BigInteger d = new BigInteger("5617843187844953170308463622230283376298685");
Charset c = Charsets.UTF_8;
String plainText = "Rosetta Code";
System.out.println("PlainText : " + plainText);
byte[] bytes = plainText.getBytes();
BigInteger plainNum = new BigInteger(bytes);
System.out.println("As number : " + plainNum);
BigInteger Bytes = new BigInteger(bytes);
if (Bytes.compareTo(n) == 1) {
System.out.println("Plaintext is too long");
return;
}
BigInteger enc = plainNum.modPow(e, n);
System.out.println("Encoded: " + enc);
BigInteger dec = enc.modPow(d, n);
System.out.println("Decoded: " + dec);
String decText = new String(dec.toByteArray(), c);
System.out.println("As text: " + decText);
} </lang>
Julia
<lang julia>function rsaencode(clearmsg::AbstractString, nmod::Integer, expub::Integer)
bytes = parse(BigInt, "0x" * bytes2hex(collect(UInt8, clearmsg))) return powermod(bytes, expub, nmod)
end
function rsadecode(cryptmsg::Integer, nmod::Integer, dsecr::Integer)
decoded = powermod(encoded, dsecr, nmod) return join(Char.(hex2bytes(hex(decoded))))
end
msg = "Rosetta Code." nmod = big"9516311845790656153499716760847001433441357" expub = 65537 dsecr = big"5617843187844953170308463622230283376298685"
encoded = rsaencode(msg, nmod, expub) decoded = rsadecode(encoded, nmod, dsecr) println("\n# $msg\n -> ENCODED: $encoded\n -> DECODED: $decoded")</lang>
- Output:
# Rosetta Code. -> ENCODED: 2440331969632134446717000067136916252596373 -> DECODED: Rosetta Code.
Kotlin
<lang scala>// version 1.1.4-3
import java.math.BigInteger
fun main(args: Array<String>) {
val n = BigInteger("9516311845790656153499716760847001433441357")
val e = BigInteger("65537")
val d = BigInteger("5617843187844953170308463622230283376298685")
val c = Charsets.UTF_8
val plainText = "Rosetta Code"
println("PlainText : $plainText")
val bytes = plainText.toByteArray(c)
val plainNum = BigInteger(bytes)
println("As number : $plainNum")
if (plainNum > n) {
println("Plaintext is too long")
return
}
val enc = plainNum.modPow(e, n)
println("Encoded : $enc")
val dec = enc.modPow(d, n)
println("Decoded : $dec")
val decText = dec.toByteArray().toString(c)
println("As text : $decText")
}</lang>
- Output:
PlainText : Rosetta Code As number : 25512506514985639724585018469 Encoded : 916709442744356653386978770799029131264344 Decoded : 25512506514985639724585018469 As text : Rosetta Code
Mathematica
Does not support blocking. <lang>toNumPlTxt[s_] := FromDigits[ToCharacterCode[s], 256]; fromNumPlTxt[plTxt_] := FromCharacterCode[IntegerDigits[plTxt, 256]]; enc::longmess = "Message '``' is too long for n = ``."; enc[n_, _, mess_] /;
toNumPlTxt[mess] >= n := (Message[enc::longmess, mess, n]; $Failed);
enc[n_, e_, mess_] := PowerMod[toNumPlTxt[mess], e, n]; dec[n_, d_, en_] := fromNumPlTxt[PowerMod[en, d, n]]; text = "The cake is a lie!"; n = 9516311845790656153499716760847001433441357; e = 65537; d = 5617843187844953170308463622230283376298685; en = enc[n, e, text]; de = dec[n, d, en]; Print["Text: '" <> text <> "'"]; Print["n = " <> IntegerString[n]]; Print["e = " <> IntegerString[e]]; Print["d = " <> IntegerString[d]]; Print["Numeric plaintext: " <> IntegerString[toNumPlTxt[text]]]; Print["Encoded: " <> IntegerString[en]]; Print["Decoded: '" <> de <> "'"];</lang>
- Output:
Text: 'The cake is a lie!' n = 9516311845790656153499716760847001433441357 e = 65537 d = 5617843187844953170308463622230283376298685 Numeric plaintext: 7352955804624388987810264523908743852287265 Encoded: 199505409518408949879682159958576932863989 Decoded: 'The cake is a lie!'
PARI/GP
<lang parigp>stigid(V,b)=subst(Pol(V),'x,b); \\ inverse function digits(...)
n = 9516311845790656153499716760847001433441357; e = 65537; d = 5617843187844953170308463622230283376298685;
text = "Rosetta Code"
inttext = stigid(Vecsmall(text),256) \\ message as an integer encoded = lift(Mod(inttext, n) ^ e) \\ encrypted message decoded = lift(Mod(encoded, n) ^ d) \\ decrypted message message = Strchr(digits(decoded, 256)) \\ readable message</lang>
Output:
text: "Rosetta Code" inttext: 25512506514985639724585018469 encoded: 916709442744356653386978770799029131264344 decoded: 25512506514985639724585018469 message: "Rosetta Code"
If inttext is equal or greater than b = 2^(log(n)/log(2)\1) use block = inttext % b; inttext /= b; to break inttext into blocks and encode piece by piece. Decode in reverse order.
As a check: it's easy to crack this weak encrypted message without knowing secret key 'd' <lang parigp>f = factor(n); \\ factorize public key 'n'
crack = Strchr(digits(lift(Mod(encoded,n) ^ lift(Mod(1,(f[1,1]-1)*(f[2,1]-1)) / e)),256))</lang>
Output:
crack: "Rosetta Code"
Perl
<lang perl>use bigint;
$n = 9516311845790656153499716760847001433441357; $e = 65537; $d = 5617843187844953170308463622230283376298685;
package Message {
my @alphabet;
push @alphabet, $_ for 'A' .. 'Z', ' ';
my $rad = +@alphabet;
$code{$alphabet[$_]} = $_ for 0..$rad-1;
sub encode {
my($t) = @_;
my $cnt = my $sum = 0;
for (split , reverse $t) {
$sum += $code{$_} * $rad**$cnt;
$cnt++;
}
$sum;
}
sub decode {
my($n) = @_;
my(@i);
while () {
push @i, $n % $rad;
last if $n < $rad;
$n = int $n / $rad;
}
reverse join , @alphabet[@i];
}
sub expmod {
my($a, $b, $n) = @_;
my $c = 1;
do {
($c *= $a) %= $n if $b % 2;
($a *= $a) %= $n;
} while ($b = int $b/2);
$c;
}
}
my $secret_message = "ROSETTA CODE";
$numeric_message = Message::encode $secret_message; $numeric_cipher = Message::expmod $numeric_message, $e, $n; $text_cipher = Message::decode $numeric_cipher; $numeric_cipher2 = Message::encode $text_cipher; $numeric_message2 = Message::expmod $numeric_cipher2, $d, $n; $secret_message2 = Message::decode $numeric_message2;
print <<"EOT"; Secret message is $secret_message Secret message in integer form is $numeric_message After exponentiation with public exponent we get: $numeric_cipher This turns into the string $text_cipher If we re-encode it in integer form we get $numeric_cipher2 After exponentiation with SECRET exponent we get: $numeric_message2 This turns into the string $secret_message2 EOT</lang>
- Output:
Secret message is ROSETTA CODE Secret message in integer form is 97525102075211938 After exponentiation with public exponent we get: 8326171774113983822045243488956318758396426 This turns into the string ZULYDCEZOWTFXFRRNLIMGNUPHVCJSX If we re-encode it in integer form we get 8326171774113983822045243488956318758396426 After exponentiation with SECRET exponent we get: 97525102075211938 This turns into the string ROSETTA CODE
Perl 6
No blocking here. Algorithm doesn't really work if either red or black text begins with 'A'. <lang perl6>constant $n = 9516311845790656153499716760847001433441357; constant $e = 65537; constant $d = 5617843187844953170308463622230283376298685;
my $secret-message = "ROSETTA CODE";
package Message {
my @alphabet = slip('A' .. 'Z'), ' ';
my $rad = +@alphabet;
my %code = @alphabet Z=> 0 .. *;
subset Text of Str where /^^ @alphabet+ $$/;
our sub encode(Text $t) {
[+] %code{$t.flip.comb} Z* (1, $rad, $rad*$rad ... *);
}
our sub decode(Int $n is copy) {
@alphabet[ gather loop { take $n % $rad; last if $n < $rad; $n div= $rad; } ].join.flip;
}
}
use Test; plan 1;
say "Secret message is $secret-message"; say "Secret message in integer form is $_" given
my $numeric-message = Message::encode $secret-message;
say "After exponentiation with public exponent we get: $_" given
my $numeric-cipher = expmod $numeric-message, $e, $n;
say "This turns into the string $_" given
my $text-cipher = Message::decode $numeric-cipher;
say "If we re-encode it in integer form we get $_" given
my $numeric-cipher2 = Message::encode $text-cipher;
say "After exponentiation with SECRET exponent we get: $_" given
my $numeric-message2 = expmod $numeric-cipher2, $d, $n;
say "This turns into the string $_" given
my $secret-message2 = Message::decode $numeric-message2;
is $secret-message, $secret-message2, "the message has been correctly decrypted";</lang>
- Output:
1..1 Secret message is ROSETTA CODE Secret message in integer form is 97525102075211938 After exponentiation with public exponent we get: 8326171774113983822045243488956318758396426 This turns into the string ZULYDCEZOWTFXFRRNLIMGNUPHVCJSX If we re-encode it in integer form we get 8326171774113983822045243488956318758396426 After exponentiation with SECRET exponent we get: 97525102075211938 This turns into the string ROSETTA CODE ok 1 - the message has been correctly decrypted
Phix
<lang Phix>include builtins/bigint.e -- (0.8.0+, not yet properly documented)
string plaintext = "Rosetta Code"
bigint n = bi_new("9516311845790656153499716760847001433441357"),
e = bi_new(65537),
d = bi_new("5617843187844953170308463622230283376298685"),
pt = bi_new_bin(plaintext,true,0)
if bi_compare(pt,n)>0 then ?9/0 end if
bigint ct = bi_mod_exp(pt, e, n),
dc = bi_mod_exp(ct, d, n)
printf(1,"Original: %s\n",{plaintext}) printf(1,"As Number: %s\n",{bi_sprint(pt)}) printf(1,"Encoded: %s\n",{bi_sprint(ct)}) printf(1,"Decoded: %s\n",{bi_sprint(dc)}) printf(1,"As ASCII: %s\n",{bi_bin(dc)})</lang>
- Output:
Original: Rosetta Code As Number: 25512506514985639724585018469 Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 As ASCII: Rosetta Code
PicoLisp
PicoLisp comes with an RSA library: <lang PicoLisp>### This is a copy of "lib/rsa.l" ###
- Generate long random number
(de longRand (N)
(use (R D)
(while (=0 (setq R (abs (rand)))))
(until (> R N)
(unless (=0 (setq D (abs (rand))))
(setq R (* R D)) ) )
(% R N) ) )
- X power Y modulus N
(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)) ) ) )
- Probabilistic prime check
(de prime? (N)
(and
(> N 1)
(bit? 1 N)
(let (Q (dec N) K 0)
(until (bit? 1 Q)
(setq
Q (>> 1 Q)
K (inc K) ) )
(do 50
(NIL (_prim? N Q K))
T ) ) ) )
- (Knuth Vol.2, p.379)
(de _prim? (N Q K)
(use (X J Y)
(while (> 2 (setq X (longRand N))))
(setq
J 0
Y (**Mod X Q N) )
(loop
(T
(or
(and (=0 J) (= 1 Y))
(= Y (dec N)) )
T )
(T
(or
(and (> J 0) (= 1 Y))
(<= K (inc 'J)) )
NIL )
(setq Y (% (* Y Y) N)) ) ) )
- Find a prime number with `Len' digits
(de prime (Len)
(let P (longRand (** 10 (*/ Len 2 3)))
(unless (bit? 1 P)
(inc 'P) )
(until (prime? P) # P: Prime number of size 2/3 Len
(inc 'P 2) )
# R: Random number of size 1/3 Len
(let (R (longRand (** 10 (/ Len 3))) K (+ R (% (- P R) 3)))
(when (bit? 1 K)
(inc 'K 3) )
(until (prime? (setq R (inc (* K P))))
(inc 'K 6) )
R ) ) )
- Generate RSA key
(de rsaKey (N) #> (Encrypt . Decrypt)
(let (P (prime (*/ N 5 10)) Q (prime (*/ N 6 10)))
(cons
(* P Q)
(/
(inc (* 2 (dec P) (dec Q)))
3 ) ) ) )
- Encrypt a list of characters
(de encrypt (Key Lst)
(let Siz (>> 1 (size Key))
(make
(while Lst
(let N (char (pop 'Lst))
(while (> Siz (size N))
(setq N (>> -16 N))
(inc 'N (char (pop 'Lst))) )
(link (**Mod N 3 Key)) ) ) ) ) )
- Decrypt a list of numbers
(de decrypt (Keys Lst)
(mapcan
'((N)
(let Res NIL
(setq N (**Mod N (cdr Keys) (car Keys)))
(until (=0 N)
(push 'Res (char (& `(dec (** 2 16)) N)))
(setq N (>> 16 N)) )
Res ) )
Lst ) )
- End of "lib/rsa.l" ###
- Generate 100-digit keys (private . public)
- (setq Keys (rsaKey 100))
-> (14394597526321726957429995133376978449624406217727317004742182671030....
- Encrypt
- (setq CryptText
(encrypt (car Keys)
(chop "The quick brown fox jumped over the lazy dog's back") ) )
-> (72521958974980041245760752728037044798830723189142175108602418861716...
- Decrypt
- (pack (decrypt Keys CryptText))
-> "The quick brown fox jumped over the lazy dog's back"</lang>
Python
<lang python>import binascii
n = 9516311845790656153499716760847001433441357 # p*q = modulus e = 65537 d = 5617843187844953170308463622230283376298685
message='Rosetta Code!' print('message ', message)
hex_data = binascii.hexlify(message.encode()) print('hex data ', hex_data)
plain_text = int(hex_data, 16) print('plain text integer ', plain_text)
if plain_text > n:
raise Exception('plain text too large for key')
encrypted_text = pow(plain_text, e, n) print('encrypted text integer ', encrypted_text)
decrypted_text = pow(encrypted_text, d, n) print('decrypted text integer ', decrypted_text)
print('message ', binascii.unhexlify(hex(decrypted_text)[2:]).decode()) # [2:] slicing, to strip the 0x part
</lang>
- Output:
message Rosetta Code! hex data b'526f736574746120436f646521' plain text integer 6531201667836323769493764728097 encrypted text integer 5307878626309103053766094186556322974789734 decrypted text integer 6531201667836323769493764728097 message Rosetta Code!
Racket
This implementation does key generation and demonstrates digital signature as a freebie.
Thanks again to the wonderful math/number-theory package (distributed as standard).
Cutting messages into blocks has not been done.
<lang racket>#lang racket (require math/number-theory) (define-logger rsa) (current-logger rsa-logger)
- -| STRING TO NUMBER MAPPING |----------------------------------------------------------------------
(define (bytes->number B) ; We'll need our data in numerical form ..
(for/fold ((rv 0)) ((b B)) (+ b (* rv 256))))
(define (number->bytes N) ; .. and back again
(define (inr n b) (if (zero? n) b (inr (quotient n 256) (bytes-append (bytes (modulo n 256)) b)))) (inr N (bytes)))
- -| RSA PUBLIC / PRIVATE FUNCTIONS |----------------------------------------------------------------
- The basic definitions... pretty well lifted from the text book!
(define ((C e n) p)
;; Just do the arithmetic to demonstrate RSA... ;; breaking large messages into blocks is something for another day. (unless (< p n) (raise-argument-error 'C (format "(and/c integer? (</c ~a))" n) p)) (modular-expt p e n))
(define ((P d n) c)
(modular-expt c d n))
- -| RSA KEY GENERATION |----------------------------------------------------------------------------
- Key generation
- Full description of the steps can be found on Wikipedia
(define (RSA-keyset function-base-name)
(log-info "RSA-keyset: ~s" function-base-name) (define max-k 4294967087) ;; I'm guessing this RNG is about as cryptographically strong as replacing spaces with tabs. (define (big-random n-rolls) (for/fold ((rv 1)) ((roll (in-range n-rolls 0 -1))) (+ (* rv (add1 max-k)) 1 (random max-k)))) (define (big-random-prime) (define start-number (big-random (/ 1024 32))) (log-debug "got large (possibly non-prime) number, finding next prime") (next-prime (match start-number ((? odd? o) o) ((app add1 e) e)))) ;; [1] Choose two distinct prime numbers p and q. (log-debug "generating p") (define p (big-random-prime)) (log-debug "p generated") (log-debug "generating q") (define q (big-random-prime)) (log-debug "q generated") (log-info "primes generated") ;; [2] Compute n = pq. (define n (* p q)) ;; [3] Compute φ(n) = φ(p)φ(q) = (p − 1)(q − 1) = n - (p + q -1), ;; where φ is Euler's totient function. (define φ (- n (+ p q -1))) ;; [4] Choose an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1; i.e., e and φ(n) are ;; coprime. ... most commonly 2^16 + 1 = 65,537 ... (define e (+ (expt 2 16) 1)) ;; [5] Determine d as d ≡ e−1 (mod φ(n)); i.e., d is the multiplicative inverse of e (modulo φ(n)). (log-debug "generating d") (define d (modular-inverse e φ)) (log-info "d generated") (values n e d))
- -| GIVE A USABLE SET OF PRIVATE STUFF TO A USER |--------------------------------------------------
- six values
- the public (encrypt) function (numeric)
- the private (decrypt) function (numeric)
- the public (encrypt) function (bytes)
- the private (decrypt) function (bytes)
- private (list n e d)
- public (list n e)
(define (RSA-key-pack #:function-base-name function-base-name)
(define (rnm-fn f s) (procedure-rename f (string->symbol (format "~a-~a" function-base-name s)))) (define-values (n e d) (RSA-keyset function-base-name)) (define my-C (rnm-fn (C e n) "C")) (define my-P (rnm-fn (P d n) "P")) (define my-encrypt (rnm-fn (compose number->bytes my-C bytes->number) "encrypt")) (define my-decrypt (rnm-fn (compose number->bytes my-P bytes->number) "decrypt")) (values my-C my-P my-encrypt my-decrypt (list n e d) (list n e)))
- -| HEREON IS JUST A LOAD OF CHATTY DEMOS |---------------------------------------------------------
(define (narrated-encrypt-bytes C who plain-text)
(define plain-n (bytes->number plain-text)) (define cypher-n (C plain-n)) (define cypher-text (number->bytes cypher-n)) (printf #<<EOS
~a wants to send plain text: ~s
as number: ~s cyphered number: ~s
sent by ~a over the public interwebs: ~s ...
EOS
who plain-text plain-n cypher-n who cypher-text) cypher-text)
(define (narrated-decrypt-bytes P who cypher-text)
(define cypher-n (bytes->number cypher-text)) (define plain-n (P cypher-n)) (define plain-text (number->bytes plain-n)) (printf #<<EOS
... ~s
received by ~a as number: ~s decyphered (with P) number: ~s
decyphered text: ~s
EOS
cypher-text who cypher-n plain-n plain-text) plain-text)
- ENCRYPT AND DECRYPT A MESSAGE WITH THE e.g. KEYS
(define-values (given-n given-e given-d)
(values 9516311845790656153499716760847001433441357
65537
5617843187844953170308463622230283376298685))
- Get the keys specific RSA functions
(for ((message-text (list #"hello world" #"TOP SECRET!")))
(define Bobs-public-function (C given-e given-n)) (define Bobs-private-function (P given-d given-n)) (define cypher-text (narrated-encrypt-bytes Bobs-public-function "Alice" message-text)) (define plain-text (narrated-decrypt-bytes Bobs-private-function "Bob" cypher-text)) plain-text)
- Demonstrate with larger keys.
- (And include a free recap on digital signatures, too)
(define-values (A-pub-C A-pvt-P A-pub-encrypt A-pvt-decrypt A-pvt-keys A-pub-keys)
(RSA-key-pack #:function-base-name 'Alice))
(define-values (B-pub-C B-pvt-P B-pub-encrypt B-pvt-decrypt B-pvt-keys B-pub-keys)
(RSA-key-pack #:function-base-name 'Bob))
- Since p and q are random, it is possible that message' = "message modulo {A,B}-key-n" will be too
- big for "message' modulo {B,A}-key-n", if that happens then I run the program again until it
- works. Strictly, we need blocking of the signed message -- which is not yet implemented.
(let* ((plain-A-to-B #"Dear Bob, meet you in Lymm at 1200, Alice")
(signed-A-to-B (A-pvt-decrypt plain-A-to-B))
(unsigned-A-to-B (A-pub-encrypt signed-A-to-B))
(crypt-signed-A-to-B (B-pub-encrypt signed-A-to-B))
(decrypt-signed-A-to-B (B-pvt-decrypt crypt-signed-A-to-B))
(decrypt-verified-B (A-pub-encrypt decrypt-signed-A-to-B)))
(printf
#<<EOS
Alice wants to send ~s to Bob. She "encrypts" with her private "decryption" key. (A-prv msg) -> ~s Only she could have done this (only she has the her private key data) -- so this is a signature on the message. Anyone can verify the signature by "decrypting" the message with the public "encryption" key. (A-pub (A-prv msg)) -> ~s But anyone is able to do this, so there is no privacy here. Everyone knows that it can only be Alice at Lymm at noon, but this message is for Bob's eyes only. We need to encrypt this with his public key: (B-pub (A-prv msg)) -> ~s Which is what gets posted to alt.chat.secret-rendezvous Bob decrypts this to get the signed message from Alice: (B-prv (B-pub (A-prv msg))) -> ~s And verifies Alice's signature: (A-pub (B-prv (B-pub (A-prv msg)))) -> ~s Alice genuinely sent the message. And nobody else (on a.c.s-r, at least) has read it.
KEYS:
Alice's full set: ~s Bob's full set: ~s
EOS
plain-A-to-B signed-A-to-B unsigned-A-to-B crypt-signed-A-to-B decrypt-signed-A-to-B decrypt-verified-B A-pvt-keys B-pvt-keys))</lang>
- Output:
Alice wants to send plain text: #"hello world"
as number: 126207244316550804821666916
cyphered number: 4109627268073579506944196826730512948879423
sent by Alice over the public interwebs:
#"/-\e\355\225\327\244\222<R@\20\4\233\275\333`?"
...
...
#"/-\e\355\225\327\244\222<R@\20\4\233\275\333`?"
received by Bob
as number: 4109627268073579506944196826730512948879423
decyphered (with P) number: 126207244316550804821666916
decyphered text:
#"hello world"
Alice wants to send plain text: #"TOP SECRET!"
as number: 101924313868583037137409057
cyphered number: 5346093164296793050289700489360581430628365
sent by Alice over the public interwebs:
#"=^\301{\17p\201AE\341D\357 \237IPP\r"
...
...
#"=^\301{\17p\201AE\341D\357 \237IPP\r"
received by Bob
as number: 5346093164296793050289700489360581430628365
decyphered (with P) number: 101924313868583037137409057
decyphered text:
#"TOP SECRET!"
Alice wants to send #"Dear Bob, meet you in Lymm at 1200, Alice" to Bob.
She "encrypts" with her private "decryption" key.
(A-prv msg) -> #"B}\4<G\373-\217\350;\0214\226\233\236\333\215\226\225=\236\350\277X\241*J\356\302\250\350fO\5\375u\367\365\315\270\312\334\204U\332\224\322\357u=\262\326\274e\31\301\321\210:i\361\361g\361\16\5a\304X\306\313\350(^\374 \353\350t\2662\305\346a\300\244b\337JI\343\335\21j\202\236\242\335<rA\a\233\375\23\t\32(d`\237i\267\336\270\340L\26\f\260\346&\t\301\326\331k@\253\242\241VKw\365 \204U\270*\r,\334h=\257\230\320V\357\304\242\4B\240\356\200\204\252\35\20c\220LJ}\275x!\25\23\262\325{\246\304?\36\272\343\17\230\2449Q[y\334(m1\252N<\253?^#\236p\311\3006\f\245M*<\273H\333\225\256\317\322\363\273\335\303\243\354\a\253\342\312\302\372vTQ\247\r\210\343\264\323*E\364\2\ba\305Z79\273M\327\310F\301,\235\32\323"
Only she could have done this (only she has the her private key data) -- so this is a signature on the
message. Anyone can verify the signature by "decrypting" the message with the public "encryption" key.
(A-pub (A-prv msg)) -> #"Dear Bob, meet you in Lymm at 1200, Alice"
But anyone is able to do this, so there is no privacy here.
Everyone knows that it can only be Alice at Lymm at noon, but this message is for Bob's eyes only.
We need to encrypt this with his public key:
(B-pub (A-prv msg)) -> #"\eXq\4/\207\250hs\244<ym\3716\210\357'0\351E\202D\360\177\361\325\24\310+s\340j1\36\0\213\353\254\314*\212a;\300\210\\\347\371z`\226\230 \230A\337d\31\262nwp\6m\312\320D@h\232d]{sN\312xAW\216'c\27V\5\270\267>@\305\312\210\262|tGU?\266\325\250\227\270X\235\6C\307\323D\301q{\266S\351,i\211,~X\341\225z4\320F\353\361I\313M\270&d\267m\207 \2736s_\272\307\275\31T\301\247\317@\16D\263X\"\340\262\204\277g\30\337\311o\205\236\34\370)\323\275\5\1\301>\226Q,\255\213\\\2\307\215c\342\323\16\226a\3U\254\214\275\274\214\325\f\226\347\325\225\354~\32z)\340re5I\321\254\34'T\n\220p\316#\1\347\6;*\347\303\351\342\221\244\eey\31\275y\271\2605y\344\261\202B\321E\335\212"
Which is what gets posted to alt.chat.secret-rendezvous
Bob decrypts this to get the signed message from Alice:
(B-prv (B-pub (A-prv msg))) -> #"B}\4<G\373-\217\350;\0214\226\233\236\333\215\226\225=\236\350\277X\241*J\356\302\250\350fO\5\375u\367\365\315\270\312\334\204U\332\224\322\357u=\262\326\274e\31\301\321\210:i\361\361g\361\16\5a\304X\306\313\350(^\374 \353\350t\2662\305\346a\300\244b\337JI\343\335\21j\202\236\242\335<rA\a\233\375\23\t\32(d`\237i\267\336\270\340L\26\f\260\346&\t\301\326\331k@\253\242\241VKw\365 \204U\270*\r,\334h=\257\230\320V\357\304\242\4B\240\356\200\204\252\35\20c\220LJ}\275x!\25\23\262\325{\246\304?\36\272\343\17\230\2449Q[y\334(m1\252N<\253?^#\236p\311\3006\f\245M*<\273H\333\225\256\317\322\363\273\335\303\243\354\a\253\342\312\302\372vTQ\247\r\210\343\264\323*E\364\2\ba\305Z79\273M\327\310F\301,\235\32\323"
And verifies Alice's signature:
(A-pub (B-prv (B-pub (A-prv msg)))) -> #"Dear Bob, meet you in Lymm at 1200, Alice"
Alice genuinely sent the message.
And nobody else (on a.c.s-r, at least) has read it.
KEYS:
Alice's full set: (104685401856522903402850023081275254628934665755538824520013952439504139625115997713509448983980532229245117213463882915048453185855818576471069794363975118091990355601850994380420233190180031768156385491949949615002445375960925665247160785747787333124376845288066200576472845984390877385677186819996627676676634639097055255729270941154875343472575964213139374388405182305309760553726485659960423106167598201105611690471457085541574734821426421348095213778701793437599877207476950721505461275161623234401077010666082709757697115644957960465476529769011591060857755043525709942747005873747546230596592939772042294065813 65537 2717089103223300710869400327467677924284264864888101995949520623458451584636500177162207191689440855119183734075439291369721208922299577316283774359722319847940485449116507338466741179127763462435479373816422239271238530669843516739939583694050479174276592059981393826091830737132442474221232201364332570578846365932454353567371199951546467136016124284306002875511872761969350668537869576342139916560099770887726733315363999637008436783521647128919138309876487426046116064403375745886449924998134881929018589019826525209406457786746758225025770316010686323085528641486882271019011944746635302695471929361680924424193)
Bob's full set: (66453628687555934925745945778628604864426900808865010224295559035622434292221884343034084372211796572669001844069872620681089178469116242843088008182594366670325110214571956032739850447309116856946000976960287962443982329056724158946509616005091553738944083845350969457626620995817549009173612137879065054069514627974682257866567361928526254468372096924822844681482943840826594855542477801468633973217959659919448029041762191501507078772425126221108904380853918736930072466664169841898502073182029626689571748335731893924787111612947107622008163390629359277338946499436753837554940480109336784185532913066463754681017 65537 40065645823449313467522156271281139875308606308813084959528059327930012759030216763756439504389958618427304726562596348031980052624474573194667691034359998325290386803002604616043604539794698176121997293172282195706991070204897106862588071733664686557012032668284371518061563321600604452438728297053809260134398419618573423334221995863281726034127469856978901426632431740449516515840848181614406772903883730243825332015822633960435748349249507976445585019837204822017018926054553157019477128528700400452557614873387735038266042290443427594857847964184247833486305612600841625002197933394323058877499979801727736278613)
The burden seems to be in finding (proving) the next large prime after the big random number. However, once keys are generated, things are pretty snappy.
Ruby
<lang ruby>
- !/usr/bin/ruby
require 'openssl' # for mod_exp only require 'prime'
def rsa_encode blocks, e, n
blocks.map{|b| b.to_bn.mod_exp(e, n).to_i}
end
def rsa_decode ciphers, d, n
rsa_encode ciphers, d, n
end
- all numbers in blocks have to be < modulus, or information is lost
- for secure encryption only use big modulus and blocksizes
def text_to_blocks text, blocksize=64 # 1 hex = 4 bit => default is 256bit
text.each_byte.reduce(""){|acc,b| acc << b.to_s(16).rjust(2, "0")} # convert text to hex (preserving leading 0 chars)
.each_char.each_slice(blocksize).to_a # slice hexnumbers in pieces of blocksize
.map{|a| a.join("").to_i(16)} # convert each slice into internal number
end
def blocks_to_text blocks
blocks.map{|d| d.to_s(16)}.join("") # join all blocks into one hex-string
.each_char.each_slice(2).to_a # group into pairs
.map{|s| s.join("").to_i(16)} # number from 2 hexdigits is byte .flatten.pack("C*") # pack bytes into ruby-string .force_encoding(Encoding::default_external) # reset encoding end
def generate_keys p1, p2
n = p1 * p2 t = (p1 - 1) * (p2 - 1) e = 2.step.each do |i| break i if i.gcd(t) == 1 end d = 1.step.each do |i| break i if (i * e) % t == 1 end return e, d, n
end
p1, p2 = Prime.take(100).last(2) public_key, private_key, modulus =
generate_keys p1, p2
print "Message: " message = gets blocks = text_to_blocks message, 4 # very small primes print "Numbers: "; p blocks encoded = rsa_encode(blocks, public_key, modulus) print "Encrypted as: "; p encoded decoded = rsa_decode(encoded, private_key, modulus) print "Decrypted to: "; p decoded final = blocks_to_text(decoded) print "Decrypted Message: "; puts final </lang>
- Output:
% echo "☆ rosettacode.org ✓" | ./rsa.rb Message: Numbers: [58008, 34336, 29295, 29541, 29812, 24931, 28516, 25902, 28530, 26400, 58012, 37642] Encrypted as: [8509, 99626, 21784, 139807, 81066, 67678, 183438, 147659, 261822, 85962, 150227, 167121] Decrypted to: [58008, 34336, 29295, 29541, 29812, 24931, 28516, 25902, 28530, 26400, 58012, 37642] Decrypted Message: ☆ rosettacode.org ✓
Scala
The code below demonstrates RSA encryption and decryption in Scala. Text to integer encryption using ASCII code. <lang Scala> object RSA_saket{
val d = BigInt("5617843187844953170308463622230283376298685")
val n = BigInt("9516311845790656153499716760847001433441357")
val e = 65537
val text = "Rosetta Code"
val encode = (msg:BigInt) => pow_mod(msg,e,n)
val decode = (msg:BigInt) => pow_mod(msg,d,n)
val getmsg = (txt:String) => BigInt(txt.map(x => "%03d".format(x.toInt)).reduceLeft(_+_))
def pow_mod(p:BigInt, q:BigInt, n:BigInt):BigInt = {
if(q==0) BigInt(1)
else if(q==1) p
else if(q%2 == 1) pow_mod(p,q-1,n)*p % n
else pow_mod(p*p % n,q/2,n)
}
def gettxt(num:String) = {
if(num.size%3==2)
("0" + num).grouped(3).toList.foldLeft("")(_ + _.toInt.toChar)
else
num.grouped(3).toList.foldLeft("")(_ + _.toInt.toChar)
}
def main(args: Array[String]): Unit = {
println(f"Original String \t: "+text)
val msg = getmsg(text)
println(f"Converted Signal \t: "+msg)
val enc_sig = encode(msg)
println("Encoded Signal \t\t: "+ enc_sig)
val dec_sig = decode(enc_sig)
println("Decoded String \t\t: "+ dec_sig)
val rec_msg = gettxt(dec_sig.toString)
println("Retrieved Signal \t: "+rec_msg)
}
} </lang>
- Output:
Ouput: Original String : Rosetta Code Converted String : 82111115101116116097032067111100101 Encoded Signal : 5902559240035849005218240192859088445397686 Decoded String : 82111115101116116097032067111100101 Retrieved String : Rosetta Code
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "bigint.s7i"; include "bytedata.s7i";
const proc: main is func
local
const string: plainText is "Rosetta Code";
# Use a key big enough to hold 16 bytes of plain text in a single block.
const bigInteger: modulus is 9516311845790656153499716760847001433441357_;
const bigInteger: encode is 65537_;
const bigInteger: decode is 5617843187844953170308463622230283376298685_;
var bigInteger: plainTextNumber is 0_;
var bigInteger: encodedNumber is 0_;
var bigInteger: decodedNumber is 0_;
var string: decodedText is "";
begin
writeln("Plain text: " <& plainText);
plainTextNumber := bytes2BigInt(plainText, UNSIGNED, BE);
if plainTextNumber >= modulus then
writeln("Plain text message too long");
else
writeln("Plain text as a number: " <& plainTextNumber);
encodedNumber := modPow(plainTextNumber, encode, modulus);
writeln("Encoded: " <& encodedNumber);
decodedNumber := modPow(encodedNumber, decode, modulus);
writeln("Decoded: " <& decodedNumber);
decodedText := bytes(decodedNumber, UNSIGNED, BE);
writeln("Decoded number as text: " <& decodedText);
end if;
end func;</lang>
- Output:
Plain text: Rosetta Code Plain text as a number: 25512506514985639724585018469 Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 Decoded number as text: Rosetta Code
Sidef
<lang ruby>const n = 9516311845790656153499716760847001433441357 const e = 65537 const d = 5617843187844953170308463622230283376298685
module Message {
var alphabet = [('A' .. 'Z')..., ' ']
var rad = alphabet.len
var code = Hash(^rad -> map {|i| (alphabet[i], i) }...)
func encode(String t) {
[code{t.reverse.chars...}] ~Z* t.len.range.map { |i| rad**i } -> sum(0)
}
func decode(Number n) {
.join(alphabet[
gather {
loop {
var (d, m) = n.divmod(rad)
take(m)
break if (n < rad)
n = d
}
}...]
).reverse
}
}
var secret_message = "ROSETTA CODE" say "Secret message is #{secret_message}"
var numeric_message = Message::encode(secret_message) say "Secret message in integer form is #{numeric_message}"
var numeric_cipher = expmod(numeric_message, e, n) say "After exponentiation with public exponent we get: #{numeric_cipher}"
var text_cipher = Message::decode(numeric_cipher) say "This turns into the string #{text_cipher}"
var numeric_cipher2 = Message::encode(text_cipher) say "If we re-encode it in integer form we get #{numeric_cipher2}"
var numeric_message2 = expmod(numeric_cipher2, d, n) say "After exponentiation with SECRET exponent we get: #{numeric_message2}"
var secret_message2 = Message::decode(numeric_message2) say "This turns into the string #{secret_message2}"</lang>
- Output:
Secret message is ROSETTA CODE Secret message in integer form is 97525102075211938 After exponentiation with public exponent we get: 8326171774113983822045243488956318758396426 This turns into the string ZULYDCEZOWTFXFRRNLIMGNUPHVCJSX If we re-encode it in integer form we get 8326171774113983822045243488956318758396426 After exponentiation with SECRET exponent we get: 97525102075211938 This turns into the string ROSETTA CODE
Tcl
This code is careful to avoid the assumption that the input string is in a single-byte encoding, instead forcing the encryption to be performed on the UTF-8 form of the text. <lang tcl>package require Tcl 8.5
- This is a straight-forward square-and-multiply implementation that relies on
- Tcl 8.5's bignum support (based on LibTomMath) for speed.
proc modexp {b expAndMod} {
lassign $expAndMod -> e n
if {$b >= $n} {puts stderr "WARNING: modulus too small"}
for {set r 1} {$e != 0} {set e [expr {$e >> 1}]} {
if {$e & 1} { set r [expr {($r * $b) % $n}] } set b [expr {($b ** 2) % $n}]
} return $r
}
- Assumes that messages are shorter than the modulus
proc rsa_encrypt {message publicKey} {
if {[lindex $publicKey 0] ne "publicKey"} {error "key handling"}
set toEnc 0
foreach char [split [encoding convertto utf-8 $message] ""] {
set toEnc [expr {$toEnc * 256 + [scan $char "%c"]}]
} return [modexp $toEnc $publicKey]
}
proc rsa_decrypt {encrypted privateKey} {
if {[lindex $privateKey 0] ne "privateKey"} {error "key handling"}
set toDec [modexp $encrypted $privateKey]
for {set message ""} {$toDec > 0} {set toDec [expr {$toDec >> 8}]} {
append message [format "%c" [expr {$toDec & 255}]]
} return [encoding convertfrom utf-8 [string reverse $message]]
}
- Assemble packaged public and private keys
set e 65537 set n 9516311845790656153499716760847001433441357 set d 5617843187844953170308463622230283376298685 set publicKey [list "publicKey" $e $n] set privateKey [list "privateKey" $d $n]
- Test on some input strings
foreach input {"Rosetta Code" "UTF-8 \u263a test"} {
set enc [rsa_encrypt $input $publicKey] set dec [rsa_decrypt $enc $privateKey] puts "$input -> $enc -> $dec"
}</lang> Output:
Rosetta Code -> 916709442744356653386978770799029131264344 -> Rosetta Code UTF-8 ☺ test -> 3905697186829810541171404594906488782823186 -> UTF-8 ☺ test
Visual Basic .NET
<lang vbnet>Imports System Imports System.Numerics Imports System.Text
Module Module1
Sub Main()
Dim n As BigInteger = BigInteger.Parse("9516311845790656153499716760847001433441357")
Dim e As BigInteger = 65537
Dim d As BigInteger = BigInteger.Parse("5617843187844953170308463622230283376298685")
Dim plainTextStr As String = "Hello, Rosetta!"
Dim plainTextBA As Byte() = ASCIIEncoding.ASCII.GetBytes(plainTextStr)
Dim pt As BigInteger = New BigInteger(plainTextBA)
If pt > n Then Throw New Exception() ' Blocking not implemented
Dim ct As BigInteger = BigInteger.ModPow(pt, e, n)
Console.WriteLine(" Encoded: " & ct.ToString("X"))
Dim dc As BigInteger = BigInteger.ModPow(ct, d, n)
Console.WriteLine(" Decoded: " & dc.ToString("X"))
Dim decoded As String = ASCIIEncoding.ASCII.GetString(dc.ToByteArray())
Console.WriteLine("As ASCII: " & decoded)
End Sub
End Module</lang>
- Output:
Encoded: 6219A470D8B319A31C8E13F612B31337098F Decoded: 2161747465736F52202C6F6C6C6548 As ASCII: Hello, Rosetta!
zkl
No blocking. <lang zkl>var BN=Import.lib("zklBigNum");
n:=BN("9516311845790656153499716760847001433441357"); e:=BN("65537"); d:=BN("5617843187844953170308463622230283376298685");
const plaintext="Rossetta Code"; pt:=BN(Data(Int,0,plaintext)); // convert string (as stream of bytes) to big int if(pt>n) throw(Exception.ValueError("Message is too large"));
println("Plain text: ",plaintext); println("As Int: ",pt); ct:=pt.powm(e,n); println("Encoded: ",ct); pt =ct.powm(d,n); println("Decoded: ",pt); txt:=pt.toData().text; // convert big int to bytes, treat as string println("As String: ",txt);</lang>
- Output:
Plain text: Rossetta Code As Int: 6531201733672758787904906421349 Encoded: 5278143020249600501803788468419399384934220 Decoded: 6531201733672758787904906421349 As String: Rossetta Code