A Description of the RC2(r) Encryption Algorithm

Network Working Group                                          R. Rivest
Request for Comments: 2268           MIT Laboratory for Computer Science
Category: Informational                      and RSA Data Security, Inc.
                                                              March 1998


            A Description of the RC2(r) Encryption Algorithm

Status of this Memo

   This memo provides information for the Internet community.  It does
   not specify an Internet standard of any kind.  Distribution of this
   memo is unlimited.

Copyright Notice

   Copyright (C) The Internet Society (1998).  All Rights Reserved.

1. Introduction

   This memo is an RSA Laboratories Technical Note.  It is meant for
   informational use by the Internet community.

   This memo describes a conventional (secret-key) block encryption
   algorithm, called RC2, which may be considered as a proposal for a
   DES replacement. The input and output block sizes are 64 bits each.
   The key size is variable, from one byte up to 128 bytes, although the
   current implementation uses eight bytes.

   The algorithm is designed to be easy to implement on 16-bit
   microprocessors. On an IBM AT, the encryption runs about twice as
   fast as DES (assuming that key expansion has been done).

1.1 Algorithm description

   We use the term "word" to denote a 16-bit quantity. The symbol + will
   denote twos-complement addition. The symbol & will denote the bitwise
   "and" operation. The term XOR will denote the bitwise "exclusive-or"
   operation. The symbol ~ will denote bitwise complement.  The symbol ^
   will denote the exponentiation operation.  The term MOD will denote
   the modulo operation.

   There are three separate algorithms involved:

     Key expansion. This takes a (variable-length) input key and
     produces an expanded key consisting of 64 words K[0],...,K[63].

     Encryption. This takes a 64-bit input quantity stored in words
     R[0], ..., R[3] and encrypts it "in place" (the result is left in
     R[0], ..., R[3]).

     Decryption. The inverse operation to encryption.

2. Key expansion

   Since we will be dealing with eight-bit byte operations as well as
   16-bit word operations, we will use two alternative notations

   for referring to the key buffer:

        For word operations, we will refer to the positions of the
             buffer as K[0], ..., K[63]; each K[i] is a 16-bit word.

        For byte operations,  we will refer to the key buffer as
             L[0], ..., L[127]; each L[i] is an eight-bit byte.

   These are alternative views of the same data buffer. At all times it
   will be true that

                       K[i] = L[2*i] + 256*L[2*i+1].

   (Note that the low-order byte of each K word is given before the
   high-order byte.)

   We will assume that exactly T bytes of key are supplied, for some T
   in the range 1 <= T <= 128. (Our current implementation uses T = 8.)
   However, regardless of T, the algorithm has a maximum effective key
   length in bits, denoted T1. That is, the search space is 2^(8*T), or
   2^T1, whichever is smaller.

   The purpose of the key-expansion algorithm is to modify the key
   buffer so that each bit of the expanded key depends in a complicated
   way on every bit of the supplied input key.

   The key expansion algorithm begins by placing the supplied T-byte key
   into bytes L[0], ..., L[T-1] of the key buffer.

   The key expansion algorithm then computes the effective key length in
   bytes T8 and a mask TM based on the effective key length in bits T1.
   It uses the following operations:

   T8 = (T1+7)/8;
   TM = 255 MOD 2^(8 + T1 - 8*T8);

   Thus TM has its 8 - (8*T8 - T1) least significant bits set.

   For example, with an effective key length of 64 bits, T1 = 64, T8 = 8
   and TM = 0xff.  With an effective key length of 63 bits, T1 = 63, T8
   = 8 and TM = 0x7f.

   Here PITABLE[0], ..., PITABLE[255] is an array of "random" bytes
   based on the digits of PI = 3.14159... . More precisely, the array
   PITABLE is a random permutation of the values 0, ..., 255. Here is
   the PITABLE in hexadecimal notation:

        0  1  2  3  4  5  6  7  8  9  a  b  c  d  e  f
   00: d9 78 f9 c4 19 dd b5 ed 28 e9 fd 79 4a a0 d8 9d
   10: c6 7e 37 83 2b 76 53 8e 62 4c 64 88 44 8b fb a2
   20: 17 9a 59 f5 87 b3 4f 13 61 45 6d 8d 09 81 7d 32
   30: bd 8f 40 eb 86 b7 7b 0b f0 95 21 22 5c 6b 4e 82
   40: 54 d6 65 93 ce 60 b2 1c 73 56 c0 14 a7 8c f1 dc
   50: 12 75 ca 1f 3b be e4 d1 42 3d d4 30 a3 3c b6 26
   60: 6f bf 0e da 46 69 07 57 27 f2 1d 9b bc 94 43 03
   70: f8 11 c7 f6 90 ef 3e e7 06 c3 d5 2f c8 66 1e d7
   80: 08 e8 ea de 80 52 ee f7 84 aa 72 ac 35 4d 6a 2a
   90: 96 1a d2 71 5a 15 49 74 4b 9f d0 5e 04 18 a4 ec
   a0: c2 e0 41 6e 0f 51 cb cc 24 91 af 50 a1 f4 70 39
   b0: 99 7c 3a 85 23 b8 b4 7a fc 02 36 5b 25 55 97 31
   c0: 2d 5d fa 98 e3 8a 92 ae 05 df 29 10 67 6c ba c9
   d0: d3 00 e6 cf e1 9e a8 2c 63 16 01 3f 58 e2 89 a9
   e0: 0d 38 34 1b ab 33 ff b0 bb 48 0c 5f b9 b1 cd 2e
   f0: c5 f3 db 47 e5 a5 9c 77 0a a6 20 68 fe 7f c1 ad

   The key expansion operation consists of the following two loops and
   intermediate step:

   for i = T, T+1, ..., 127 do
     L[i] = PITABLE[L[i-1] + L[i-T]];

   L[128-T8] = PITABLE[L[128-T8] & TM];

   for i = 127-T8, ..., 0 do
     L[i] = PITABLE[L[i+1] XOR L[i+T8]];

   (In the first loop, the addition of L[i-1] and L[i-T] is performed
   modulo 256.)

   The "effective key" consists of the values L[128-T8],..., L[127].
   The intermediate step's bitwise "and" operation reduces the search
   space for L[128-T8] so that the effective number of key bits is T1.
   The expanded key depends only on the effective key bits, regardless
   of the supplied key K. Since the expanded key is not itself modified
   during encryption or decryption, as a pragmatic matter one can expand
   the key just once when encrypting or decrypting a large block of
   data.

3. Encryption algorithm

   The encryption operation is defined in terms of primitive "mix" and
   "mash" operations.

   Here the expression "x rol k" denotes the 16-bit word x rotated left
   by k bits, with the bits shifted out the top end entering the bottom
   end.

3.1 Mix up R[i]

   The primitive "Mix up R[i]" operation is defined as follows, where
   s[0] is 1, s[1] is 2, s[2] is 3, and s[3] is 5, and where the indices
   of the array R are always to be considered "modulo 4," so that R[i-1]
   refers to R[3] if i is 0 (these values are

   "wrapped around" so that R always has a subscript in the range 0 to 3
   inclusive):

   R[i] = R[i] + K[j] + (R[i-1] & R[i-2]) + ((~R[i-1]) & R[i-3]);
   j = j + 1;
   R[i] = R[i] rol s[i];

   In words: The next key word K[j] is added to R[i], and j is advanced.
   Then R[i-1] is used to create a "composite" word which is added to
   R[i]. The composite word is identical with R[i-2] in those positions
   where R[i-1] is one, and identical to R[i-3] in those positions where
   R[i-1] is zero. Then R[i] is rotated left by s[i] bits (bits rotated
   out the left end of R[i] are brought back in at the right). Here j is
   a "global" variable so that K[j] is always the first key word in the
   expanded key which has not yet been used in a "mix" operation.

3.2 Mixing round

   A "mixing round" consists of the following operations:

   Mix up R[0]
   Mix up R[1]
   Mix up R[2]
   Mix up R[3]

3.3 Mash R[i]

   The primitive "Mash R[i]" operation is defined as follows (using the
   previous conventions regarding subscripts for R):

   R[i] = R[i] + K[R[i-1] & 63];

   In words: R[i] is "mashed" by adding to it one of the words of the
   expanded key. The key word to be used is determined by looking at the
   low-order six bits of R[i-1], and using that as an index into the key
   array K.

3.4 Mashing round

   A "mashing round" consists of:

   Mash R[0]
   Mash R[1]
   Mash R[2]
   Mash R[3]

3.5 Encryption operation

   The entire encryption operation can now be described as follows. Here
   j is a global integer variable which is affected by the mixing
   operations.

        1. Initialize words R[0], ..., R[3] to contain the
           64-bit input value.

        2. Expand the key, so that words K[0], ..., K[63] become
           defined.

        3. Initialize j to zero.

        4. Perform five mixing rounds.

        5. Perform one mashing round.

        6. Perform six mixing rounds.

        7. Perform one mashing round.

        8. Perform five mixing rounds.

   Note that each mixing round uses four key words, and that there are
   16 mixing rounds altogether, so that each key word is used exactly
   once in a mixing round. The mashing rounds will refer to up to eight
   of the key words in a data-dependent manner. (There may be
   repetitions, and the actual set of words referred to will vary from
   encryption to encryption.)

4. Decryption algorithm

   The decryption operation is defined in terms of primitive operations
   that undo the "mix" and "mash" operations of the encryption
   algorithm. They are named "r-mix" and "r-mash" (r- denotes the
   reverse operation).

   Here the expression "x ror k" denotes the 16-bit word x rotated right
   by k bits, with the bits shifted out the bottom end entering the top
   end.

4.1 R-Mix up R[i]

   The primitive "R-Mix up R[i]" operation is defined as follows, where
   s[0] is 1, s[1] is 2, s[2] is 3, and s[3] is 5, and where the indices
   of the array R are always to be considered "modulo 4," so that R[i-1]
   refers to R[3] if i is 0 (these values are "wrapped around" so that R
   always has a subscript in the range 0 to 3 inclusive):

   R[i] = R[i] ror s[i];
   R[i] = R[i] - K[j] - (R[i-1] & R[i-2]) - ((~R[i-1]) & R[i-3]);
   j = j - 1;

   In words: R[i] is rotated right by s[i] bits (bits rotated out the
   right end of R[i] are brought back in at the left). Here j is a
   "global" variable so that K[j] is always the key word with greatest
   index in the expanded key which has not yet been used in a "r-mix"
   operation. The key word K[j] is subtracted from R[i], and j is
   decremented. R[i-1] is used to create a "composite" word which is
   subtracted from R[i].  The composite word is identical with R[i-2] in
   those positions where R[i-1] is one, and identical to R[i-3] in those
   positions where R[i-1] is zero.

4.2 R-Mixing round

   An "r-mixing round" consists of the following operations:

   R-Mix up R[3]
   R-Mix up R[2]
   R-Mix up R[1]
   R-Mix up R[0]

4.3 R-Mash R[i]

   The primitive "R-Mash R[i]" operation is defined as follows (using
   the previous conventions regarding subscripts for R):

   R[i] = R[i] - K[R[i-1] & 63];

   In words: R[i] is "r-mashed" by subtracting from it one of the words
   of the expanded key. The key word to be used is determined by looking
   at the low-order six bits of R[i-1], and using that as an index into
   the key array K.

4.4 R-Mashing round

   An "r-mashing round" consists of:

   R-Mash R[3]
   R-Mash R[2]
   R-Mash R[1]
   R-Mash R[0]

4.5 Decryption operation

   The entire decryption operation can now be described as follows.
   Here j is a global integer variable which is affected by the mixing
   operations.

        1. Initialize words R[0], ..., R[3] to contain the 64-bit
           ciphertext value.

        2. Expand the key, so that words K[0], ..., K[63] become
           defined.

        3. Initialize j to 63.

        4. Perform five r-mixing rounds.

        5. Perform one r-mashing round.

        6. Perform six r-mixing rounds.

        7. Perform one r-mashing round.

        8. Perform five r-mixing rounds.

5. Test vectors

   Test vectors for encryption with RC2 are provided below.

   All quantities are given in hexadecimal notation.

   Key length (bytes) = 8
   Effective key length (bits) = 63
   Key = 00000000 00000000
   Plaintext = 00000000 00000000
   Ciphertext = ebb773f9 93278eff

   Key length (bytes) = 8
   Effective key length (bits) = 64
   Key = ffffffff ffffffff
   Plaintext = ffffffff ffffffff
   Ciphertext = 278b27e4 2e2f0d49

   Key length (bytes) = 8
   Effective key length (bits) = 64
   Key = 30000000 00000000
   Plaintext = 10000000 00000001
   Ciphertext = 30649edf 9be7d2c2

   Key length (bytes) = 1
   Effective key length (bits) = 64
   Key = 88
   Plaintext = 00000000 00000000
   Ciphertext = 61a8a244 adacccf0

   Key length (bytes) = 7
   Effective key length (bits) = 64
   Key = 88bca90e 90875a
   Plaintext = 00000000 00000000
   Ciphertext = 6ccf4308 974c267f

   Key length (bytes) = 16
   Effective key length (bits) = 64
   Key = 88bca90e 90875a7f 0f79c384 627bafb2
   Plaintext = 00000000 00000000
   Ciphertext = 1a807d27 2bbe5db1

   Key length (bytes) = 16
   Effective key length (bits) = 128
   Key = 88bca90e 90875a7f 0f79c384 627bafb2
   Plaintext = 00000000 00000000
   Ciphertext = 2269552a b0f85ca6

   Key length (bytes) = 33
   Effective key length (bits) = 129
   Key = 88bca90e 90875a7f 0f79c384 627bafb2 16f80a6f 85920584
         c42fceb0 be255daf 1e

   Plaintext = 00000000 00000000
   Ciphertext = 5b78d3a4 3dfff1f1

6. RC2 Algorithm Object Identifier

   The Object Identifier for RC2 in cipher block chaining mode is

   rc2CBC OBJECT IDENTIFIER
     ::= {iso(1) member-body(2) US(840) rsadsi(113549)
          encryptionAlgorithm(3) 2}

   RC2-CBC takes parameters

   RC2-CBCParameter ::= CHOICE {
     iv IV,
     params SEQUENCE {
       version RC2Version,
       iv IV
     }
   }

   where

   IV ::= OCTET STRING -- 8 octets
   RC2Version ::= INTEGER -- 1-1024

   RC2 in CBC mode has two parameters: an 8-byte initialization vector
   (IV) and a version number in the range 1-1024 which specifies in a
   roundabout manner the number of effective key bits to be used for the
   RC2 encryption/decryption.

   The correspondence between effective key bits and version number is
   as follows:

   1. If the number EKB of effective key bits is in the range 1-255,
      then the version number is given by Table[EKB], where the 256-byte
      translation table Table[] is specified below. Table[] specifies a
      permutation on the numbers 0-255; note that it is not the same
      table that appears in the key expansion phase of RC2.

   2. If the number EKB of effective key bits is in the range
      256-1024, then the version number is simply EKB.

      The default number of effective key bits for RC2 is 32. If RC2-CBC
      is being performed with 32 effective key bits, the parameters
      should be supplied as a simple IV, rather than as a SEQUENCE
      containing a version and an IV.

        0  1  2  3  4  5  6  7  8  9  a  b  c  d  e  f

   00: bd 56 ea f2 a2 f1 ac 2a b0 93 d1 9c 1b 33 fd d0
   10: 30 04 b6 dc 7d df 32 4b f7 cb 45 9b 31 bb 21 5a
   20: 41 9f e1 d9 4a 4d 9e da a0 68 2c c3 27 5f 80 36
   30: 3e ee fb 95 1a fe ce a8 34 a9 13 f0 a6 3f d8 0c
   40: 78 24 af 23 52 c1 67 17 f5 66 90 e7 e8 07 b8 60
   50: 48 e6 1e 53 f3 92 a4 72 8c 08 15 6e 86 00 84 fa
   60: f4 7f 8a 42 19 f6 db cd 14 8d 50 12 ba 3c 06 4e
   70: ec b3 35 11 a1 88 8e 2b 94 99 b7 71 74 d3 e4 bf
   80: 3a de 96 0e bc 0a ed 77 fc 37 6b 03 79 89 62 c6
   90: d7 c0 d2 7c 6a 8b 22 a3 5b 05 5d 02 75 d5 61 e3
   a0: 18 8f 55 51 ad 1f 0b 5e 85 e5 c2 57 63 ca 3d 6c
   b0: b4 c5 cc 70 b2 91 59 0d 47 20 c8 4f 58 e0 01 e2
   c0: 16 38 c4 6f 3b 0f 65 46 be 7e 2d 7b 82 f9 40 b5
   d0: 1d 73 f8 eb 26 c7 87 97 25 54 b1 28 aa 98 9d a5
   e0: 64 6d 7a d4 10 81 44 ef 49 d6 ae 2e dd 76 5c 2f
   f0: a7 1c c9 09 69 9a 83 cf 29 39 b9 e9 4c ff 43 ab

A. Intellectual Property Notice

   RC2 is a registered trademark of RSA Data Security, Inc. RSA's
   copyrighted RC2 software is available under license from RSA Data
   Security, Inc.

B. Author's Address

   Ron Rivest
   RSA Laboratories
   100 Marine Parkway, #500
   Redwood City, CA  94065  USA

   Phone: (650) 595-7703
   EMail: rsa-labs@rsa.com

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