From: pgut01@cs.auckland.ac.nz (Peter Gutmann)
Newsgroups: sci.crypt
Subject: Specification for Ron Rivests Cipher No.2
Date: 11 Feb 1996 06:45:03 GMT
Organization: University of Auckland
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                           Ron Rivest's Cipher No.2
                           ------------------------
 
Ron Rivest's Cipher No.2 (hereafter referred to as RRC.2, other people may
refer to it by other names) is word oriented, operating on a block of 64 bits
divided into four 16-bit words, with a key table of 64 words.  All data units
are little-endian.  This functional description of the algorithm is based in
the paper "The RC5 Encryption Algorithm" (RC5 is a trademark of RSADSI), using
the same general layout, terminology, and pseudocode style.
 
 
Notation and RRC.2 Primitive Operations
 
RRC.2 uses the following primitive operations:
 
1. Two's-complement addition of words, denoted by "+".  The inverse operation,
   subtraction, is denoted by "-".
2. Bitwise exclusive OR, denoted by "^".
3. Bitwise AND, denoted by "&".
4. Bitwise NOT, denoted by "~".
5. A left-rotation of words; the rotation of word x left by y is denoted
   x <<< y.  The inverse operation, right-rotation, is denoted x >>> y.
 
These operations are directly and efficiently supported by most processors.
 
 
The RRC.2 Algorithm
 
RRC.2 consists of three components, a *key expansion* algorithm, an
*encryption* algorithm, and a *decryption* algorithm.
 
 
Key Expansion
 
The purpose of the key-expansion routine is to expand the user's key K to fill
the expanded key array S, so S resembles an array of random binary words
determined by the user's secret key K.
 
Initialising the S-box
 
RRC.2 uses a single 256-byte S-box derived from the ciphertext contents of
Beale Cipher No.1 XOR'd with a one-time pad.  The Beale Ciphers predate modern
cryptography by enough time that there should be no concerns about trapdoors
hidden in the data.  They have been published widely, and the S-box can be
easily recreated from the one-time pad values and the Beale Cipher data taken
from a standard source.  To initialise the S-box:
 
  for i = 0 to 255 do
    sBox[ i ] = ( beale[ i ] mod 256 ) ^ pad[ i ]
 
The contents of Beale Cipher No.1 and the necessary one-time pad are given as
an appendix at the end of this document.  For efficiency, implementors may wish
to skip the Beale Cipher expansion and store the sBox table directly.
 
Expanding the Secret Key to 128 Bytes
 
The secret key is first expanded to fill 128 bytes (64 words).  The expansion
consists of taking the sum of the first and last bytes in the user key, looking
up the sum (modulo 256) in the S-box, and appending the result to the key.  The
operation is repeated with the second byte and new last byte of the key until
all 128 bytes have been generated.  Note that the following pseudocode treats
the S array as an array of 128 bytes rather than 64 words.
 
  for j = 0 to length-1 do
    S[ j ] = K[ j ]
  for j = length to 127 do
    s[ j ] = sBox[ ( S[ j-length ] + S[ j-1 ] ) mod 256 ];
 
At this point it is possible to perform a truncation of the effective key
length to ease the creation of espionage-enabled software products.  However
since the author cannot conceive why anyone would want to do this, it will not
be considered further.
 
The final phase of the key expansion involves replacing the first byte of S
with the entry selected from the S-box:
 
  S[ 0 ] = sBox[ S[ 0 ] ]
 
 
Encryption
 
The cipher has 16 full rounds, each divided into 4 subrounds.  Two of the full
rounds perform an additional transformation on the data.  Note that the
following pseudocode treats the S array as an array of 64 words rather than 128
bytes.
 
  for i = 0 to 15 do
    j = i * 4;
    word0 = ( word0 + ( word1 & ~word3 ) + ( word2 & word3 ) + S[ j+0 ] ) <<< 1
    word1 = ( word1 + ( word2 & ~word0 ) + ( word3 & word0 ) + S[ j+1 ] ) <<< 2
    word2 = ( word2 + ( word3 & ~word1 ) + ( word0 & word1 ) + S[ j+2 ] ) <<< 3
    word3 = ( word3 + ( word0 & ~word2 ) + ( word1 & word2 ) + S[ j+3 ] ) <<< 5
 
In addition the fifth and eleventh rounds add the contents of the S-box indexed
by one of the data words to another of the data words following the four
subrounds as follows:
 
    word0 = word0 + S[ word3 & 63 ];
    word1 = word1 + S[ word0 & 63 ];
    word2 = word2 + S[ word1 & 63 ];
    word3 = word3 + S[ word2 & 63 ];
 
 
Decryption
 
The decryption operation is simply the inverse of the encryption operation.
Note that the following pseudocode treats the S array as an array of 64 words
rather than 128 bytes.
 
  for i = 15 downto 0 do
    j = i * 4;
    word3 = ( word3 >>> 5 ) - ( word0 & ~word2 ) - ( word1 & word2 ) - S[ j+3 ]
    word2 = ( word2 >>> 3 ) - ( word3 & ~word1 ) - ( word0 & word1 ) - S[ j+2 ]
    word1 = ( word1 >>> 2 ) - ( word2 & ~word0 ) - ( word3 & word0 ) - S[ j+1 ]
    word0 = ( word0 >>> 1 ) - ( word1 & ~word3 ) - ( word2 & word3 ) - S[ j+0 ]
 
In addition the fifth and eleventh rounds subtract the contents of the S-box
indexed by one of the data words from another one of the data words following
the four subrounds as follows:
 
    word3 = word3 - S[ word2 & 63 ]
    word2 = word2 - S[ word1 & 63 ]
    word1 = word1 - S[ word0 & 63 ]
    word0 = word0 - S[ word3 & 63 ]
 
 
Test Vectors
 
The following test vectors may be used to test the correctness of an RRC.2
implementation:
 
  Key:      0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
  Plain:    0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
  Cipher:   0x1C, 0x19, 0x8A, 0x83, 0x8D, 0xF0, 0x28, 0xB7
 
  Key:      0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01
  Plain:    0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
  Cipher:   0x21, 0x82, 0x9C, 0x78, 0xA9, 0xF9, 0xC0, 0x74
 
  Key:      0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
  Plain:    0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF
  Cipher:   0x13, 0xDB, 0x35, 0x17, 0xD3, 0x21, 0x86, 0x9E
 
  Key:      0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07,
            0x08, 0x09, 0x0A, 0x0B, 0x0C, 0x0D, 0x0E, 0x0F
  Plain:    0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
  Cipher:   0x50, 0xDC, 0x01, 0x62, 0xBD, 0x75, 0x7F, 0x31
 
 
Appendix: Beale Cipher No.1, "The Locality of the Vault", and One-time Pad for
          Creating the S-Box
 
Beale Cipher No.1.
 
  71, 194,  38,1701,  89,  76,  11,  83,1629,  48,  94,  63, 132,  16, 111,  95,
  84, 341, 975,  14,  40,  64,  27,  81, 139, 213,  63,  90,1120,   8,  15,   3,
 126,2018,  40,  74, 758, 485, 604, 230, 436, 664, 582, 150, 251, 284, 308, 231,
 124, 211, 486, 225, 401, 370,  11, 101, 305, 139, 189,  17,  33,  88, 208, 193,
 145,   1,  94,  73, 416, 918, 263,  28, 500, 538, 356, 117, 136, 219,  27, 176,
 130,  10, 460,  25, 485,  18, 436,  65,  84, 200, 283, 118, 320, 138,  36, 416,
 280,  15,  71, 224, 961,  44,  16, 401,  39,  88,  61, 304,  12,  21,  24, 283,
 134,  92,  63, 246, 486, 682,   7, 219, 184, 360, 780,  18,  64, 463, 474, 131,
 160,  79,  73, 440,  95,  18,  64, 581,  34,  69, 128, 367, 460,  17,  81,  12,
 103, 820,  62, 110,  97, 103, 862,  70,  60,1317, 471, 540, 208, 121, 890, 346,
  36, 150,  59, 568, 614,  13, 120,  63, 219, 812,2160,1780,  99,  35,  18,  21,
 136, 872,  15,  28, 170,  88,   4,  30,  44, 112,  18, 147, 436, 195, 320,  37,
 122, 113,   6, 140,   8, 120, 305,  42,  58, 461,  44, 106, 301,  13, 408, 680,
  93,  86, 116, 530,  82, 568,   9, 102,  38, 416,  89,  71, 216, 728, 965, 818,
   2,  38, 121, 195,  14, 326, 148, 234,  18,  55, 131, 234, 361, 824,   5,  81,
 623,  48, 961,  19,  26,  33,  10,1101, 365,  92,  88, 181, 275, 346, 201, 206
 
One-time Pad.
 
 158, 186, 223,  97,  64, 145, 190, 190, 117, 217, 163,  70, 206, 176, 183, 194,
 146,  43, 248, 141,   3,  54,  72, 223, 233, 153,  91, 210,  36, 131, 244, 161,
 105, 120, 113, 191, 113,  86,  19, 245, 213, 221,  43,  27, 242, 157,  73, 213,
 193,  92, 166,  10,  23, 197, 112, 110, 193,  30, 156,  51, 125,  51, 158,  67,
 197, 215,  59, 218, 110, 246, 181,   0, 135,  76, 164,  97,  47,  87, 234, 108,
 144, 127,   6,   6, 222, 172,  80, 144,  22, 245, 207,  70, 227, 182, 146, 134,
 119, 176,  73,  58, 135,  69,  23, 198,   0, 170,  32, 171, 176, 129,  91,  24,
 126,  77, 248,   0, 118,  69,  57,  60, 190, 171, 217,  61, 136, 169, 196,  84,
 168, 167, 163, 102, 223,  64, 174, 178, 166, 239, 242, 195, 249,  92,  59,  38,
 241,  46, 236,  31,  59, 114,  23,  50, 119, 186,   7,  66, 212,  97, 222, 182,
 230, 118, 122,  86, 105,  92, 179, 243, 255, 189, 223, 164, 194, 215,  98,  44,
  17,  20,  53, 153, 137, 224, 176, 100, 208, 114,  36, 200, 145, 150, 215,  20,
  87,  44, 252,  20, 235, 242, 163, 132,  63,  18,   5, 122,  74,  97,  34,  97,
 142,  86, 146, 221, 179, 166, 161,  74,  69, 182,  88, 120, 128,  58,  76, 155,
  15,  30,  77, 216, 165, 117, 107,  90, 169, 127, 143, 181, 208, 137, 200, 127,
 170, 195,  26,  84, 255, 132, 150,  58, 103, 250, 120, 221, 237,  37,   8,  99
 
 
Implementation
 
A non-US based programmer who has never seen any encryption code before will
shortly be implementing RRC.2 based solely on this specification and not on
knowledge of any other encryption algorithms.  Stand by.