Network Working Group D. Eastlake 3rd Request for Comments: 4634 Motorola Labs Updates: 3174 T. Hansen Category: Informational AT&T Labs July 2006 US Secure Hash Algorithms (SHA and HMAC-SHA) 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 (2006).## Abstract

The United States of America has adopted a suite of Secure Hash Algorithms (SHAs), including four beyond SHA-1, as part of a Federal Information Processing Standard (FIPS), specifically SHA-224 (RFC 3874), SHA-256, SHA-384, and SHA-512. The purpose of this document is to make source code performing these hash functions conveniently available to the Internet community. The sample code supports input strings of arbitrary bit length. SHA-1's sample code from RFC 3174 has also been updated to handle input strings of arbitrary bit length. Most of the text herein was adapted by the authors from FIPS 180-2. Code to perform SHA-based HMACs, with arbitrary bit length text, is also included.

## Table of Contents

1. Overview of Contents ............................................3 1.1. License ....................................................4 2. Notation for Bit Strings and Integers ...........................4 3. Operations on Words .............................................5 4. Message Padding and Parsing .....................................6 4.1. SHA-224 and SHA-256 ........................................7 4.2. SHA-384 and SHA-512 ........................................8 5. Functions and Constants Used ....................................9 5.1. SHA-224 and SHA-256 ........................................9 5.2. SHA-384 and SHA-512 .......................................10 6. Computing the Message Digest ...................................11 6.1. SHA-224 and SHA-256 Initialization ........................11 6.2. SHA-224 and SHA-256 Processing ............................11 6.3. SHA-384 and SHA-512 Initialization ........................13 6.4. SHA-384 and SHA-512 Processing ............................14 7. SHA-Based HMACs ................................................15 8. C Code for SHAs ................................................15 8.1. The .h File ...............................................18 8.2. The SHA Code ..............................................24 8.2.1. sha1.c .............................................24 8.2.2. sha224-256.c .......................................33 8.2.3. sha384-512.c .......................................45 8.2.4. usha.c .............................................67 8.2.5. sha-private.h ......................................72 8.3. The HMAC Code .............................................73 8.4. The Test Driver ...........................................78 9. Security Considerations .......................................106 10. Normative References .........................................106 11. Informative References .......................................106

## 1. Overview of Contents

NOTE: Much of the text below is taken from [FIPS180-2] and assertions therein of the security of the algorithms described are made by the US Government, the author of [FIPS180-2], and not by the authors of this document. The text below specifies Secure Hash Algorithms, SHA-224 [RFC3874], SHA-256, SHA-384, and SHA-512, for computing a condensed representation of a message or a data file. (SHA-1 is specified in [RFC3174].) When a message of any length < 2^64 bits (for SHA-224 and SHA-256) or < 2^128 bits (for SHA-384 and SHA-512) is input to one of these algorithms, the result is an output called a message digest. The message digests range in length from 224 to 512 bits, depending on the algorithm. Secure hash algorithms are typically used with other cryptographic algorithms, such as digital signature algorithms and keyed hash authentication codes, or in the generation of random numbers [RFC4086]. The four algorithms specified in this document are called secure because it is computationally infeasible to (1) find a message that corresponds to a given message digest, or (2) find two different messages that produce the same message digest. Any change to a message in transit will, with very high probability, result in a different message digest. This will result in a verification failure when the secure hash algorithm is used with a digital signature algorithm or a keyed-hash message authentication algorithm. The code provided herein supports input strings of arbitrary bit length. SHA-1's sample code from [RFC3174] has also been updated to handle input strings of arbitrary bit length. See Section 1.1 for license information for this code. Section 2 below defines the terminology and functions used as building blocks to form these algorithms. Section 3 describes the fundamental operations on words from which these algorithms are built. Section 4 describes how messages are padded up to an integral multiple of the required block size and then parsed into blocks. Section 5 defines the constants and the composite functions used to specify these algorithms. Section 6 gives the actual specification for the SHA-224, SHA-256, SHA-384, and SHA-512 functions. Section 7 provides pointers to the specification of HMAC keyed message authentication codes based on the SHA algorithms. Section 8 gives sample code for the SHA algorithms and Section 9 code for SHA-based HMACs. The SHA-based HMACs will accept arbitrary bit length text.

## 1.1. License

Permission is granted for all uses, commercial and non-commercial, of the sample code found in Section 8. Royalty free license to use, copy, modify and distribute the software found in Section 8 is granted, provided that this document is identified in all material mentioning or referencing this software, and provided that redistributed derivative works do not contain misleading author or version information. The authors make no representations concerning either the merchantability of this software or the suitability of this software for any particular purpose. It is provided "as is" without express or implied warranty of any kind.## 2. Notation for Bit Strings and Integers

The following terminology related to bit strings and integers will be used: a. A hex digit is an element of the set {0, 1, ... , 9, A, ... , F}. A hex digit is the representation of a 4-bit string. Examples: 7 = 0111, A = 1010. b. A word equals a 32-bit or 64-bit string, which may be represented as a sequence of 8 or 16 hex digits, respectively. To convert a word to hex digits, each 4-bit string is converted to its hex equivalent as described in (a) above. Example: 1010 0001 0000 0011 1111 1110 0010 0011 = A103FE23. Throughout this document, the "big-endian" convention is used when expressing both 32-bit and 64-bit words, so that within each word the most significant bit is shown in the left-most bit position. c. An integer may be represented as a word or pair of words. An integer between 0 and 2^32 - 1 inclusive may be represented as a 32-bit word. The least significant four bits of the integer are represented by the right-most hex digit of the word representation. Example: the integer 291 = 2^8+2^5+2^1+2^0 = 256+32+2+1 is represented by the hex word 00000123. The same holds true for an integer between 0 and 2^64-1 inclusive, which may be represented as a 64-bit word.

If Z is an integer, 0 <= z < 2^64, then z = (2^32)x + y where 0 <= x < 2^32 and 0 <= y < 2^32. Since x and y can be represented as words X and Y, respectively, z can be represented as the pair of words (X,Y). d. block = 512-bit or 1024-bit string. A block (e.g., B) may be represented as a sequence of 32-bit or 64-bit words.## 3. Operations on Words

The following logical operators will be applied to words in all four hash operations specified herein. SHA-224 and SHA-256 operate on 32-bit words, while SHA-384 and SHA-512 operate on 64-bit words. In the operations below, x<<n is obtained as follows: discard the left-most n bits of x and then pad the result with n zeroed bits on the right (the result will still be the same number of bits). a. Bitwise logical word operations X AND Y = bitwise logical "and" of X and Y. X OR Y = bitwise logical "inclusive-or" of X and Y. X XOR Y = bitwise logical "exclusive-or" of X and Y. NOT X = bitwise logical "complement" of X. Example: 01101100101110011101001001111011 XOR 01100101110000010110100110110111 -------------------------------- = 00001001011110001011101111001100 b. The operation X + Y is defined as follows: words X and Y represent w-bit integers x and y, where 0 <= x < 2^w and 0 <= y < 2^w. For positive integers n and m, let n mod m be the remainder upon dividing n by m. Compute z = (x + y) mod 2^w. Then 0 <= z < 2^w. Convert z to a word, Z, and define Z = X + Y.

c. The right shift operation SHR^n(x), where x is a w-bit word and n is an integer with 0 <= n < w, is defined by SHR^n(x) = x>>n d. The rotate right (circular right shift) operation ROTR^n(x), where x is a w-bit word and n is an integer with 0 <= n < w, is defined by ROTR^n(x) = (x>>n) OR (x<<(w-n)) e. The rotate left (circular left shift) operation ROTL^n(x), where x is a w-bit word and n is an integer with 0 <= n < w, is defined by ROTL^n(X) = (x<<n) OR (x>>w-n) Note the following equivalence relationships, where w is fixed in each relationship: ROTL^n(x) = ROTR^(w-x)(x) ROTR^n(x) = ROTL^(w-n)(x)## 4. Message Padding and Parsing

The hash functions specified herein are used to compute a message digest for a message or data file that is provided as input. The message or data file should be considered to be a bit string. The length of the message is the number of bits in the message (the empty message has length 0). If the number of bits in a message is a multiple of 8, for compactness we can represent the message in hex. The purpose of message padding is to make the total length of a padded message a multiple of 512 for SHA-224 and SHA-256 or a multiple of 1024 for SHA-384 and SHA-512. The following specifies how this padding shall be performed. As a summary, a "1" followed by a number of "0"s followed by a 64-bit or 128-bit integer are appended to the end of the message to produce a padded message of length 512*n or 1024*n. The minimum number of "0"s necessary to meet this criterion is used. The appended integer is the length of the original message. The padded message is then processed by the hash function as n 512-bit or 1024-bit blocks.

## 4.1. SHA-224 and SHA-256

Suppose a message has length L < 2^64. Before it is input to the hash function, the message is padded on the right as follows: a. "1" is appended. Example: if the original message is "01010000", this is padded to "010100001". b. K "0"s are appended where K is the smallest, non-negative solution to the equation L + 1 + K = 448 (mod 512) c. Then append the 64-bit block that is L in binary representation. After appending this block, the length of the message will be a multiple of 512 bits. Example: Suppose the original message is the bit string 01100001 01100010 01100011 01100100 01100101 After step (a), this gives 01100001 01100010 01100011 01100100 01100101 1 Since L = 40, the number of bits in the above is 41 and K = 407 "0"s are appended, making the total now 448. This gives the following in hex: 61626364 65800000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 The 64-bit representation of L = 40 is hex 00000000 00000028. Hence the final padded message is the following hex: 61626364 65800000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000028

## 4.2. SHA-384 and SHA-512

Suppose a message has length L < 2^128. Before it is input to the hash function, the message is padded on the right as follows: a. "1" is appended. Example: if the original message is "01010000", this is padded to "010100001". b. K "0"s are appended where K is the smallest, non-negative solution to the equation L + 1 + K = 896 (mod 1024) c. Then append the 128-bit block that is L in binary representation. After appending this block, the length of the message will be a multiple of 1024 bits. Example: Suppose the original message is the bit string 01100001 01100010 01100011 01100100 01100101 After step (a) this gives 01100001 01100010 01100011 01100100 01100101 1 Since L = 40, the number of bits in the above is 41 and K = 855 "0"s are appended, making the total now 896. This gives the following in hex: 61626364 65800000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 The 128-bit representation of L = 40 is hex 00000000 00000000 00000000 00000028. Hence the final padded message is the following hex: 61626364 65800000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000028## 5. Functions and Constants Used

The following subsections give the six logical functions and the table of constants used in each of the hash functions.## 5.1. SHA-224 and SHA-256

SHA-224 and SHA-256 use six logical functions, where each function operates on 32-bit words, which are represented as x, y, and z. The result of each function is a new 32-bit word. CH( x, y, z) = (x AND y) XOR ( (NOT x) AND z) MAJ( x, y, z) = (x AND y) XOR (x AND z) XOR (y AND z) BSIG0(x) = ROTR^2(x) XOR ROTR^13(x) XOR ROTR^22(x) BSIG1(x) = ROTR^6(x) XOR ROTR^11(x) XOR ROTR^25(x) SSIG0(x) = ROTR^7(x) XOR ROTR^18(x) XOR SHR^3(x) SSIG1(x) = ROTR^17(x) XOR ROTR^19(x) XOR SHR^10(x) SHA-224 and SHA-256 use the same sequence of sixty-four constant 32-bit words, K0, K1, ..., K63. These words represent the first thirty-two bits of the fractional parts of the cube roots of the first sixty-four prime numbers. In hex, these constant words are as follows (from left to right): 428a2f98 71374491 b5c0fbcf e9b5dba5 3956c25b 59f111f1 923f82a4 ab1c5ed5 d807aa98 12835b01 243185be 550c7dc3 72be5d74 80deb1fe 9bdc06a7 c19bf174 e49b69c1 efbe4786 0fc19dc6 240ca1cc 2de92c6f 4a7484aa 5cb0a9dc 76f988da 983e5152 a831c66d b00327c8 bf597fc7 c6e00bf3 d5a79147 06ca6351 14292967 27b70a85 2e1b2138 4d2c6dfc 53380d13 650a7354 766a0abb 81c2c92e 92722c85 a2bfe8a1 a81a664b c24b8b70 c76c51a3 d192e819 d6990624 f40e3585 106aa070 19a4c116 1e376c08 2748774c 34b0bcb5

391c0cb3 4ed8aa4a 5b9cca4f 682e6ff3 748f82ee 78a5636f 84c87814 8cc70208 90befffa a4506ceb bef9a3f7 c67178f2## 5.2. SHA-384 and SHA-512

SHA-384 and SHA-512 each use six logical functions, where each function operates on 64-bit words, which are represented as x, y, and z. The result of each function is a new 64-bit word. CH( x, y, z) = (x AND y) XOR ( (NOT x) AND z) MAJ( x, y, z) = (x AND y) XOR (x AND z) XOR (y AND z) BSIG0(x) = ROTR^28(x) XOR ROTR^34(x) XOR ROTR^39(x) BSIG1(x) = ROTR^14(x) XOR ROTR^18(x) XOR ROTR^41(x) SSIG0(x) = ROTR^1(x) XOR ROTR^8(x) XOR SHR^7(x) SSIG1(x) = ROTR^19(x) XOR ROTR^61(x) XOR SHR^6(x) SHA-384 and SHA-512 use the same sequence of eighty constant 64-bit words, K0, K1, ... K79. These words represent the first sixty-four bits of the fractional parts of the cube roots of the first eighty prime numbers. In hex, these constant words are as follows (from left to right): 428a2f98d728ae22 7137449123ef65cd b5c0fbcfec4d3b2f e9b5dba58189dbbc 3956c25bf348b538 59f111f1b605d019 923f82a4af194f9b ab1c5ed5da6d8118 d807aa98a3030242 12835b0145706fbe 243185be4ee4b28c 550c7dc3d5ffb4e2 72be5d74f27b896f 80deb1fe3b1696b1 9bdc06a725c71235 c19bf174cf692694 e49b69c19ef14ad2 efbe4786384f25e3 0fc19dc68b8cd5b5 240ca1cc77ac9c65 2de92c6f592b0275 4a7484aa6ea6e483 5cb0a9dcbd41fbd4 76f988da831153b5 983e5152ee66dfab a831c66d2db43210 b00327c898fb213f bf597fc7beef0ee4 c6e00bf33da88fc2 d5a79147930aa725 06ca6351e003826f 142929670a0e6e70 27b70a8546d22ffc 2e1b21385c26c926 4d2c6dfc5ac42aed 53380d139d95b3df 650a73548baf63de 766a0abb3c77b2a8 81c2c92e47edaee6 92722c851482353b a2bfe8a14cf10364 a81a664bbc423001 c24b8b70d0f89791 c76c51a30654be30 d192e819d6ef5218 d69906245565a910 f40e35855771202a 106aa07032bbd1b8 19a4c116b8d2d0c8 1e376c085141ab53 2748774cdf8eeb99 34b0bcb5e19b48a8 391c0cb3c5c95a63 4ed8aa4ae3418acb 5b9cca4f7763e373 682e6ff3d6b2b8a3 748f82ee5defb2fc 78a5636f43172f60 84c87814a1f0ab72 8cc702081a6439ec 90befffa23631e28 a4506cebde82bde9 bef9a3f7b2c67915 c67178f2e372532b ca273eceea26619c d186b8c721c0c207 eada7dd6cde0eb1e f57d4f7fee6ed178 06f067aa72176fba 0a637dc5a2c898a6 113f9804bef90dae 1b710b35131c471b 28db77f523047d84 32caab7b40c72493 3c9ebe0a15c9bebc 431d67c49c100d4c 4cc5d4becb3e42b6 597f299cfc657e2a 5fcb6fab3ad6faec 6c44198c4a475817

## 6. Computing the Message Digest

The output of each of the secure hash functions, after being applied to a message of N blocks, is the hash quantity H(N). For SHA-224 and SHA-256, H(i) can be considered to be eight 32-bit words, H(i)0, H(i)1, ... H(i)7. For SHA-384 and SHA-512, it can be considered to be eight 64-bit words, H(i)0, H(i)1, ..., H(i)7. As described below, the hash words are initialized, modified as each message block is processed, and finally concatenated after processing the last block to yield the output. For SHA-256 and SHA-512, all of the H(N) variables are concatenated while the SHA-224 and SHA-384 hashes are produced by omitting some from the final concatenation.## 6.1. SHA-224 and SHA-256 Initialization

For SHA-224, the initial hash value, H(0), consists of the following 32-bit words in hex: H(0)0 = c1059ed8 H(0)1 = 367cd507 H(0)2 = 3070dd17 H(0)3 = f70e5939 H(0)4 = ffc00b31 H(0)5 = 68581511 H(0)6 = 64f98fa7 H(0)7 = befa4fa4 For SHA-256, the initial hash value, H(0), consists of the following eight 32-bit words, in hex. These words were obtained by taking the first thirty-two bits of the fractional parts of the square roots of the first eight prime numbers. H(0)0 = 6a09e667 H(0)1 = bb67ae85 H(0)2 = 3c6ef372 H(0)3 = a54ff53a H(0)4 = 510e527f H(0)5 = 9b05688c H(0)6 = 1f83d9ab H(0)7 = 5be0cd19## 6.2. SHA-224 and SHA-256 Processing

SHA-224 and SHA-256 perform identical processing on messages blocks and differ only in how H(0) is initialized and how they produce their final output. They may be used to hash a message, M, having a length of L bits, where 0 <= L < 2^64. The algorithm uses (1) a message

schedule of sixty-four 32-bit words, (2) eight working variables of 32 bits each, and (3) a hash value of eight 32-bit words. The words of the message schedule are labeled W0, W1, ..., W63. The eight working variables are labeled a, b, c, d, e, f, g, and h. The words of the hash value are labeled H(i)0, H(i)1, ..., H(i)7, which will hold the initial hash value, H(0), replaced by each successive intermediate hash value (after each message block is processed), H(i), and ending with the final hash value, H(N), after all N blocks are processed. They also use two temporary words, T1 and T2. The input message is padded as described in Section 4.1 above then parsed into 512-bit blocks, which are considered to be composed of 16 32-bit words M(i)0, M(i)1, ..., M(i)15. The following computations are then performed for each of the N message blocks. All addition is performed modulo 2^32. For i = 1 to N 1. Prepare the message schedule W: For t = 0 to 15 Wt = M(i)t For t = 16 to 63 Wt = SSIG1(W(t-2)) + W(t-7) + SSIG0(t-15) + W(t-16) 2. Initialize the working variables: a = H(i-1)0 b = H(i-1)1 c = H(i-1)2 d = H(i-1)3 e = H(i-1)4 f = H(i-1)5 g = H(i-1)6 h = H(i-1)7 3. Perform the main hash computation: For t = 0 to 63 T1 = h + BSIG1(e) + CH(e,f,g) + Kt + Wt T2 = BSIG0(a) + MAJ(a,b,c) h = g g = f f = e e = d + T1 d = c c = b b = a a = T1 + T2

4. Compute the intermediate hash value H(i): H(i)0 = a + H(i-1)0 H(i)1 = b + H(i-1)1 H(i)2 = c + H(i-1)2 H(i)3 = d + H(i-1)3 H(i)4 = e + H(i-1)4 H(i)5 = f + H(i-1)5 H(i)6 = g + H(i-1)6 H(i)7 = h + H(i-1)7 After the above computations have been sequentially performed for all of the blocks in the message, the final output is calculated. For SHA-256, this is the concatenation of all of H(N)0, H(N)1, through H(N)7. For SHA-224, this is the concatenation of H(N)0, H(N)1, through H(N)6.## 6.3. SHA-384 and SHA-512 Initialization

For SHA-384, the initial hash value, H(0), consists of the following eight 64-bit words, in hex. These words were obtained by taking the first sixty-four bits of the fractional parts of the square roots of the ninth through sixteenth prime numbers. H(0)0 = cbbb9d5dc1059ed8 H(0)1 = 629a292a367cd507 H(0)2 = 9159015a3070dd17 H(0)3 = 152fecd8f70e5939 H(0)4 = 67332667ffc00b31 H(0)5 = 8eb44a8768581511 H(0)6 = db0c2e0d64f98fa7 H(0)7 = 47b5481dbefa4fa4 For SHA-512, the initial hash value, H(0), consists of the following eight 64-bit words, in hex. These words were obtained by taking the first sixty-four bits of the fractional parts of the square roots of the first eight prime numbers. H(0)0 = 6a09e667f3bcc908 H(0)1 = bb67ae8584caa73b H(0)2 = 3c6ef372fe94f82b H(0)3 = a54ff53a5f1d36f1 H(0)4 = 510e527fade682d1 H(0)5 = 9b05688c2b3e6c1f H(0)6 = 1f83d9abfb41bd6b H(0)7 = 5be0cd19137e2179

## 6.4. SHA-384 and SHA-512 Processing

SHA-384 and SHA-512 perform identical processing on message blocks and differ only in how H(0) is initialized and how they produce their final output. They may be used to hash a message, M, having a length of L bits, where 0 <= L < 2^128. The algorithm uses (1) a message schedule of eighty 64-bit words, (2) eight working variables of 64 bits each, and (3) a hash value of eight 64-bit words. The words of the message schedule are labeled W0, W1, ..., W79. The eight working variables are labeled a, b, c, d, e, f, g, and h. The words of the hash value are labeled H(i)0, H(i)1, ..., H(i)7, which will hold the initial hash value, H(0), replaced by each successive intermediate hash value (after each message block is processed), H(i), and ending with the final hash value, H(N) after all N blocks are processed. The input message is padded as described in Section 4.2 above, then parsed into 1024-bit blocks, which are considered to be composed of 16 64-bit words M(i)0, M(i)1, ..., M(i)15. The following computations are then performed for each of the N message blocks. All addition is performed modulo 2^64. For i = 1 to N 1. Prepare the message schedule W: For t = 0 to 15 Wt = M(i)t For t = 16 to 79 Wt = SSIG1(W(t-2)) + W(t-7) + SSIG0(t-15) + W(t-16) 2. Initialize the working variables: a = H(i-1)0 b = H(i-1)1 c = H(i-1)2 d = H(i-1)3 e = H(i-1)4 f = H(i-1)5 g = H(i-1)6 h = H(i-1)7 3. Perform the main hash computation: For t = 0 to 79 T1 = h + BSIG1(e) + CH(e,f,g) + Kt + Wt T2 = BSIG0(a) + MAJ(a,b,c) h = g g = f f = e

e = d + T1 d = c c = b b = a a = T1 + T2 4. Compute the intermediate hash value H(i): H(i)0 = a + H(i-1)0 H(i)1 = b + H(i-1)1 H(i)2 = c + H(i-1)2 H(i)3 = d + H(i-1)3 H(i)4 = e + H(i-1)4 H(i)5 = f + H(i-1)5 H(i)6 = g + H(i-1)6 H(i)7 = h + H(i-1)7 After the above computations have been sequentially performed for all of the blocks in the message, the final output is calculated. For SHA-512, this is the concatenation of all of H(N)0, H(N)1, through H(N)7. For SHA-384, this is the concatenation of H(N)0, H(N)1, through H(N)5.

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