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Network Working Group R. Baldwin Request for Comments: 2040 RSA Data Security, Inc. Category: Informational R. Rivest MIT Laboratory for Computer Science and RSA Data Security, Inc. October 1996 The RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS Algorithms Status of this Memo This memo provides information for the Internet community. This memo does not specify an Internet standard of any kind. Distribution of this memo is unlimited. Acknowledgments We would like to thank Steve Dusse, Victor Chang, Tim Mathews, Brett Howard, and Burt Kaliski for helpful suggestions. Table of Contents 1. Executive Summary ....................... 1 2. Overview ................................ 2 3. Terminology and Notation ................ 3 4. Description of RC5 Keys ................. 4 5. Description of RC5 Key Expansion ........ 6 6. Description of RC5 Block Cipher ......... 10 7. Description of RC5-CBC and RC5-CBC-Pad .. 12 8. Description of RC5-CTS .................. 18 9. Test Program and Vectors ................ 19 10. Security Considerations ................. 26 11. ASN.1 Identifiers ....................... 28 References ........................................ 28 Authors' Addresses ................................ 29 1. Executive Summary This document defines four ciphers with enough detail to ensure interoperability between different implementations. The first cipher is the raw RC5 block cipher. The RC5 cipher takes a fixed size input block and produces a fixed sized output block using a transformation that depends on a key. The second cipher, RC5-CBC, is the Cipher Block Chaining (CBC) mode for RC5. It can process messages whose length is a multiple of the RC5 block size. The third cipher, RC5- CBC-Pad, handles plaintext of any length, though the ciphertext will be longer than the plaintext by at most the size of a single RC5 Baldwin & Rivest Informational [Page 1]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 block. The RC5-CTS cipher is the Cipher Text Stealing mode of RC5, which handles plaintext of any length and the ciphertext length matches the plaintext length. The RC5 cipher was invented by Professor Ronald L. Rivest of the Massachusetts Institute of Technology in 1994. It is a very fast and simple algorithm that is parameterized by the block size, the number of rounds, and key length. These parameters can be adjusted to meet different goals for security, performance, and exportability. RSA Data Security Incorporated has filed a patent application on the RC5 cipher and for trademark protection for RC5, RC5-CBC, RC5-CBC- Pad, RC5-CTS and assorted variations. 2. Overview This memo is a restatement of existing published material. The description of RC5 follows the notation and order of explanation found in the original RC5 paper by Professor Rivest [2]. The CBC mode appears in reference works such as the one by Bruce Schneier [6]. The CBC-Pad mode is the same as in the Public Key Cryptography Standard (PKCS) number five [5]. Sample C code [8] is included for clarity only and is equivalent to the English language descriptions. The ciphers will be explained in a bottom up object-oriented fashion. First, RC5 keys will be presented along with the key expansion algorithm. Second, the RC5 block cipher is explained, and finally, the RC5-CBC and RC5-CBC-Pad ciphers are specified. For brevity, only the encryption process is described. Decryption is achieved by inverting the steps of encryption. The object-oriented description found here should make it easier to implement interoperable systems, though it is not as terse as the functional descriptions found in the references. There are two classes of objects, keys and cipher algorithms. Both classes share operations that create and destroy these objects in a manner that ensures that secret information is not returned to the memory manager. Keys also have a "set" operation that copies a secret key into the object. The "set" operation for the cipher objects defines the number of rounds, and the initialization vector. There are four operations for the cipher objects described in this memo. There is binding a key to a cipher object, setting a new initialization vector for a cipher object without changing the key, encrypting part of a message (this would be performed multiple times for long messages), and processing the last part of a message which Baldwin & Rivest Informational [Page 2]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 may add padding or check the length of the message. In summary, the cipher will be explained in terms of these operations: RC5_Key_Create - Create a key object. RC5_Key_Destroy - Destroy a key object. RC5_Key_Set - Bind a user key to a key object. RC5_CBC_Create - Create a cipher object. RC5_CBC_Destroy - Destroy a cipher object. RC5_CBC_Encrypt_Init - Bind a key object to a cipher object. RC5_CBC_SetIV - Set a new IV without changing the key. RC5_CBC_Encrypt_Update - Process part of a message. RC5_CBC_Encrypt_Final - Process the end of a message. 3. Terminology and Notation The term "word" refers to a string of bits of a particular length that can be operated on as either an unsigned integer or as a bit vector. For example a "word" might be 32 or 64 bits long depending on the desired block size for the RC5 cipher. A 32 bit word will produce a 64 bit block size. For best performance the RC5 word size should match the register size of the CPU. The term "byte" refers to eight bits. The following variables will be used throughout this memo with these meanings: W This is the word size for RC5 measured in bits. It is half the block size. The word sizes covered by this memo are 32 and 64. WW This is the word size for RC5 measured in bytes. B This is the block size for RC5 measured in bits. It is twice the word size. When RC5 is used as a 64 bit block cipher, B is 64 and W is 32. 0 < B < 257. In the sample code, B, is used as a variable instead of a cipher system parameter, but this usage should be obvious from context. BB This is the block size for RC5 measured in bytes. BB = B / 8. Baldwin & Rivest Informational [Page 3]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 b This is the byte length of the secret key. 0 <= b < 256. K This is the secret key which is treated as a sequence of b bytes indexed by: K[0], ..., K[b-1]. R This is the number of rounds of the inner RC5 transform. 0 <= R < 256. T This is the number of words in the expanded key table. It is always 2*(R + 1). 1 < T < 513. S This is the expanded key table which is treated as a sequence of words indexed by: S[0], ..., S[T-1]. N This is the byte length of the plaintext message. P This is the plaintext message which is treated as a sequence of N bytes indexed by: P[0], ..., P[N-1]. C This is the ciphertext output which is treated as a sequence of bytes indexed by: C[0], C[1], ... I This is the initialization vector for the CBC mode which is treated as a sequence of bytes indexed by: I[0], ..., I[BB-1]. 4. Description of RC5 Keys Like most block ciphers, RC5 expands a small user key into a table of internal keys. The byte length of the user key is one of the parameters of the cipher, so the RC5 user key object must be able to hold variable length keys. A possible structure for this in C is: /* Definition of RC5 user key object. */ typedef struct rc5UserKey { int keyLength; /* In Bytes. */ unsigned char *keyBytes; } rc5UserKey; The basic operations on a key are to create, destroy and set. To avoid exposing key material to other parts of an application, the destroy operation zeros the memory allocated for the key before releasing it to the memory manager. A general key object may support other operations such as generating a new random key and deriving a key from key-agreement information. Baldwin & Rivest Informational [Page 4]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 4.1 Creating an RC5 Key To create a key, the memory for the key object must be allocated and initialized. The C code below assumes that a function called "malloc" will return a block of uninitialized memory from the heap, or zero indicating an error. /* Allocate and initialize an RC5 user key. * Return 0 if problems. */ rc5UserKey *RC5_Key_Create () { rc5UserKey *pKey; pKey = (rc5UserKey *) malloc (sizeof(*pKey)); if (pKey != ((rc5UserKey *) 0)) { pKey->keyLength = 0; pKey->keyBytes = (unsigned char *) 0; } return (pKey); } 4.2 Destroying an RC5 Key To destroy a key, the memory must be zeroed and released to the memory manager. The C code below assumes that a function called "free" will return a block of memory to the heap. /* Zero and free an RC5 user key. */ void RC5_Key_Destroy (pKey) rc5UserKey *pKey; { unsigned char *to; int count; if (pKey == ((rc5UserKey *) 0)) return; if (pKey->keyBytes == ((unsigned char *) 0)) return; to = pKey->keyBytes; for (count = 0 ; count < pKey->keyLength ; count++) *to++ = (unsigned char) 0; free (pKey->keyBytes); pKey->keyBytes = (unsigned char *) 0; pKey->keyLength = 0; free (pKey); Baldwin & Rivest Informational [Page 5]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 } 4.3 Setting an RC5 Key Setting the key object makes a copy of the secret key into a block of memory allocated from the heap. /* Set the value of an RC5 user key. * Copy the key bytes so the caller can zero and * free the original. * Return zero if problems */ int RC5_Key_Set (pKey, keyLength, keyBytes) rc5UserKey *pKey; int keyLength; unsigned char *keyBytes; { unsigned char *keyBytesCopy; unsigned char *from, *to; int count; keyBytesCopy = (unsigned char *) malloc (keyLength); if (keyBytesCopy == ((unsigned char *) 0)) return (0); from = keyBytes; to = keyBytesCopy; for (count = 0 ; count < keyLength ; count++) *to++ = *from++; pKey->keyLength = count; pKey->keyBytes = keyBytesCopy; return (1); } 5. Description of RC5 Key Expansion This section describes the key expansion algorithm. To be specific, the sample code assumes that the block size is 64 bits. Several programming parameters depend on the block size. /* Definitions for RC5 as a 64 bit block cipher. */ /* The "unsigned int" will be 32 bits on all but */ /* the oldest compilers, which will make it 16 bits. */ /* On a DEC Alpha "unsigned long" is 64 bits, not 32. */ #define RC5_WORD unsigned int #define W (32) #define WW (W / 8) #define ROT_MASK (W - 1) #define BB ((2 * W) / 8) /* Bytes per block */ Baldwin & Rivest Informational [Page 6]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 /* Define macros used in multiple procedures. */ /* These macros assumes ">>" is an unsigned operation, */ /* and that x and s are of type RC5_WORD. */ #define SHL(x,s) ((RC5_WORD)((x)<<((s)&ROT_MASK))) #define SHR(x,s,w) ((RC5_WORD)((x)>>((w)-((s)&ROT_MASK)))) #define ROTL(x,s,w) ((RC5_WORD)(SHL((x),(s))|SHR((x),(s),(w)))) 5.1 Definition of initialization constants Two constants, Pw and Qw, are defined for any word size W by the expressions: Pw = Odd((e-2)*2**W) Qw = Odd((phi-1)*2**W) where e is the base of the natural logarithm (2.71828 ...), and phi is the golden ratio (1.61803 ...), and 2**W is 2 raised to the power of W, and Odd(x) is equal to x if x is odd, or equal to x plus one if x is even. For W equal to 16, 32, and 64, the Pw and Qw constants are the following hexadecimal values: #define P16 0xb7e1 #define Q16 0x9e37 #define P32 0xb7e15163 #define Q32 0x9e3779b9 #define P64 0xb7e151628aed2a6b #define Q64 0x9e3779b97f4a7c15 #if W == 16 #define Pw P16 /* Select 16 bit word size */ #define Qw Q16 #endif #if W == 32 #define Pw P32 /* Select 32 bit word size */ #define Qw Q32 #endif #if W == 64 #define Pw P64 /* Select 64 bit word size */ #define Qw Q64 #endif Baldwin & Rivest Informational [Page 7]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 5.2 Interface definition The key expansion routine converts the b-byte secret key, K, into an expanded key, S, which is a sequence of T = 2*(R+1) words. The expansion algorithm uses two constants that are derived from the constants, e, and phi. These are used to initialize S, which is then modified using K. A C code procedure header for this routine could be: /* Expand an RC5 user key. */ void RC5_Key_Expand (b, K, R, S) int b; /* Byte length of secret key */ char *K; /* Secret key */ int R; /* Number of rounds */ RC5_WORD *S; /* Expanded key buffer, 2*(R+1) words */ { 5.3 Convert secret key from bytes to words This step converts the b-byte key into a sequence of words stored in the array L. On a little-endian processor this is accomplished by zeroing the L array and copying in the b bytes of K. The following C code will achieve this effect on all processors: int i, j, k, LL, t, T; RC5_WORD L[256/WW]; /* Based on max key size */ RC5_WORD A, B; /* LL is number of elements used in L. */ LL = (b + WW - 1) / WW; for (i = 0 ; i < LL ; i++) { L[i] = 0; } for (i = 0 ; i < b ; i++) { t = (K[i] & 0xFF) << (8*(i%4)); /* 0, 8, 16, 24*/ L[i/WW] = L[i/WW] + t; } 5.4 Initialize the expanded key table This step fills in the S table with a fixed (key independent) pseudo-random pattern using an arithmetic progression based on Pw and Qw modulo 2**W. The element S[i] equals i*Qw + Pw modulo 2**W. This table could be precomputed and copied as needed or computed on the fly. In C code it can be computed by: Baldwin & Rivest Informational [Page 8]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 T = 2*(R+1); S[0] = Pw; for (i = 1 ; i < T ; i++) { S[i] = S[i-1] + Qw; } 5.5 Mix in the secret key This step mixes the secret key, K, into the expanded key, S. First the number of iterations of the mixing function, k, is set to three times the maximum of the number of initialized elements of L, called LL, and the number of elements in S, called T. Each iteration is similar to an interation of the encryption inner loop in that two variables A and B are updated by the first and second halves of the iteration. Initially A and B are zero as are the indexes into the S array, i, and the L array, j. In the first half of the iteration, a partial result is computed by summing S[i], A and B. The new value for A is this partial result rotated left three bits. The A value is then placed into S[i]. The second half of the iteration computes a second partial result that is the sum of L[j], A and B. The second partial result is then rotated left by A+B bit positions and set to be the new value for B. The new B value is then placed into L[j]. At the end of the iteration, i and j are incremented modulo the size of their respective arrays. In C code: i = j = 0; A = B = 0; if (LL > T) k = 3 * LL; /* Secret key len > expanded key. */ else k = 3 * T; /* Secret key len < expanded key. */ for ( ; k > 0 ; k--) { A = ROTL(S[i] + A + B, 3, W); S[i] = A; B = ROTL(L[j] + A + B, A + B, W); L[j] = B; i = (i + 1) % T; j = (j + 1) % LL; } return; } /* End of RC5_Key_Expand */ Baldwin & Rivest Informational [Page 9]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 6. Description of RC5 Block Cipher This section describes the RC5 block cipher by explaining the steps required to perform an encryption of a single input block. The decryption process is the reverse of these steps so it will not be explained. The RC5 cipher is parameterized by a version number, V, a round count, R, and a word size in bits, W. This description corresponds to original version of RC5 (V = 16 decimal) and covers any positive value for R and the values 16, 32, and 64 for W. The inputs to this process are the expanded key table, S, the number of rounds, R, the input buffer pointer, in, and the output buffer pointer, out. A possible C code procedure header for this would be: void RC5_Block_Encrypt (S, R, in, out) RC5_WORD *S; int R; char *in; char *out; { 6.1 Loading A and B values This step converts input bytes into two unsigned integers called A and B. When RC5 is used as a 64 bit block cipher A and B are 32 bit values. The first input byte becomes the least significant byte of A, the fourth input byte becomes the most significant byte of A, the fifth input byte becomes the least significant byte of B and the last input byte becomes the most significant byte of B. This conversion can be very efficient for little-endian processors such as the Intel family. In C code this could be expressed as: int i; RC5_WORD A, B; A = in[0] & 0xFF; A += (in[1] & 0xFF) << 8; A += (in[2] & 0xFF) << 16; A += (in[3] & 0xFF) << 24; B = in[4] & 0xFF; B += (in[5] & 0xFF) << 8; B += (in[6] & 0xFF) << 16; B += (in[7] & 0xFF) << 24; Baldwin & Rivest Informational [Page 10]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 6.2 Iterating the round function This step mixes the expanded key with the input to perform the fundamental encryption operation. The first two words of the expanded key are added to A and B respectively, and then the round function is repeated R times. The first half of the round function computes a new value for A based on the values of A, B, and the next unused word in the expanded key table. Specifically, A is XOR'ed with B and then this first partial result is rotated to the left by an amount specified by B to form the second partial result. The rotation is performed on a W bit boundary (i.e., 32 bit rotation for the version of RC5 that has a 64 bit block size). The actual rotation amount only depends on the least significant log base-2 of W bits of B. The next unused word of the expanded key table is then added to the second partial result and this becomes the new value for A. The second half of the round function is identical except the roles of A and B are switched. Specifically, B is exclusive or'ed with A and then this first partial result is rotated to the left by an amount specified by A to form the second partial result. The next unused word of the expanded key table is then added to the second partial result and this becomes the new value for B. One way to express this in C code is: A = A + S[0]; B = B + S[1]; for (i = 1 ; i <= R ; i++) { A = A ^ B; A = ROTL(A, B, W) + S[2*i]; B = B ^ A; B = ROTL(B, A, W) + S[(2*i)+1]; } 6.3 Storing the A and B values The final step is to convert A and B back into a sequence of bytes. This is the inverse of the load operation. An expression of this in C code could be: out[0] = (A >> 0) & 0xFF; out[1] = (A >> 8) & 0xFF; out[2] = (A >> 16) & 0xFF; out[3] = (A >> 24) & 0xFF; out[4] = (B >> 0) & 0xFF; out[5] = (B >> 8) & 0xFF; Baldwin & Rivest Informational [Page 11]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 out[6] = (B >> 16) & 0xFF; out[7] = (B >> 24) & 0xFF; return; } /* End of RC5_Block_Encrypt */ 7. Description of RC5-CBC and RC5-CBC-Pad This section describes the CBC and CBC-Pad modes of the RC5 cipher. This description is based on the RC5 key objects and RC5 block cipher described earlier. 7.1 Creating cipher objects The cipher object needs to keep track of the padding mode, the number of rounds, the expanded key, the initialization vector, the CBC chaining block, and an input buffer. A possible structure definition for this in C code would be: /* Definition of the RC5 CBC algorithm object. */ typedef struct rc5CBCAlg { int Pad; /* 1 = RC5-CBC-Pad, 0 = RC5-CBC. */ int R; /* Number of rounds. */ RC5_WORD *S; /* Expanded key. */ unsigned char I[BB]; /* Initialization vector. */ unsigned char chainBlock[BB]; unsigned char inputBlock[BB]; int inputBlockIndex; /* Next inputBlock byte. */ } rc5CBCAlg; To create a cipher algorithm object, the parameters must be checked and then space allocated for the expanded key table. The expanded key is initialized using the method described earlier. Finally, the state variables (padding mode, number of rounds, and the input buffer) are set to their initial values. In C this could be accomplished by: /* Allocate and initialize the RC5 CBC algorithm object. * Return 0 if problems. */ rc5CBCAlg *RC5_CBC_Create (Pad, R, Version, bb, I) int Pad; /* 1 = RC5-CBC-Pad, 0 = RC5-CBC. */ int R; /* Number of rounds. */ int Version; /* RC5 version number. */ int bb; /* Bytes per RC5 block == IV len. */ char *I; /* CBC IV, bb bytes long. */ { Baldwin & Rivest Informational [Page 12]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 rc5CBCAlg *pAlg; int index; if ((Version != RC5_FIRST_VERSION) || (bb != BB) || (R < 0) || (255 < R)) return ((rc5CBCAlg *) 0); pAlg = (rc5CBCAlg *) malloc (sizeof(*pAlg)); if (pAlg == ((rc5CBCAlg *) 0)) return ((rc5CBCAlg *) 0); pAlg->S = (RC5_WORD *) malloc (BB * (R + 1)); if (pAlg->S == ((RC5_WORD *) 0)) { free (pAlg); return ((rc5CBCAlg *) 0); } pAlg->Pad = Pad; pAlg->R = R; pAlg->inputBlockIndex = 0; for (index = 0 ; index < BB ; index++) pAlg->I[index] = I[index]; return (pAlg); } 7.2 Destroying cipher objects Destroying the cipher object is the inverse of creating it with care being take to zero memory before returning it to the memory manager. In C this could be accomplished by: /* Zero and free an RC5 algorithm object. */ void RC5_CBC_Destroy (pAlg) rc5CBCAlg *pAlg; { RC5_WORD *to; int count; if (pAlg == ((rc5CBCAlg *) 0)) return; if (pAlg->S == ((RC5_WORD *) 0)) return; to = pAlg->S; for (count = 0 ; count < (1 + pAlg->R) ; count++) { *to++ = 0; /* Two expanded key words per round. */ *to++ = 0; } free (pAlg->S); for (count = 0 ; count < BB ; count++) Baldwin & Rivest Informational [Page 13]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 { pAlg->I[count] = (unsigned char) 0; pAlg->inputBlock[count] = (unsigned char) 0; pAlg->chainBlock[count] = (unsigned char) 0; } pAlg->Pad = 0; pAlg->R = 0; pAlg->inputBlockIndex = 0; free (pAlg); } 7.3 Setting the IV for cipher objects For CBC cipher objects, the state of the algorithm depends on the expanded key, the CBC chain block, and any internally buffered input. Often the same key is used with many messages that each have a unique initialization vector. To avoid the overhead of creating a new cipher object, it makes more sense to provide an operation that allows the caller to change the initialization vector for an existing cipher object. In C this could be accomplished by the following code: /* Setup a new initialization vector for a CBC operation * and reset the CBC object. * This can be called after Final without needing to * call Init or Create again. * Return zero if problems. */ int RC5_CBC_SetIV (pAlg, I) rc5CBCAlg *pAlg; char *I; /* CBC Initialization vector, BB bytes. */ { int index; pAlg->inputBlockIndex = 0; for (index = 0 ; index < BB ; index++) { pAlg->I[index] = pAlg->chainBlock[index] = I[index]; pAlg->inputBlock[index] = (unsigned char) 0; } return (1); } 7.4 Binding a key to a cipher object The operation that binds a key to a cipher object performs the key expansion. Key expansion could be an operation on keys, but that would not work correctly for ciphers that modify the expanded key as Baldwin & Rivest Informational [Page 14]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 they operate. After expanding the key, this operation must initialize the CBC chain block from the initialization vector and prepare the input buffer to receive the first character. In C this could be done by: /* Initialize the encryption object with the given key. * After this routine, the caller frees the key object. * The IV for this CBC object can be changed by calling * the SetIV routine. The only way to change the key is * to destroy the CBC object and create a new one. * Return zero if problems. */ int RC5_CBC_Encrypt_Init (pAlg, pKey) rc5CBCAlg *pAlg; rc5UserKey *pKey; { if ((pAlg == ((rc5CBCAlg *) 0)) || (pKey == ((rc5UserKey *) 0))) return (0); RC5_Key_Expand (Key->keyLength, pKey->keyBytes, pAlg->R, pAlg->S); return (RC5_CBC_SetIV(pAlg, pAlg->I)); } 7.5 Processing part of a message The encryption process described here uses the Init-Update-Final paradigm. The update operation can be performed on a sequence of message parts in order to incrementally produce the ciphertext. After the last part is processed, the Final operation is called to pick up any plaintext bytes or padding that are buffered inside the cipher object. An appropriate procedure header for this operation would be: /* Encrypt a buffer of plaintext. * The plaintext and ciphertext buffers can be the same. * The byte len of the ciphertext is put in *pCipherLen. * Call this multiple times passing successive * parts of a large message. * After the last part has been passed to Update, * call Final. * Return zero if problems like output buffer too small. */ int RC5_CBC_Encrypt_Update (pAlg, N, P, pCipherLen, maxCipherLen, C) rc5CBCAlg *pAlg; /* Cipher algorithm object. */ int N; /* Byte length of P. */ char *P; /* Plaintext buffer. */ Baldwin & Rivest Informational [Page 15]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 int *pCipherLen;/* Gets byte len of C. */ int maxCipherLen; /* Size of C. */ char *C; /* Ciphertext buffer. */ { 7.5.1 Output buffer size check. The first step of plaintext processing is to make sure that the output buffer is big enough hold the ciphertext. The ciphertext will be produced in multiples of the block size and depends on the number of plaintext characters passed to this operation plus any characters that are in the cipher object's internal buffer. In C code this would be: int plainIndex, cipherIndex, j; /* Check size of the output buffer. */ if (maxCipherLen < (((pAlg->inputBlockIndex+N)/BB)*BB)) { *pCipherLen = 0; return (0); } 7.5.2 Divide plaintext into blocks The next step is to add characters to the internal buffer until a full block has been constructed. When that happens, the buffer pointers are reset and the input buffer is exclusive-or'ed (XORed) with the CBC chaining block. The byte order of the chaining block is the same as the input block. For example, the ninth input byte is XOR'ed with the first ciphertext byte. The result is then passed to the RC5 block cipher which was described earlier. To reduce data movement and byte alignment problems, the output of RC5 can be directly written into the CBC chaining block. Finally, this output is copied to the ciphertext buffer provided by the user. Before returning, the actual size of the ciphertext is passed back to the caller. In C, this step can be performed by: plainIndex = cipherIndex = 0; while (plainIndex < N) { if (pAlg->inputBlockIndex < BB) { pAlg->inputBlock[pAlg->inputBlockIndex] = P[plainIndex]; pAlg->inputBlockIndex++; plainIndex++; } Baldwin & Rivest Informational [Page 16]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 if (pAlg->inputBlockIndex == BB) { /* Have a complete input block, process it. */ pAlg->inputBlockIndex = 0; for (j = 0 ; j < BB ; j++) { /* XOR in the chain block. */ pAlg->inputBlock[j] = pAlg->inputBlock[j] ^ pAlg->chainBlock[j]; } RC5_Block_Encrypt(pAlg->S, pAlg->R pAlg->inputBlock, pAlg->chainBlock); for (j = 0 ; j < BB ; j++) { /* Output the ciphertext. */ C[cipherIndex] = pAlg->chainBlock[j]; cipherIndex++; } } } *pCipherLen = cipherIndex; return (1); } /* End of RC5_CBC_Encrypt_Update */ 7.6 Final block processing This step handles the last block of plaintext. For RC5-CBC, this step just performs error checking to ensure that the plaintext length was indeed a multiple of the block length. For RC5-CBC-Pad, padding bytes are added to the plaintext. The pad bytes are all the same and are set to a byte that represents the number of bytes of padding. For example if there are eight bytes of padding, the bytes will all have the hexadecimal value 0x08. There will be between one and BB padding bytes, inclusive. In C code this would be: /* Produce the final block of ciphertext including any * padding, and then reset the algorithm object. * Return zero if problems. */ int RC5_CBC_Encrypt_Final (pAlg, pCipherLen, maxCipherLen, C) rc5CBCAlg *pAlg; int *pCipherLen; /* Gets byte len of C. */ int maxCipherLen; /* Len of C buffer. */ char *C; /* Ciphertext buffer. */ { int cipherIndex, j; int padLength; /* For non-pad mode error if input bytes buffered. */ *pCipherLen = 0; Baldwin & Rivest Informational [Page 17]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 if ((pAlg->Pad == 0) && (pAlg->inputBlockIndex != 0)) return (0); if (pAlg->Pad == 0) return (1); if (maxCipherLen < BB) return (0); padLength = BB - pAlg->inputBlockIndex; for (j = 0 ; j < padLength ; j++) { pAlg->inputBlock[pAlg->inputBlockIndex] = (unsigned char) padLength; pAlg->inputBlockIndex++; } for (j = 0 ; j < BB ; j++) { /* XOR the chain block into the plaintext block. */ pAlg->inputBlock[j] = pAlg->inputBlock[j] ^ pAlg->chainBlock[j]; } RC5_Block_Encrypt(pAlg->S, pAlg->R, pAlg->inputBlock, pAlg->chainBlock); cipherIndex = 0; for (j = 0 ; j < BB ; j++) { /* Output the ciphertext. */ C[cipherIndex] = pAlg->chainBlock[j]; cipherIndex++; } *pCipherLen = cipherIndex; /* Reset the CBC algorithm object. */ return (RC5_CBC_SetIV(pAlg, pAlg->I)); } /* End of RC5_CBC_Encrypt_Final */ 8. Description of RC5-CTS The Cipher Text Stealing (CTS) mode for block ciphers is described by Schneier on pages 195 and 196 of [6]. This mode handles any length of plaintext and produces ciphertext whose length matches the plaintext length. The CTS mode behaves like the CBC mode for all but the last two blocks of the plaintext. The following steps describe how to handle the last two portions of the plaintext, called Pn-1 and Pn, where the length of Pn-1 equals the block size, BB, and the length of the last block, Pn, is Ln bytes. Notice that Ln ranges from 1 to BB, inclusive, so Pn could in fact be a complete block. Baldwin & Rivest Informational [Page 18]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 1. Exclusive-or Pn-1 with the previous ciphertext block, Cn-2, to create Xn-1. 2. Encrypt Xn-1 to create En-1. 3. Select the first Ln bytes of En-1 to create Cn. 4. Pad Pn with zeros at the end to create P of length BB. 5. Exclusive-or En-1 with P to create to create Dn. 6. Encrypt Dn to create Cn-1 7. The last two parts of the ciphertext are Cn-1 and Cn respectively. To implement CTS encryption, the RC5-CTS object must hold on to (buffer) at most 2*BB bytes of plaintext and process them specially when the RC5_CTS_Encrypt_Final routine is called. The following steps describe how to decrypt Cn-1 and Cn. 1. Decrypt Cn-1 to create Dn. 2. Pad Cn with zeros at the end to create C of length BB. 3. Exclusive-or Dn with C to create Xn. 4. Select the first Ln bytes of Xn to create Pn. 5. Append the tail (BB minus Ln) bytes of Xn to Cn to create En. 6. Decrypt En to create Pn-1. 7. The last two parts of the plaintext are Pn-1 and Pn respectively. 9. Test Program and Vectors To help confirm the correctness of an implementation, this section gives a test program and results from a set of test vectors. 9.1 Test Program The following test program written in C reads test vectors from its input stream and writes results on its output stream. The following subsections give a set of test vectors for inputs and the resulting Baldwin & Rivest Informational [Page 19]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 outputs. #include <stdio.h> #define BLOCK_LENGTH (8 /* bytes */) #define MAX_KEY_LENGTH (64 /* bytes */) #define MAX_PLAIN_LENGTH (128 /* bytes */) #define MAX_CIPHER_LENGTH(MAX_PLAIN_LENGTH + BLOCK_LENGTH) #define MAX_ROUNDS (20) #define MAX_S_LENGTH (2 * (MAX_ROUNDS + 1)) typedef struct test_vector { int padding_mode; int rounds; char keytext[2*MAX_KEY_LENGTH+1]; int key_length; char key[MAX_KEY_LENGTH]; char ivtext[2*BLOCK_LENGTH+1]; int iv_length; char iv[BLOCK_LENGTH]; char plaintext[2*MAX_PLAIN_LENGTH+1]; int plain_length; char plain[MAX_PLAIN_LENGTH]; char ciphertext[2*MAX_CIPHER_LENGTH+1]; int cipher_length; char cipher[MAX_CIPHER_LENGTH]; RC5_WORD S[MAX_S_LENGTH]; } test_vector; void show_banner() { (void) printf("RC5 CBC Tester.\n"); (void) printf("Each input line should contain the following\n"); (void) printf("test parameters separated by a single space:\n"); (void) printf("- Padding mode flag. Use 1 for RC5_CBC_Pad, else 0.\n"); (void) printf("- Number of rounds for RC5.\n"); (void) printf("- Key bytes in hexadecimal. Two characters per byte like '01'.\n"); (void) printf("- IV bytes in hexadecimal. Must be 16 hex characters.\n"); (void) printf("- Plaintext bytes in hexadecimal.\n"); (void) printf("An end of file or format error terminates the tester.\n"); (void) printf("\n"); } Baldwin & Rivest Informational [Page 20]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 /* Convert a buffer from ascii hex to bytes. * Set pTo_length to the byte length of the result. * Return 1 if everything went OK. */ int hex_to_bytes (from, to, pTo_length) char *from, *to; int *pTo_length; { char *pHex; /* Ptr to next hex character. */ char *pByte; /* Ptr to next resulting byte. */ int byte_length = 0; int value; pByte = to; for (pHex = from ; *pHex != 0 ; pHex += 2) { if (1 != sscanf(pHex, "%02x", &value)) return (0); *pByte++ = ((char)(value & 0xFF)); byte_length++; } *pTo_length = byte_length; return (1); } /* Convert a buffer from bytes to ascii hex. * Return 1 if everything went OK. */ int bytes_to_hex (from, from_length, to) char *from, *to; int from_length; { char *pHex; /* Ptr to next hex character. */ char *pByte; /* Ptr to next resulting byte. */ int value; pHex = to; for (pByte = from ; from_length > 0 ; from_length--) { value = *pByte++ & 0xFF; (void) sprintf(pHex, "%02x", value); pHex += 2; } return (1); } /* Return 1 if get a valid test vector. */ int get_test_vector(ptv) test_vector *ptv; { Baldwin & Rivest Informational [Page 21]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 if (1 != scanf("%d", &ptv->padding_mode)) return (0); if (1 != scanf("%d", &ptv->rounds)) return (0); if ((ptv->rounds < 0) || (MAX_ROUNDS < ptv->rounds)) return (0); if (1 != scanf("%s", &ptv->keytext)) return (0); if (1 != hex_to_bytes(ptv->keytext, ptv->key, &ptv->key_length)) return (0); if (1 != scanf("%s", &ptv->ivtext)) return (0); if (1 != hex_to_bytes(ptv->ivtext, ptv->iv, &ptv->iv_length)) return (0); if (BLOCK_LENGTH != ptv->iv_length) return (0); if (1 != scanf("%s", &ptv->plaintext)) return (0); if (1 != hex_to_bytes(ptv->plaintext, ptv->plain, &ptv->plain_length)) return (0); return (1); } void run_test (ptv) test_vector *ptv; { rc5UserKey *pKey; rc5CBCAlg *pAlg; int numBytesOut; pKey = RC5_Key_Create (); RC5_Key_Set (pKey, ptv->key_length, ptv->key); pAlg = RC5_CBC_Create (ptv->padding_mode, ptv->rounds, RC5_FIRST_VERSION, BB, ptv->iv); (void) RC5_CBC_Encrypt_Init (pAlg, pKey); ptv->cipher_length = 0; (void) RC5_CBC_Encrypt_Update (pAlg, ptv->plain_length, ptv->plain, &(numBytesOut), MAX_CIPHER_LENGTH - ptv->cipher_length, &(ptv->cipher[ptv->cipher_length])); Baldwin & Rivest Informational [Page 22]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 ptv->cipher_length += numBytesOut; (void) RC5_CBC_Encrypt_Final (pAlg, &(numBytesOut), MAX_CIPHER_LENGTH - ptv->cipher_length, &(ptv->cipher[ptv->cipher_length])); ptv->cipher_length += numBytesOut; bytes_to_hex (ptv->cipher, ptv->cipher_length, ptv->ciphertext); RC5_Key_Destroy (pKey); RC5_CBC_Destroy (pAlg); } void show_results (ptv) test_vector *ptv; { if (ptv->padding_mode) printf ("RC5_CBC_Pad "); else printf ("RC5_CBC "); printf ("R = %2d ", ptv->rounds); printf ("Key = %s ", ptv->keytext); printf ("IV = %s ", ptv->ivtext); printf ("P = %s ", ptv->plaintext); printf ("C = %s", ptv->ciphertext); printf ("\n"); } int main(argc, argv) int argc; char *argv[]; { test_vector tv; test_vector *ptv = &tv; show_banner(); while (get_test_vector(ptv)) { run_test(ptv); show_results(ptv); } return (0); } Baldwin & Rivest Informational [Page 23]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 9.2 Test vectors The following text is an input file to the test program presented in the previous subsection. The output is given in the next subsection. 0 00 00 0000000000000000 0000000000000000 0 00 00 0000000000000000 ffffffffffffffff 0 00 00 0000000000000001 0000000000000000 0 00 00 0000000000000000 0000000000000001 0 00 00 0102030405060708 1020304050607080 0 01 11 0000000000000000 0000000000000000 0 02 00 0000000000000000 0000000000000000 0 02 00000000 0000000000000000 0000000000000000 0 08 00 0000000000000000 0000000000000000 0 08 00 0102030405060708 1020304050607080 0 12 00 0102030405060708 1020304050607080 0 16 00 0102030405060708 1020304050607080 0 08 01020304 0000000000000000 ffffffffffffffff 0 12 01020304 0000000000000000 ffffffffffffffff 0 16 01020304 0000000000000000 ffffffffffffffff 0 12 0102030405060708 0000000000000000 ffffffffffffffff 0 08 0102030405060708 0102030405060708 1020304050607080 0 12 0102030405060708 0102030405060708 1020304050607080 0 16 0102030405060708 0102030405060708 1020304050607080 0 08 01020304050607081020304050607080 0102030405060708 1020304050607080 0 12 01020304050607081020304050607080 0102030405060708 1020304050607080 0 16 01020304050607081020304050607080 0102030405060708 1020304050607080 0 12 0102030405 0000000000000000 ffffffffffffffff 0 08 0102030405 0000000000000000 ffffffffffffffff 0 08 0102030405 7875dbf6738c6478 0808080808080808 1 08 0102030405 0000000000000000 ffffffffffffffff 0 08 0102030405 0000000000000000 0000000000000000 0 08 0102030405 7cb3f1df34f94811 1122334455667701 1 08 0102030405 0000000000000000 ffffffffffffffff7875dbf6738c647811223344556677 Baldwin & Rivest Informational [Page 24]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 9.3 Test results The following text is the output produced by the test program run on the inputs given in the previous subsection. RC5 CBC Tester. Each input line should contain the following test parameters separated by a single space: - Padding mode flag. Use 1 for RC5_CBC_Pad, else 0. - Number of rounds for RC5. - Key bytes in hexadecimal. Two characters per byte like '01'. - IV bytes in hexadecimal. Must be 16 hex characters. - Plaintext bytes in hexadecimal. An end of file or format error terminates the tester. RC5_CBC R = 0 Key = 00 IV = 0000000000000000 P = 0000000000000000 C = 7a7bba4d79111d1e RC5_CBC R = 0 Key = 00 IV = 0000000000000000 P = ffffffffffffffff C = 797bba4d78111d1e RC5_CBC R = 0 Key = 00 IV = 0000000000000001 P = 0000000000000000 C = 7a7bba4d79111d1f RC5_CBC R = 0 Key = 00 IV = 0000000000000000 P = 0000000000000001 C = 7a7bba4d79111d1f RC5_CBC R = 0 Key = 00 IV = 0102030405060708 P = 1020304050607080 C = 8b9ded91ce7794a6 RC5_CBC R = 1 Key = 11 IV = 0000000000000000 P = 0000000000000000 C = 2f759fe7ad86a378 RC5_CBC R = 2 Key = 00 IV = 0000000000000000 P = 0000000000000000 C = dca2694bf40e0788 RC5_CBC R = 2 Key = 00000000 IV = 0000000000000000 P = 0000000000000000 C = dca2694bf40e0788 RC5_CBC R = 8 Key = 00 IV = 0000000000000000 P = 0000000000000000 C = dcfe098577eca5ff RC5_CBC R = 8 Key = 00 IV = 0102030405060708 P = 1020304050607080 C = 9646fb77638f9ca8 RC5_CBC R = 12 Key = 00 IV = 0102030405060708 P = 1020304050607080 C = b2b3209db6594da4 RC5_CBC R = 16 Key = 00 IV = 0102030405060708 P = 1020304050607080 C = 545f7f32a5fc3836 RC5_CBC R = 8 Key = 01020304 IV = 0000000000000000 P = ffffffffffffffff C = 8285e7c1b5bc7402 RC5_CBC R = 12 Key = 01020304 IV = 0000000000000000 P = ffffffffffffffff C = fc586f92f7080934 RC5_CBC R = 16 Key = 01020304 IV = 0000000000000000 P = ffffffffffffffff C = cf270ef9717ff7c4 RC5_CBC R = 12 Key = 0102030405060708 IV = 0000000000000000 P = ffffffffffffffff C = e493f1c1bb4d6e8c Baldwin & Rivest Informational [Page 25]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 RC5_CBC R = 8 Key = 0102030405060708 IV = 0102030405060708 P = 1020304050607080 C = 5c4c041e0f217ac3 RC5_CBC R = 12 Key = 0102030405060708 IV = 0102030405060708 P = 1020304050607080 C = 921f12485373b4f7 RC5_CBC R = 16 Key = 0102030405060708 IV = 0102030405060708 P = 1020304050607080 C = 5ba0ca6bbe7f5fad RC5_CBC R = 8 Key = 01020304050607081020304050607080 IV = 0102030405060708 P = 1020304050607080 C = c533771cd0110e63 RC5_CBC R = 12 Key = 01020304050607081020304050607080 IV = 0102030405060708 P = 1020304050607080 C = 294ddb46b3278d60 RC5_CBC R = 16 Key = 01020304050607081020304050607080 IV = 0102030405060708 P = 1020304050607080 C = dad6bda9dfe8f7e8 RC5_CBC R = 12 Key = 0102030405 IV = 0000000000000000 P = ffffffffffffffff C = 97e0787837ed317f RC5_CBC R = 8 Key = 0102030405 IV = 0000000000000000 P = ffffffffffffffff C = 7875dbf6738c6478 RC5_CBC R = 8 Key = 0102030405 IV = 7875dbf6738c6478 P = 0808080808080808 C = 8f34c3c681c99695 RC5_CBC_Pad R = 8 Key = 0102030405 IV = 0000000000000000 P = ffffffffffffffff C = 7875dbf6738c64788f34c3c681c99695 RC5_CBC R = 8 Key = 0102030405 IV = 0000000000000000 P = 0000000000000000 C = 7cb3f1df34f94811 RC5_CBC R = 8 Key = 0102030405 IV = 7cb3f1df34f94811 P = 1122334455667701 C = 7fd1a023a5bba217 RC5_CBC_Pad R = 8 Key = 0102030405 IV = 0000000000000000 P = ffffffffffffffff7875dbf6738c647811223344556677 C = 7875dbf6738c64787cb3f1df34f948117fd1a023a5bba217 10. Security Considerations The RC5 cipher is relatively new so critical reviews are still being performed. However, the cipher's simple structure makes it easy to analyze and hopefully easier to assess its strength. Reviews so far are very promising. Early results [1] suggest that for RC5 with a 64 bit block size (32 bit word size), 12 rounds will suffice to resist linear and differential cyptanalysis. The 128 bit block version has not been studied as much as the 64 bit version, but it appears that 16 rounds would be an appropriate minimum. Block sizes less than 64 bits are academically interesting but should not be used for cryptographic security. Greater security can be achieved by increasing the number of rounds at the cost of decreasing the throughput of the cipher. Baldwin & Rivest Informational [Page 26]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 The length of the secret key helps determine the cipher's resistance to brute force key searching attacks. A key length of 128 bits should give adequate protection against brute force key searching by a well funded opponent for a couple decades [7]. For RC5 with 12 rounds, the key setup time and data encryption time are the same for all key lengths less than 832 bits, so there is no performance reason for choosing short keys. For larger keys, the key expansion step will run slower because the user key table, L, will be longer than the expanded key table, S. However, the encryption time will be unchanged since it is only a function of the number of rounds. To comply with export regulations it may be necessary to choose keys that only have 40 unknown bits. A poor way to do this would be to choose a simple 5 byte key. This should be avoided because it would be easy for an opponent to pre-compute key searching information. Another common mechanism is to pick a 128 bit key and publish the first 88 bits. This method reveals a large number of the entries in the user key table, L, and the question of whether RC5 key expansion provides adequate security in this situation has not been studied, though it may be fine. A conservative way to conform to a 40 bit limitation is to pick a seed value of 128 bits, publish 88 bits of this seed, run the entire seed through a hash function like MD5 [4], and use the 128 bit output of the hash function as the RC5 key. In the case of 40 unknown key bits with 88 known key bits (i.e., 88 salt bits) there should still be 12 or more rounds for the 64 bit block version of RC5, otherwise the value of adding salt bits to the key is likely to be lost. The lifetime of the key also influences security. For high security applications, the key to any 64 bit block cipher should be changed after encrypting 2**32 blocks (2**64 blocks for a 128 bit block cipher). This helps to guard against linear and differential cryptanalysis. For the case of 64 bit blocks, this rule would recommend changing the key after 2**40 (i.e. 10**12) bytes are encrypted. See Schneier [6] page 183 for further discussion. Baldwin & Rivest Informational [Page 27]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 11. ASN.1 Identifiers For applications that use ASN.1 descriptions, it is necessary to define the algorithm identifier for these ciphers along with their parameter block formats. The ASN.1 definition of an algorithm identifier already exists and is listed below for reference. AlgorithmIdentifier ::= SEQUENCE { algorithm OBJECT IDENTIFIER, parameters ANY DEFINED BY algorithm OPTIONAL } The values for the algorithm field are: RC5_CBC OBJECT IDENTIFIER ::= { iso (1) member-body (2) US (840) rsadsi (113549) encryptionAlgorithm (3) RC5CBC (8) } RC5_CBC_Pad OBJECT IDENTIFIER ::= { iso (1) member-body (2) US (840) rsadsi (113549) encryptionAlgorithm (3) RC5CBCPAD (9) } The structure of the parameters field for these algorithms is given below. NOTE: if the iv field is not included, then the initialization vector defaults to a block of zeros whose size depends on the blockSizeInBits field. RC5_CBC_Parameters ::= SEQUENCE { version INTEGER (v1_0(16)), rounds INTEGER (8..127), blockSizeInBits INTEGER (64, 128), iv OCTET STRING OPTIONAL } References [1] Kaliski, Burton S., and Yinqun Lisa Yin, "On Differential and Linear Cryptanalysis of the RC5 Encryption Algorithm", In Advances in Cryptology - Crypto '95, pages 171-184, Springer-Verlag, New York, 1995. [2] Rivest, Ronald L., "The RC5 Encryption Algorithm", In Proceedings of the Second International Workshop on Fast Software Encryption, pages 86-96, Leuven Belgium, December 1994. [3] Rivest, Ronald L., "RC5 Encryption Algorithm", In Dr. Dobbs Journal, number 226, pages 146-148, January 1995. Baldwin & Rivest Informational [Page 28]
RFC 2040 RC5, RC5-CBC, RC5-CBC-Pad, and RC5-CTS October 1996 [4] Rivest, Ronald L., "The MD5 Message-Digest Algorithm", RFC 1321. [5] RSA Laboratories, "Public Key Cryptography Standards (PKCS)", RSA Data Security Inc. See ftp.rsa.com. [6] Schneier, Bruce, "Applied Cryptography", Second Edition, John Wiley and Sons, New York, 1996. Errata: on page 195, line 13, the reference number should be [402]. [7] Business Software Alliance, Matt Blaze et al., "Minimum Key Length for Symmetric Ciphers to Provide Adequate Commercial Security", http://www.bsa.org/bsa/cryptologists.html. [8] RSA Data Security Inc., "RC5 Reference Code in C", See the web site: www.rsa.com, for availability. Not available with the first draft of this document. Authors' Addresses Robert W. Baldwin RSA Data Security, Inc. 100 Marine Parkway Redwood City, CA 94065 Phone: (415) 595-8782 Fax: (415) 595-1873 EMail: baldwin@rsa.com, or baldwin@lcs.mit.edu Ronald L. Rivest Massachusetts Institute of Technology Laboratory for Computer Science NE43-324 545 Technology Square Cambridge, MA 02139-1986 Phone: (617) 253-5880 EMail: rivest@theory.lcs.mit.edu Baldwin & Rivest Informational [Page 29]