NIST FIPS 180-4 日本語訳

Secure Hash Standard (SHS)

Information Technology Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-8900

This publication is available free of charge from: http://dx.doi.org/10.6028/NIST.FIPS.180-4

August 2015

U.S. Department of Commerce Penny Pritzker, Secretary

National Institute of Standards and Technology Willie E. May, Under Secretary for Standards and Technology and Director

The Federal Information Processing Standards Publication Series of the National Institute of Standards and Technology (NIST) is the official series of publications relating to standards and guidelines adopted and promulgated under the provisions of the Federal Information Security Management Act (FISMA) of 2002.

Comments concerning FIPS publications are welcomed and should be addressed to the Director, Information Technology Laboratory, National Institute of Standards and Technology, 100 Bureau Drive, Stop 8900, Gaithersburg, MD 20899-8900.

Charles H. Romine, Director Information Technology Laboratory

This standard specifies hash algorithms that can be used to generate digests of messages. The digests are used to detect whether messages have been changed since the digests were generated.

Key words: computer security, cryptography, message digest, hash function, hash algorithm, Federal Information Processing Standards, Secure Hash Standard.

August 2015

Announcing the

Federal Information Processing Standards Publications (FIPS PUBS) are issued by the National Institute of Standards and Technology (NIST) after approval by the Secretary of Commerce pursuant to Section 5131 of the Information Technology Management Reform Act of 1996 (Public Law 104-106), and the Computer Security Act of 1987 (Public Law 100-235).

The hash algorithms specified in this Standard are called secure because, for a given algorithm, it is computationally infeasible 1) to find a message that corresponds to a given message digest, or 2) to find two different messages that produce the same message digest. Any change to a message will, with a 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.

This Standard supersedes FIPS 180-3 [FIPS 180-3].

Specifications for the

SECURE HASH STANDARD

1. INTRODUCTION — 3

2. DEFINITIONS — 4

This Standard specifies secure hash algorithms, SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256. All of the algorithms are iterative, one-way hash functions that can process a message to produce a condensed representation called a message digest. These algorithms enable the determination of a message’s integrity: any change to the message will, with a very high probability, result in a different message digest. This property is useful in the generation and verification of digital signatures and message authentication codes, and in the generation of random numbers or bits.

Each algorithm can be described in two stages: preprocessing and hash computation. Preprocessing involves padding a message, parsing the padded message into m-bit blocks, and setting initialization values to be used in the hash computation. The hash computation generates a message schedule from the padded message and uses that schedule, along with functions, constants, and word operations to iteratively generate a series of hash values. The final hash value generated by the hash computation is used to determine the message digest.

The algorithms differ most significantly in the security strengths that are provided for the data being hashed. The security strengths of these hash functions and the system as a whole when each of them is used with other cryptographic algorithms, such as digital signature algorithms and keyed-hash message authentication codes, can be found in [SP 800-57] and [SP 800-107].

Additionally, the algorithms differ in terms of the size of the blocks and words of data that are used during hashing or message digest sizes. Figure 1 presents the basic properties of these hash algorithms.

Figure 1: Secure Hash Algorithm Properties

Bit — A binary digit having a value of 0 or 1.

Byte — A group of eight bits.

FIPS — Federal Information Processing Standard.

NIST — National Institute of Standards and Technology.

SHA — Secure Hash Algorithm.

SP — Special Publication

Word — A group of either 32 bits (4 bytes) or 64 bits (8 bytes), depending on the secure hash algorithm.

The following parameters are used in the secure hash algorithm specifications in this Standard.

\(a, b, c, \ldots, h\) — Working variables that are the \(w\)-bit words used in the computation of the hash values, \(H^{(i)}\).

\(H^{(i)}\) — The \(i^{\text{th}}\) hash value. \(H^{(0)}\) is the initial hash value; \(H^{(N)}\) is the final hash value and is used to determine the message digest.

\(H_j^{(i)}\) — The \(j^{\text{th}}\) word of the \(i^{\text{th}}\) hash value, where \(H_0^{(i)}\) is the left-most word of hash value \(i\).

\(K_t\) — Constant value to be used for the iteration \(t\) of the hash computation.

\(k\) — Number of zeroes appended to a message during the padding step.

\(\ell\) — Length of the message, \(M\), in bits.

\(m\) — Number of bits in a message block, \(M^{(i)}\).

\(M\) — Message to be hashed.

\(M^{(i)}\) — Message block \(i\), with a size of \(m\) bits.

\(M_j^{(i)}\) — The \(j^{\text{th}}\) word of the \(i^{\text{th}}\) message block, where \(M_0^{(i)}\) is the left-most word of message block \(i\).

\(n\) — Number of bits to be rotated or shifted when a word is operated upon.

\(N\) — Number of blocks in the padded message.

\(T\) — Temporary \(w\)-bit word used in the hash computation.

\(w\) — Number of bits in a word.

\(W_t\) — The \(t^{\text{th}}\) \(w\)-bit word of the message schedule.

The following symbols are used in the secure hash algorithm specifications; each operates on \(w\)-bit words.

\(\wedge\) — Bitwise AND operation.

\(\vee\) — Bitwise OR (“inclusive-OR”) operation.

\(\oplus\) — Bitwise XOR (“exclusive-OR”) operation.

\(\neg\) — Bitwise complement operation.

\(+\) — Addition modulo \(2^w\).

\(<<\) — Left-shift operation, where \(x << n\) is obtained by discarding the left-most \(n\) bits of the word \(x\) and then padding the result with \(n\) zeroes on the right.

\(>>\) — Right-shift operation, where \(x >> n\) is obtained by discarding the right-most \(n\) bits of the word \(x\) and then padding the result with \(n\) zeroes on the left.

The following operations are used in the secure hash algorithm specifications:

\(\boldsymbol{ROTL}^{n}(\boldsymbol{x})\) — The rotate left (circular left shift) operation, where \(x\) is a \(w\)-bit word and \(n\) is an integer with \(0 \le n < w\), is defined by \(ROTL^{n}(x) = (x << n) \vee (x >> w - n)\).

\(\boldsymbol{ROTR}^{n}(\boldsymbol{x})\) — The rotate right (circular right shift) operation, where \(x\) is a \(w\)-bit word and \(n\) is an integer with \(0 \le n < w\), is defined by \(ROTR^{n}(x) = (x >> n) \vee (x << w - n)\).

\(\boldsymbol{SHR}^{n}(\boldsymbol{x})\) — The right shift operation, where \(x\) is a \(w\)-bit word and \(n\) is an integer with \(0 \le n < w\), is defined by \(SHR^{n}(x) = x >> n\).

The following terminology related to bit strings and integers will be used.

1010 0001 0000 0011 1111 1110 0010 0011

can be expressed as “a103fe23”, and the 64-bit string

1010 0001 0000 0011 1111 1110 0010 0011 0011 0010 1110 1111 0011 0000 0001 1010

can be expressed as “a103fe2332ef301a”.

Throughout this specification, the “big-endian” convention is used when expressing both 32- and 64-bit words, so that within each word, the most significant bit is stored in the left-most bit position.

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. For 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, \(0 \le Z < 2^{64}\), then \(Z=2^{32}X + Y\), where \(0 \le X < 2^{32}\) and \(0 \le Y < 2^{32}\). Since \(X\) and \(Y\) can be represented as 32-bit words \(x\) and \(y\), respectively, the integer \(Z\) can be represented as the pair of words \((x, y)\). This property is used for SHA-1, SHA-224 and SHA-256.

If \(Z\) is an integer, \(0 \le Z < 2^{128}\), then \(Z=2^{64}X + Y\), where \(0 \le X < 2^{64}\) and \(0 \le Y < 2^{64}\). Since \(X\) and \(Y\) can be represented as 64-bit words \(x\) and \(y\), respectively, the integer \(Z\) can be represented as the pair of words \((x, y)\). This property is used for SHA-384, SHA-512, SHA-512/224 and SHA-512/256.

b) For SHA-384, SHA-512, SHA-512/224 and SHA-512/256 each message block has 1024 bits, which are represented as a sequence of sixteen 64-bit words.

The following operations are applied to \(w\)-bit words in all five secure hash algorithms. SHA-1, SHA-224 and SHA-256 operate on 32-bit words (\(w=32\)), and SHA-384, SHA-512, SHA-512/224 and SHA-512/256 operate on 64-bit words (\(w=64\)).

The operation \(x + y\) is defined as follows. The words \(x\) and \(y\) represent integers \(X\) and \(Y\), where \(0 \le X < 2^w\) and \(0 \le Y < 2^w\). For positive integers \(U\) and \(V\), let \(U \bmod V\) be the remainder upon dividing \(U\) by \(V\). Compute

Then \(0 \le Z < 2^w\). Convert the integer \(Z\) to a word, \(z\), and define \(z=x + y\).

This operation is used in the SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 algorithms.

Thus, \(ROTR^{n}(x)\) is equivalent to a circular shift (rotation) of \(x\) by \(n\) positions to the right.

This operation is used by the SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 algorithms.

Thus, \(ROTL^{n}(x)\) is equivalent to a circular shift (rotation) of \(x\) by \(n\) positions to the left.

This operation is used only in the SHA-1 algorithm.

This section defines the functions that are used by each of the algorithms. Although the SHA-224, SHA-256, SHA-384,SHA-512, SHA-512/224 and SHA-512/256 algorithms all use similar functions, their descriptions are separated into sections for SHA-224 and SHA-256 (Sec. 4.1.2) and for SHA-384, SHA-512, SHA-512/224 and SHA-512/256 (Sec. 4.1.3), since the input and output for these functions are words of different sizes. Each of the algorithms include \(Ch(x, y, z)\) and \(Maj(x, y, z)\) functions; the exclusive-OR operation (\(\oplus\)) in these functions may be replaced by a bitwise OR operation (\(\lor\)) and produce identical results.

SHA-1 uses a sequence of logical functions, \(f_0, f_1, \ldots, f_{79}\). Each function \(f_t\), where \(0 \le t \le 79\), operates on three 32-bit words, \(x\), \(y\), and \(z\), and produces a 32-bit word as output. The function \(f_t(x, y, z)\) is defined as follows:

SHA-224 and SHA-256 both 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.

SHA-384, SHA-512, SHA-512/224 and SHA-512/256 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.

SHA-1 uses a sequence of eighty constant 32-bit words, \(K_0, K_1, \ldots, K_{79}\), which are given by

SHA-224 and SHA-256 use the same sequence of sixty-four constant 32-bit words, \(K_0^{\{256\}}, K_1^{\{256\}}, \ldots, K_{63}^{\{256\}}\). 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 (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

SHA-384, SHA-512, SHA-512/224 and SHA-512/256 use the same sequence of eighty constant 64-bit words, \(K_0^{\{512\}}, K_1^{\{512\}}, \ldots, K_{79}^{\{512\}}\). 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 (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

Preprocessing consists of three steps: padding the message, \(M\) (Sec. 5.1), parsing the message into message blocks (Sec. 5.2), and setting the initial hash value, \(H^{(0)}\) (Sec. 5.3).

The purpose of this padding is to ensure that the padded message is a multiple of 512 or 1024 bits, depending on the algorithm. Padding can be inserted before hash computation begins on a message, or at any other time during the hash computation prior to processing the block(s) that will contain the padding.

Suppose that the length of the message, \(M\), is \(\ell\) bits. Append the bit “1” to the end of the message, followed by \(k\) zero bits, where \(k\) is the smallest, non-negative solution to the equation \(\ell + 1 + k \equiv 448 \bmod 512\). Then append the 64-bit block that is equal to the number \(\ell\) expressed using a binary representation. For example, the (8-bit ASCII) message “abc” has length \(8 \times 3 = 24\), so the message is padded with a one bit, then \(448 - (24 + 1) = 423\) zero bits, and then the message length, to become the 512-bit padded message

The length of the padded message should now be a multiple of 512 bits.

Suppose the length of the message \(M\), in bits, is \(\ell\) bits. Append the bit “1” to the end of the message, followed by \(k\) zero bits, where \(k\) is the smallest non-negative solution to the equation \(\ell + 1 + k \equiv 896 \bmod 1024\). Then append the 128-bit block that is equal to the number \(\ell\) expressed using a binary representation. For example, the (8-bit ASCII) message “abc” has length \(8 \times 3 = 24\), so the message is padded with a one bit, then \(896 - (24 + 1) = 871\) zero bits, and then the message length, to become the 1024-bit padded message

The length of the padded message should now be a multiple of 1024 bits.

The message and its padding must be parsed into N m-bit blocks.

For SHA-1, SHA-224 and SHA-256, the message and its padding are parsed into \(N\) 512-bit blocks, \(M^{(1)}, M^{(2)}, \ldots, M^{(N)}\). Since the 512 bits of the input block may be expressed as sixteen 32-bit words, the first 32 bits of message block \(i\) are denoted \(M_0^{(i)}\), the next 32 bits are \(M_1^{(i)}\), and so on up to \(M_{15}^{(i)}\).

For SHA-384, SHA-512, SHA-512/224 and SHA-512/256, the message and its padding are parsed into \(N\) 1024-bit blocks, \(M^{(1)}, M^{(2)}, \ldots, M^{(N)}\). Since the 1024 bits of the input block may be expressed as sixteen 64-bit words, the first 64 bits of message block \(i\) are denoted \(M_0^{(i)}\), the next 64 bits are \(M_1^{(i)}\), and so on up to \(M_{15}^{(i)}\).

Before hash computation begins for each of the secure hash algorithms, the initial hash value, \(H^{(0)}\), must be set. The size and number of words in \(H^{(0)}\) depends on the message digest size.

For SHA-1, the initial hash value, \(H^{(0)}\), shall consist of the following five 32-bit words, in hex:

For SHA-224, the initial hash value, \(H^{(0)}\), shall consist of the following eight 32-bit words, in hex:

For SHA-256, the initial hash value, \(H^{(0)}\), shall consist 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.

For SHA-384, the initial hash value, \(H^{(0)}\), shall consist 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.

For SHA-512, the initial hash value, \(H^{(0)}\), shall consist 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.

“SHA-512/t” is the general name for a t-bit hash function based on SHA-512 whose output is truncated to t bits. Each hash function requires a distinct initial hash value. This section provides a procedure for determining the initial value for SHA-512/ t for a given value of t.

For SHA-512/t, t is any positive integer without a leading zero such that t < 512, and t is not 384. For example: t is 256, but not 0256, and “SHA-512/t” is “SHA-512/256” (an 11 character long ASCII string), which is equivalent to 53 48 41 2D 35 31 32 2F 32 35 36 in hexadecimal.

The initial hash value for SHA-512/t, for a given value of t, shall be generated by the SHA-512/t IV Generation Function below.

(begin:)

Denote \(H^{(0)\prime}\) to be the initial hash value of SHA-512 as specified in Section 5.3.5 above.

Denote \(H^{(0)\prime\prime}\) to be the initial hash value computed below.

\(H^{(0)}\) is the IV for SHA-512/\(t\).

\(H^{(0)}\) = SHA-512 (“SHA-512/\(t\)”) using \(H^{(0)\prime\prime}\) as the IV, where \(t\) is the specific truncation value.

(end.)

SHA-512/224 (\(t = 224\)) and SHA-512/256 (\(t = 256\)) are approved hash algorithms. Other SHA-512/\(t\) hash algorithms with different \(t\) values may be specified in [SP 800-107] in the future as the need arises. Below are the IVs for SHA-512/224 and SHA-512/256.

For SHA-512/224, the initial hash value, \(H^{(0)}\), shall consist of the following eight 64-bit words, in hex:

These words were obtained by executing the SHA-512/t IV Generation Function with \(t = 224\).

For SHA-512/256, the initial hash value, \(H^{(0)}\), shall consist of the following eight 64-bit words, in hex:

These words were obtained by executing the SHA-512/t IV Generation Function with \(t = 256\).

In the following sections, the hash algorithms are not described in ascending order of size. SHA-256 is described before SHA-224 because the specification for SHA-224 is identical to SHA-256, except that different initial hash values are used, and the final hash value is truncated to 224 bits for SHA-224. The same is true for SHA-512, SHA-384, SHA-512/224 and SHA-512/256, except that the final hash value is truncated to 224 bits for SHA-512/224, 256 bits for SHA-512/256 or 384 bits for SHA-384.

For each of the secure hash algorithms, there may exist alternate computation methods that yield identical results; one example is the alternative SHA-1 computation described in Sec. 6.1.3. Such alternate methods may be implemented in conformance to this standard.

SHA-1 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{64}\). The algorithm uses 1) a message schedule of eighty 32-bit words, 2) five working variables of 32 bits each, and 3) a hash value of five 32-bit words. The final result of SHA-1 is a 160-bit message digest.

The words of the message schedule are labeled \(W_0, W_1, \ldots, W_{79}\). The five working variables are labeled \(\boldsymbol{a}\), \(\boldsymbol{b}\), \(\boldsymbol{c}\), \(\boldsymbol{d}\), and \(\boldsymbol{e}\). The words of the hash value are labeled \(H_0^{(i)}, H_1^{(i)}, \ldots, H_4^{(i)}\), 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, \(H^{(N)}\). SHA-1 also uses a single temporary word, \(T\).

The SHA-1 hash computation uses functions and constants previously defined in Sec. 4.1.1 and Sec. 4.2.1, respectively. Addition (+) is performed modulo \(2^{32}\).

Each message block, \(M^{(1)}, M^{(2)}, \ldots, M^{(N)}\), is processed in order, using the following steps:

For \(i\)=1 to \(N\): {

}

}

After repeating steps one through four a total of \(N\) times (i.e., after processing \(M^{(N)}\)), the resulting 160-bit message digest of the message, \(M\), is

The SHA-1 hash computation method described in Sec. 6.1.2 assumes that the message schedule \(W_0, W_1, \ldots, W_{79}\) is implemented as an array of eighty 32-bit words. This is efficient from the standpoint of the minimization of execution time, since the addresses of \(W_{t-3}, \ldots, W_{t-16}\) in step (2) of Sec. 6.1.2 are easily computed.

However, if memory is limited, an alternative is to regard \(\{W_t\}\) as a circular queue that may be implemented using an array of sixteen 32-bit words, \(W_0, W_1, \ldots, W_{15}\). The alternate method that is described in this section yields the same message digest as the SHA-1 computation method described in Sec. 6.1.2. Although this alternate method saves sixty-four 32-bit words of storage, it is likely to lengthen the execution time due to the increased complexity of the address computations for the \(\{W_t\}\) in step (3).

For this alternate SHA-1 method, let \(\mathit{MASK}\)=0000000f (in hex). As in Sec. 6.1.1, addition is performed modulo \(2^{32}\). Assuming that the preprocessing as described in Sec. 6.1.1 has been performed, the processing of \(M^{(i)}\) is as follows:

For \(i\)=1 to \(N\): {

}

After repeating steps one through four a total of \(N\) times (i.e., after processing \(M^{(N)}\)), the resulting 160-bit message digest of the message, \(M\), is

SHA-256 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{64}\). 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 final result of SHA-256 is a 256-bit message digest.

The words of the message schedule are labeled \(W_0\), \(W_1\),…, \(W_{63}\). The eight working variables are labeled \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\), \(\mathbf{e}\), \(\mathbf{f}\), \(\mathbf{g}\), and \(\mathbf{h}\). The words of the hash value are labeled \(H_0^{(i)}, H_1^{(i)}, \ldots, H_7^{(i)}\), 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, \(H^{(N)}\). SHA-256 also uses two temporary words, \(T_1\) and \(T_2\).

The SHA-256 hash computation uses functions and constants previously defined in Sec. 4.1.2 and Sec. 4.2.2, respectively. Addition (+) is performed modulo \(2^{32}\).

Each message block, \(M^{(1)}\), \(M^{(2)}\), …, \(M^{(N)}\), is processed in order, using the following steps:

For \(i\)=1 to \(N\): {

}

After repeating steps one through four a total of \(N\) times (i.e., after processing \(M^{(N)}\)), the resulting 256-bit message digest of the message, \(M\), is

SHA-224 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{64}\). The function is defined in the exact same manner as SHA-256 (Section 6.2), with the following two exceptions:

SHA-512 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{128}\). 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 final result of SHA-512 is a 512-bit message digest.

The words of the message schedule are labeled \(W_0, W_1, \ldots, W_{79}\). The eight working variables are labeled \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\), \(\mathbf{e}\), \(\mathbf{f}\), \(\mathbf{g}\), and \(\mathbf{h}\). The words of the hash value are labeled \(H_0^{(i)}, H_1^{(i)}, \ldots, H_7^{(i)}\), 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, \(H^{(N)}\). SHA-512 also uses two temporary words, \(T_1\) and \(T_2\).

The SHA-512 hash computation uses functions and constants previously defined in Sec. 4.1.3 and Sec. 4.2.3, respectively. Addition (+) is performed modulo \(2^{64}\).

Each message block, \(M^{(1)}, M^{(2)}, \ldots, M^{(N)}\), is processed in order, using the following steps:

For \(i=1\) to \(N\): {

}

After repeating steps one through four a total of \(N\) times (i.e., after processing \(M^{(N)}\)), the resulting 512-bit message digest of the message, \(M\), is

SHA-384 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{128}\). The algorithm is defined in the exact same manner as SHA-512 (Sec. 6.4), with the following two exceptions:

SHA-512/224 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{128}\). The algorithm is defined in the exact same manner as SHA-512 (Sec. 6.4), with the following two exceptions:

SHA-512/256 may be used to hash a message, \(M\), having a length of \(\ell\) bits, where \(0 \le \ell < 2^{128}\). The algorithm is defined in the exact same manner as SHA-512 (Sec. 6.4), with the following two exceptions:

Some application may require a hash function with a message digest length different than those provided by the hash functions in this Standard. In such cases, a truncated message digest may be used, whereby a hash function with a larger message digest length is applied to the data to be hashed, and the resulting message digest is truncated by selecting an appropriate number of the leftmost bits. For guidelines on choosing the length of the truncated message digest and information about its security implications for the cryptographic application that uses it, see SP 800-107 [SP 800-107].

The security of the five hash algorithms, SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 is discussed in [SP 800-107].

Examples of SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 are available at http://csrc.nist.gov/groups/ST/toolkit/examples.html.

Object identifiers (OIDs) for the SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224 and SHA-512/256 algorithms are posted at http://csrc.nist.gov/groups/ST/crypto_apps_infra/csor/algorithms.html.

[FIPS 180-3] NIST, Federal Information Processing Standards Publication 180-3, Secure Hash Standards (SHS), October 2008.

[SP 800-57] NIST Special Publication (SP) 800-57, Part 1, Recommendation for Key Management: General, (Draft) May 2011.

[SP 800-107] NIST Special Publication (SP) 800-107, Recommendation for Applications Using Approved Hash Algorithms, (Revised), (Draft) September 2011.

The following change has been incorporated into FIPS 180-4, as of the date indicated in the table.