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Versions: 00 rfc2268                                       Informational
Internet Draft                      Ron Rivest
draft-rivest-rc2desc-00.txt         RSA Data Security, Inc.
Created 1987
Revised March 12, 1992
Revised June 23, 1997               Expires December 22, 1997

           A Description of the RC2(r) Encryption Algorithm

Status of this memo

This document is an Internet-Draft. Internet-Drafts are working
documents of the Internet Engineering Task Force (IETF), its areas,
and its working groups. Note that other groups may also distribute
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1. Introduction

This draft 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. (This will not be
   described, since it is merely the inverse operation.)

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 a
"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

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

     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.)

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
(415) 595-8782