# RSA code

RSA code
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 “${\displaystyle n}$” and “${\displaystyle e}$” (which makes up the Public Key used for encryption) to everyone. The Private Key used for decryption “${\displaystyle d}$” 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 ${\displaystyle a=01,b=02,...,z=26}$). This yields a string of numbers, generally referred to as "numerical plaintext", “${\displaystyle P}$”. For example, “Hello World” encoded with a=1,...,z=26 by hundreds would yield ${\displaystyle 08051212152315181204}$.

The plaintext must also be split into blocks so that the numerical plaintext is smaller than ${\displaystyle n}$ otherwise the decryption will fail.

The ciphertext, ${\displaystyle C}$, is then computed by taking each block of ${\displaystyle P}$, and computing

${\displaystyle C\equiv P^{e}\mod n}$

Similarly, to decode, one computes

${\displaystyle P\equiv C^{d}\mod n}$

To generate a key, one finds 2 (ideally large) primes ${\displaystyle p}$ and ${\displaystyle q}$. the value “${\displaystyle n}$” is simply: ${\displaystyle n=p\times q}$. One must then choose an “${\displaystyle e}$” such that ${\displaystyle \gcd(e,(p-1)\times (q-1))=1}$. That is to say, ${\displaystyle e}$ and ${\displaystyle (p-1)\times (q-1)}$ are relatively prime to each other.

The decryption value ${\displaystyle d}$ is then found by solving

${\displaystyle d\times e\equiv 1\mod (p-1)\times (q-1)}$

The security of the code is based on the secrecy of the Private Key (decryption exponent) “${\displaystyle d}$” and the difficulty in factoring “${\displaystyle n}$”. 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).

• 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.
 WarningRosetta Code is not a place you should rely on for examples of code in critical roles, including security.Cryptographic routines should be validated before being used.For a discussion of limitations and please refer to Talk:RSA_code#Difference_from_practical_cryptographical_version.

## ALGOL 68

The code below uses Algol 68 Genie which provides arbitrary precision arithmetic for LONG LONG modes.

 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
Output:
THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE


## C

Library: GMP
 #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;}
Output:
Encoded:   5278143020249600501803788468419399384934220
Decoded:   6531201733672758787904906421349
As String: Rossetta Code


## C#

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);    }}
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.

(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)))

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.

Translation of: Go
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);}
Output:
Plain text:             Rosetta Code
Plain text as a number: 25512506514985639724585018469
Encoded:                916709442744356653386978770799029131264344
Decoded:                25512506514985639724585018469
Decoded number as text: Rosetta Code

## F#

 //Nigel Galloway February 12th., 2018let RSA n g l = bigint.ModPow(l,n,g)let encrypt = RSA 65537I 9516311845790656153499716760847001433441357Ilet 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_inlet decrypt = RSA 5617843187844953170308463622230283376298685I 9516311845790656153499716760847001433441357Ilet g = Array.map decrypt nlet 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.GetStringprintfn "'The magic words are SQUEAMISH OSSIFRAGE' as numbers -> %A\nEncrypted -> %A\nDecrypted -> %A\nAs text -> %A" m_in n g m_out
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

Translation of: C
' 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 bufferWhile Inkey <> "" : WendPrint : Print "hit any key to end program"SleepEnd
Output:
  Encoded: 916709442744356653386978770799029131264344
Decoded: 25512506514985639724585018469
As string: Rosetta Code

## Go

Note: see the crypto/rsa package included with Go for a full implementation.

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:]))}

Output:

Plain text:             Rosetta Code
Plain text as a number: 25512506514985639724585018469
Encoded:                916709442744356653386978770799029131264344
Decoded:                25512506514985639724585018469
Decoded number as text: Rosetta Code


module RSAMaker    whereimport 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] -> Stringdecode encoded = map ( chr . fromInteger ) encoded  divisors :: Integer -> [Integer]divisors n = [m | m <- [1..n] , mod n m == 0 ] isPrime :: Integer -> BoolisPrime n = divisors n == [1,n] totient :: Integer -> Integer -> Integertotient prime1 prime2 = (prime1 - 1 ) * ( prime2 - 1 )  myE :: Integer -> IntegermyE tot = head [n | n <- [2..tot - 1] , gcd n tot == 1] myD :: Integer -> Integer -> Integer  -> IntegermyD 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 )
Output:
Enter a test text!
Rosettacode
Encrypted: 65646265111107071564791028551028551458331139502651145035156479
And now decrypted: Rosettacode


## Icon and Unicon

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 resultend procedure rsa(text,e,n)  # return rsa encryption of numerically encoded message; fail if text < nreturn mod_power(text,e,text < n)end procedure encode(text)  # numerically encode ascii text as int   every (message := 0) := ord(!text) + 256 * message   return messageend 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 Mend procedure unblock(M,b)  # reassemble blocks of size b into lg int   every (m := 0) := !M + b * m   return mend

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]

   N=: 9516311845790656153499716760847001433441357x   E=: 65537x   D=: 5617843187844953170308463622230283376298685x    ] text=: 'Rosetta Code'Rosetta Code   ] num=: 256x #. a.i.text25512506514985639724585018469   num >: N  NB. check if blocking is necessary (0 means no)0   ] enc=: N&|@^&E num916709442744356653386978770799029131264344   ] dec=: N&|@^&D enc25512506514985639724585018469   ] final=: a. {~ 256x #.inv decRosetta Code

Note: as indicated at http://www.jsoftware.com/help/dictionary/special.htm, N&|@^ does not bother with creating the exponential intermediate result.

## 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);}

## Julia

Works with: Julia version 0.6
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 = 65537dsecr = big"5617843187844953170308463622230283376298685" encoded = rsaencode(msg, nmod, expub)decoded = rsadecode(encoded, nmod, dsecr)println("\n# $msg\n -> ENCODED:$encoded\n -> DECODED: $decoded") Output: # Rosetta Code. -> ENCODED: 2440331969632134446717000067136916252596373 -> DECODED: Rosetta Code. ## Kotlin // version 1.1.4-3 import java.math.BigInteger fun main(args: Array<String>) { val n = BigInteger("9516311845790656153499716760847001433441357") val e = BigInteger("65537") val d = BigInteger("5617843187844953170308463622230283376298685") val c = Charsets.UTF_8 val plainText = "Rosetta Code" println("PlainText :$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")}
Output:
PlainText : Rosetta Code
As number : 25512506514985639724585018469
Encoded   : 916709442744356653386978770799029131264344
Decoded   : 25512506514985639724585018469
As text   : Rosetta Code


## Mathematica

Does not support blocking.

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]; $Failed);enc[n_, e_, mess_] := PowerMod[toNumPlTxt[mess], e, n];dec[n_, d_, en_] := fromNumPlTxt[PowerMod[en, d, n]];text = "The cake is a lie!";n = 9516311845790656153499716760847001433441357;e = 65537;d = 5617843187844953170308463622230283376298685;en = enc[n, e, text];de = dec[n, d, en];Print["Text: '" <> text <> "'"];Print["n = " <> IntegerString[n]];Print["e = " <> IntegerString[e]];Print["d = " <> IntegerString[d]];Print["Numeric plaintext: " <> IntegerString[toNumPlTxt[text]]];Print["Encoded: " <> IntegerString[en]];Print["Decoded: '" <> de <> "'"]; Output: Text: 'The cake is a lie!' n = 9516311845790656153499716760847001433441357 e = 65537 d = 5617843187844953170308463622230283376298685 Numeric plaintext: 7352955804624388987810264523908743852287265 Encoded: 199505409518408949879682159958576932863989 Decoded: 'The cake is a lie!' ## PARI/GP stigid(V,b)=subst(Pol(V),'x,b); \\ inverse function digits(...) n = 9516311845790656153499716760847001433441357;e = 65537;d = 5617843187844953170308463622230283376298685; text = "Rosetta Code" inttext = stigid(Vecsmall(text),256) \\ message as an integerencoded = lift(Mod(inttext, n) ^ e) \\ encrypted messagedecoded = lift(Mod(encoded, n) ^ d) \\ decrypted messagemessage = Strchr(digits(decoded, 256)) \\ readable message Output: text: "Rosetta Code" inttext: 25512506514985639724585018469 encoded: 916709442744356653386978770799029131264344 decoded: 25512506514985639724585018469 message: "Rosetta Code"  If inttext is equal or greater than b = 2^(log(n)/log(2)\1) use block = inttext % b; inttext /= b; to break inttext into blocks and encode piece by piece. Decode in reverse order. As a check: it's easy to crack this weak encrypted message without knowing secret key 'd' f = factor(n); \\ factorize public key 'n' crack = Strchr(digits(lift(Mod(encoded,n) ^ lift(Mod(1,(f[1,1]-1)*(f[2,1]-1)) / e)),256)) Output: crack: "Rosetta Code" ## Perl Translation of: Perl 6 use bigint;$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_messageSecret message in integer form is $numeric_messageAfter exponentiation with public exponent we get:$numeric_cipherThis turns into the string $text_cipherIf we re-encode it in integer form we get$numeric_cipher2After exponentiation with SECRET exponent we get: $numeric_message2This turns into the string$secret_message2EOT
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

Works with: rakudo version 2015-11-04

No blocking here. Algorithm doesn't really work if either red or black text begins with 'A'.

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";
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

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)})
Output:
Original:  Rosetta Code
As Number: 25512506514985639724585018469
Encoded:   916709442744356653386978770799029131264344
Decoded:   25512506514985639724585018469
As ASCII:  Rosetta Code


## PicoLisp

PicoLisp comes with an RSA library:

### 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"

## Python

import binascii n = 9516311845790656153499716760847001433441357    # p*q = moduluse = 65537d = 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
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(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: ~ssent 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: ~sdecyphered 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   #<<EOSAlice wants to send ~s to Bob.She "encrypts" with her private "decryption" key.(A-prv msg) -> ~sOnly she could have done this (only she has the her private key data) -- so this is a signature on themessage. Anyone can verify the signature by "decrypting" the message with the public "encryption" key.(A-pub (A-prv msg)) -> ~sBut 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)) -> ~sWhich is what gets posted to alt.chat.secret-rendezvousBob decrypts this to get the signed message from Alice:(B-prv (B-pub (A-prv msg))) -> ~sAnd verifies Alice's signature:(A-pub (B-prv (B-pub (A-prv msg)))) -> ~sAlice 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: ~sEOS   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))
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<[email protected]\20\4\233\275\333?"
...

...
#"/-\e\355\225\327\244\222<[email protected]\20\4\233\275\333?"
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"
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\[email protected]\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\[email protected]\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\[email protected]\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\[email protected]\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

 #!/usr/bin/ruby require 'openssl' # for mod_exp onlyrequire '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, nend # all numbers in blocks have to be < modulus, or information is lost# for secure encryption only use big modulus and blocksizesdef 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 numberend 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 encodingend 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, nend p1, p2 = Prime.take(100).last(2)public_key, private_key, modulus =  generate_keys p1, p2 print "Message: "message = getsblocks = text_to_blocks message, 4 # very small primesprint "Numbers: "; p blocksencoded = rsa_encode(blocks, public_key, modulus)print "Encrypted as: "; p encodeddecoded = rsa_decode(encoded, private_key, modulus)print "Decrypted to: "; p decodedfinal = blocks_to_text(decoded)print "Decrypted Message: "; puts final 
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.

 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)    }} 
Output:
    Ouput:
Original String         : Rosetta Code
Converted String        : 82111115101116116097032067111100101
Encoded Signal          : 5902559240035849005218240192859088445397686
Decoded String          : 82111115101116116097032067111100101
Retrieved String        : Rosetta Code


$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; Output: Plain text: Rosetta Code Plain text as a number: 25512506514985639724585018469 Encoded: 916709442744356653386978770799029131264344 Decoded: 25512506514985639724585018469 Decoded number as text: Rosetta Code  ## Sidef Translation of: Perl 6 const n = 9516311845790656153499716760847001433441357const e = 65537const 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}" 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. 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 $r} # Assumes that messages are shorter than the modulusproc rsa_encrypt {message publicKey} { if {[lindex$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 keysset e 65537set n 9516311845790656153499716760847001433441357set d 5617843187844953170308463622230283376298685set publicKey  [list "publicKey"  $e$n]set privateKey [list "privateKey" $d$n] # Test on some input stringsforeach input {"Rosetta Code" "UTF-8 \u263a test"} {    set enc [rsa_encrypt $input$publicKey]    set dec [rsa_decrypt $enc$privateKey]    puts "$input ->$enc -> \$dec"}

Output:

Rosetta Code -> 916709442744356653386978770799029131264344 -> Rosetta Code
UTF-8 ☺ test -> 3905697186829810541171404594906488782823186 -> UTF-8 ☺ test


## Visual Basic .NET

Translation of: C#
Imports SystemImports System.NumericsImports 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 SubEnd Module
Output:
 Encoded: 6219A470D8B319A31C8E13F612B31337098F
Decoded: 2161747465736F52202C6F6C6C6548
As ASCII: Hello, Rosetta!

## zkl

Translation of: C
Library: GMP

No blocking.

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 intif(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 stringprintln("As String:  ",txt);
Output:
Plain text: Rossetta Code
As Int:     6531201733672758787904906421349
Encoded:    5278143020249600501803788468419399384934220
Decoded:    6531201733672758787904906421349
As String:  Rossetta Code
`