# SHA hash functions

The SHA hash functions are five cryptographic hash functions designed by the National Security Agency (NSA) and published by the NIST as a U.S. Federal Information Processing Standard. SHA stands for Secure Hash Algorithm. Hash algorithms compute a fixed-length digital representation (known as a message digest) of an input data sequence (the message) of any length. They are called “secure” when (in the words of the standard), “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 will, with a very high probability, result in a different message digest.” Template:TOCnestright

The five algorithms are denoted SHA-1, SHA-224, SHA-256, SHA-384, and SHA-512. The latter four variants are sometimes collectively referred to as SHA-2. SHA-1 produces a message digest that is 160 bits long; the number in the other four algorithms' names denote the bit length of the digest they produce.

SHA-1 is employed in several widely used security applications and protocols, including TLS and SSL, PGP, SSH, S/MIME, and IPsec. It was considered to be the successor to MD5, an earlier, widely-used hash function.

The security of SHA-1 has been somewhat compromised by cryptography researchers[1]. Although no attacks have yet been reported on the SHA-2 variants, they are algorithmically similar to SHA-1 and so efforts are underway to develop improved alternative hashing algorithms.[2][3] "NIST is initiating an effort to develop one or more additional hash algorithms through a public competition, similar to the development process for the Advanced Encryption Standard (AES)."[4] It's tentatively scheduled to proclaim winner and publish final standard in 2012.

## SHA-0 and SHA-1

Image:SHA-1.svg
One iteration within the SHA-1 compression function. A, B, C, D and E are 32-bit words of the state; F is a nonlinear function that varies; n denotes a left bit rotation by n places; n varies for each operation. denotes addition modulo 232. Kt is a constant.

The original specification of the algorithm was published in 1993 as the Secure Hash Standard, FIPS PUB 180, by US government standards agency NIST (National Institute of Standards and Technology). This version is now often referred to as SHA-0. It was withdrawn by the NSA shortly after publication and was superseded by the revised version, published in 1995 in FIPS PUB 180-1 and commonly referred to as SHA-1. SHA-1 differs from SHA-0 only by a single bitwise rotation in the message schedule of its compression function; this was done, according to the NSA, to correct a flaw in the original algorithm which reduced its cryptographic security. However, the NSA did not provide any further explanation or identify what flaw was corrected. Weaknesses have subsequently been reported in both SHA-0 and SHA-1. SHA-1 appears to provide greater resistance to attacks, supporting the NSA’s assertion that the change increased the security.

SHA-1 (as well as SHA-0) produces a 160-bit digest from a message with a maximum length of (264 − 1) bits and is based on principles similar to those used by Ronald L. Rivest of MIT in the design of the MD4 and MD5 message digest algorithms.

### Cryptanalysis of SHA-0

At CRYPTO 98, two French researchers presented an attack on SHA-0 (Chabaud and Joux, 1998): collisions can be found with complexity 261, fewer than the 280 for an ideal hash function of the same size.

In 2004, Biham and Chen found near-collisions for SHA-0 — two messages that hash to nearly the same value; in this case, 142 out of the 160 bits are equal. They also found full collisions of SHA-0 reduced to 62 out of its 80 rounds.

Subsequently, on 12 August 2004, a collision for the full SHA-0 algorithm was announced by Joux, Carribault, Lemuet, and Jalby. This was done by using a generalization of the Chabaud and Joux attack. Finding the collision had complexity 251 and took about 80,000 CPU hours on a supercomputer with 256 Itanium 2 processors.

On 17 August 2004, at the Rump Session of CRYPTO 2004, preliminary results were announced by Wang, Feng, Lai, and Yu, about an attack on MD5, SHA-0 and other hash functions. The complexity of their attack on SHA-0 is 240, significantly better than the attack by Joux et al. [5][6]

In February 2005, an attack by Xiaoyun Wang, Yiqun Lisa Yin, and Hongbo Yu was announced which could find collisions in SHA-0 in 239 operations [7][8]

### Cryptanalysis of SHA-1

In light of the results on SHA-0, some experts suggested that plans for the use of SHA-1 in new cryptosystems should be reconsidered. After the CRYPTO 2004 results were published, NIST announced that they planned to phase out the use of SHA-1 by 2010 in favor of the SHA-2 variants.[9]

In early 2005, Rijmen and Oswald published an attack on a reduced version of SHA-1 — 53 out of 80 rounds — which finds collisions with a complexity of fewer than 280 operations. [10]

In February 2005, an attack by Xiaoyun Wang, Yiqun Lisa Yin, and Hongbo Yu was announced. [11] The attacks can find collisions in the full version of SHA-1, requiring fewer than 269 operations. (A brute-force search would require 280 operations.)

The authors write: "In particular, our analysis is built upon the original differential attack on SHA0 [sic], the near collision attack on SHA0, the multiblock collision techniques, as well as the message modification techniques used in the collision search attack on MD5. Breaking SHA1 would not be possible without these powerful analytical techniques" [12]. The authors have presented a collision for 58-round SHA-1, found with 233 hash operations. The paper with the full attack description was published in August 2005 at the CRYPTO conference.

In an interview, Yin states that, "Roughly, we exploit the following two weaknesses: One is that the file preprocessing step is not complicated enough; another is that certain math operations in the first 20 rounds have unexpected security problems."[13]

On 17 August 2005, an improvement on the SHA-1 attack was announced on behalf of Xiaoyun Wang, Andrew Yao and Frances Yao at the CRYPTO 2005 rump session, lowering the complexity required for finding a collision in SHA-1 to 263.[14]

Christophe De Cannière and Christian Rechberger further improved the attack on SHA-1 in "Finding SHA-1 Characteristics: General Results and Applications"[15], receiving the Best Paper Award at ASIACRYPT 2006. A two-block collison for 64-round SHA-1 was presented, found using unoptimized methods with 235 compression function evaluations.

In academic cryptography, any attack that has less computational complexity than a brute force search is considered a break.[16] This does not, however, necessarily mean that the attack can be practically exploited. It has been speculated that finding a collision for SHA-1 is within reach of massive distributed Internet search.

At the Rump Session of CRYPTO 2006, Christian Rechberger and Christophe De Cannière claimed to have discovered a collision attack on SHA-1 that would allow an attacker to select at least parts of the message. [17][18]

## SHA-2

A round of the SHA-2

NIST has published four additional hash functions in the SHA family, each with longer digests, collectively known as SHA-2. The individual variants are named after their digest lengths (in bits): SHA-224, SHA-256, SHA-384, and SHA-512. The latter three were first published in 2001 in the draft FIPS PUB 180-2, at which time review and comment were accepted. FIPS PUB 180-2, which also includes SHA-1, was released as an official standard in 2002. In February 2004, a change notice was published for FIPS PUB 180-2, specifying an additional variant, SHA-224, defined to match the key length of two-key Triple DES. These variants are patented in Template:Cite patent.

SHA-256 and SHA-512 are novel hash functions computed with 32- and 64-bit words, respectively. They use different shift amounts and additive constants, but their structures are otherwise virtually identical, differing only in the number of rounds. SHA-224 and SHA-384 are simply truncated versions of the first two, computed with different initial values.

These new hash functions have not received as much scrutiny by the public cryptographic community as SHA-1 has, and so their cryptographic security is not yet as well-established. Gilbert and Handschuh (2003) have studied the newer variants and found no weaknesses.

## SHA sizes

In the following below, internal state means the “internal hash sum” after each compression of a data block; see Merkle-Damgård construction for more details.

Algorithm Output size (bits) Internal state size (bits) Block size (bits) Max message size (bits) Word size (bits) Rounds Operations Collision
SHA-0 160 160 512 264 − 1 32 80 +,and,or,xor,rotl Yes
SHA-1 160 160 512 264 − 1 32 80 +,and,or,xor,rotl 263 attack
SHA-256/224 256/224 256 512 264 − 1 32 64 +,and,or,xor,shr,rotr None yet
SHA-512/384 512/384 512 1024 2128 − 1 64 80 +,and,or,xor,shr,rotr None yet

## Applications

SHA-1, SHA-224, SHA-256, SHA-384, and SHA-512 are the secure hash algorithms required by law for use in certain U. S. Government applications, including use within other cryptographic algorithms and protocols, for the protection of sensitive unclassified information. FIPS PUB 180-1 also encouraged adoption and use of SHA-1 by private and commercial organizations.

A prime motivation for the publication of the Secure Hash Algorithm was the Digital Signature Standard, in which it is incorporated.

The SHA hash functions have been used as the basis for the SHACAL block ciphers.

## Example hashes

The following are some examples of SHA digests. ASCII encoding is assumed for all messages.

### SHA-1 hashes

SHA1("The quick brown fox jumps over the lazy dog")
= 2fd4e1c6 7a2d28fc ed849ee1 bb76e739 1b93eb12
Even a small change in the message will, with overwhelming probability, result in a completely different hash due to the avalanche effect. For example, changing
dog
to
cog
:
SHA1("The quick brown fox jumps over the lazy cog")
= de9f2c7f d25e1b3a fad3e85a 0bd17d9b 100db4b3

The hash of the zero-length message is:

SHA1("")
= da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709

### SHA-224 hashes

SHA224("The quick brown fox jumps over the lazy dog")
= 730e109b d7a8a32b 1cb9d9a0 9aa2325d 2430587d dbc0c38b ad911525
// Avalanche effect when changing the last word to "cog":
SHA224("The quick brown fox jumps over the lazy cog")
= fee755f4 4a55f20f b3362cdc 3c493615 b3cb574e d95ce610 ee5b1e9b

The hash of the zero-length message is:

SHA224("")
= d14a028c 2a3a2bc9 476102bb 288234c4 15a2b01f 828ea62a c5b3e42f

### SHA-256 hashes

SHA256("The quick brown fox jumps over the lazy dog")
= d7a8fbb3 07d78094 69ca9abc b0082e4f 8d5651e4 6d3cdb76 2d02d0bf 37c9e592
// Avalanche effect when changing the last word to "cog":
SHA256("The quick brown fox jumps over the lazy cog")
= e4c4d8f3 bf76b692 de791a17 3e053211 50f7a345 b46484fe 427f6acc 7ecc81be

The hash of the zero-length message is:

SHA256("")
= e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c a495991b 7852b855

### SHA-384 hashes

SHA384("The quick brown fox jumps over the lazy dog")
= ca737f10 14a48f4c 0b6dd43c b177b0af d9e51693 67544c49 4011e331 7dbf9a50
9cb1e5dc 1e85a941 bbee3d7f 2afbc9b1
// Avalanche effect when changing the last word to "cog":
SHA384("The quick brown fox jumps over the lazy cog")
= 098cea62 0b0978ca a5f0befb a6ddcf22 764bea97 7e1c70b3 483edfdf 1de25f4b

The hash of the zero-length message is:

SHA384("")
= 38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743 4c0cc7bf 63f6e1da

### SHA-512 hashes

SHA512("The quick brown fox jumps over the lazy dog")
= 07e547d9 586f6a73 f73fbac0 435ed769 51218fb7 d0c8d788 a309d785 436bbb64
2e93a252 a954f239 12547d1e 8a3b5ed6 e1bfd709 7821233f a0538f3d b854fee6
// Avalanche effect when changing the last word to "cog":
SHA512("The quick brown fox jumps over the lazy cog")
= 3eeee1d0 e11733ef 152a6c29 503b3ae2 0c4f1f3c da4cb26f 1bc1a41f 91c7fe4a
b3bd8649 4049e201 c4bd5155 f31ecb7a 3c860684 3c4cc8df cab7da11 c8ae5045

The hash of the zero-length message is:

SHA512("")
= cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc 83f4a921 d36ce9ce
47d0d13c 5d85f2b0 ff8318d2 877eec2f 63b931bd 47417a81 a538327a f927da3e

## Official validation

Main article: CMVP
Implementations of all FIPS-approved security functions can be officially validated through the CMVP program, jointly run by the National Institute of Standards and Technology (NIST) and the Communications Security Establishment (CSE). For informal verification, a package to generate a high number of test vectors is made available for download on the NIST site; the resulting verification however does not replace in any way the formal CMVP validation, which is required by law for certain applications.

As of October 2006, there are more than 500 validated implementations of SHA-1, with fewer than ten of them capable of handling messages with a length in bits not a multiple of eight (see SHS Validation List). It is also important to note that some implementations available on the Internet do not digest the NIST validation vectors correctly, although they may correctly process the examples listed in the SHA-1 standard.

## SHA-1 algorithm

Pseudocode for the SHA-1 algorithm follows:

Note: All variables are unsigned 32 bits and wrap modulo 232 when calculating

Initialize variables:
h0 := 0x67452301
h1 := 0xEFCDAB89
h3 := 0x10325476
h4 := 0xC3D2E1F0

Pre-processing:
append the bit '1' to the message
append k bits '0', where k is the minimum number >= 0 such that the resulting message
length (in bits) is congruent to 448 (mod 512)
append length of message (before pre-processing), in bits, as 64-bit big-endian integer

Process the message in successive 512-bit chunks:
break message into 512-bit chunks
for each chunk
break chunk into sixteen 32-bit big-endian words w[i], 0 ≤ i ≤ 15

Extend the sixteen 32-bit words into eighty 32-bit words:
for i from 16 to 79
w[i] := (w[i-3] xor w[i-8] xor w[i-14] xor w[i-16]) leftrotate 1

Initialize hash value for this chunk:
a := h0
b := h1
c := h2
d := h3
e := h4

Main loop:
for i from 0 to 79
if 0 ≤ i ≤ 19 then
f := (b and c) or ((not b) and d)
k := 0x5A827999
else if 20 ≤ i ≤ 39
f := b xor c xor d
k := 0x6ED9EBA1
else if 40 ≤ i ≤ 59
f := (b and c) or (b and d) or (c and d)
k := 0x8F1BBCDC
else if 60 ≤ i ≤ 79
f := b xor c xor d
k := 0xCA62C1D6

temp := (a leftrotate 5) + f + e + k + w[i]
e := d
d := c
c := b leftrotate 30
b := a
a := temp

Add this chunk's hash to result so far:
h0 := h0 + a
h1 := h1 + b
h2 := h2 + c
h3 := h3 + d
h4 := h4 + e

Produce the final hash value (big-endian):
digest = hash = h0 append h1 append h2 append h3 append h4
Instead of the formulation from the original FIPS PUB 180-1 shown, the following equivalent expressions may be used to compute
f
in the main loop above:
(0  ≤ i ≤ 19): f := d xor (b and (c xor d))                (alternative 1)
(0  ≤ i ≤ 19): f := (b and c) xor ((not b) and d)          (alternative 2)

(40 ≤ i ≤ 59): f := (b and c) or (d and (b or c))          (alternative 1)
(40 ≤ i ≤ 59): f := (b and c) or (d and (b xor c))         (alternative 2)
(40 ≤ i ≤ 59): f := (b and c) + (d and (b xor c))          (alternative 3)
(40 ≤ i ≤ 59): f := (b and c) xor (b and d) xor (c and d)  (alternative 4)

## SHA-2 algorithm

Pseudocode for the SHA-256 algorithm follows. Note the great increase in mixing between bits of the
w[16..63]
words compared to SHA-1.
Note: All variables are unsigned 32 bits and wrap modulo 232 when calculating

Initialize variables
(first 32 bits of the fractional parts<em> of the square roots of the first 8 primes 2..19):</span>
h0 := 0x6a09e667
h1 := 0xbb67ae85
h2 := 0x3c6ef372
h3 := 0xa54ff53a
h4 := 0x510e527f
h5 := 0x9b05688c
h6 := 0x1f83d9ab
h7 := 0x5be0cd19

Initialize table of round constants
(first 32 bits of the <em>fractional parts</em> of the cube roots of the first 64 primes 2..311):
k[0..63] :=
0x428a2f98, 0x71374491, 0xb5c0fbcf, 0xe9b5dba5, 0x3956c25b, 0x59f111f1, 0x923f82a4, 0xab1c5ed5,
0xd807aa98, 0x12835b01, 0x243185be, 0x550c7dc3, 0x72be5d74, 0x80deb1fe, 0x9bdc06a7, 0xc19bf174,
0xe49b69c1, 0xefbe4786, 0x0fc19dc6, 0x240ca1cc, 0x2de92c6f, 0x4a7484aa, 0x5cb0a9dc, 0x76f988da,
0x983e5152, 0xa831c66d, 0xb00327c8, 0xbf597fc7, 0xc6e00bf3, 0xd5a79147, 0x06ca6351, 0x14292967,
0x27b70a85, 0x2e1b2138, 0x4d2c6dfc, 0x53380d13, 0x650a7354, 0x766a0abb, 0x81c2c92e, 0x92722c85,
0xa2bfe8a1, 0xa81a664b, 0xc24b8b70, 0xc76c51a3, 0xd192e819, 0xd6990624, 0xf40e3585, 0x106aa070,
0x19a4c116, 0x1e376c08, 0x2748774c, 0x34b0bcb5, 0x391c0cb3, 0x4ed8aa4a, 0x5b9cca4f, 0x682e6ff3,
0x748f82ee, 0x78a5636f, 0x84c87814, 0x8cc70208, 0x90befffa, 0xa4506ceb, 0xbef9a3f7, 0xc67178f2

Pre-processing:
append the bit '1' to the message
append k bits '0', where k is the minimum number >= 0 such that the resulting message
length (in bits) is congruent to 448 (mod 512)
append length of message (before pre-processing), in bits, as 64-bit big-endian integer

Process the message in successive 512-bit chunks:
break message into 512-bit chunks
for each chunk
break chunk into sixteen 32-bit big-endian words w[0..15]

Extend the sixteen 32-bit words into sixty-four 32-bit words:
for i from 16 to 63
s0 := (w[i-15] rightrotate 7) xor (w[i-15] rightrotate 18) xor (w[i-15] rightshift 3)
s1 := (w[i-2] rightrotate 17) xor (w[i-2] rightrotate 19) xor (w[i-2] rightshift 10)
w[i] := w[i-16] + s0 + w[i-7] + s1

Initialize hash value for this chunk:
a := h0
b := h1
c := h2
d := h3
e := h4
f := h5
g := h6
h := h7

Main loop:
for i from 0 to 63
s0 := (a rightrotate 2) xor (a rightrotate 13) xor (a rightrotate 22)
maj := (a and b) xor (a and c) xor (b and c)
t2 := s0 + maj
s1 := (e rightrotate 6) xor (e rightrotate 11) xor (e rightrotate 25)
ch := (e and f) xor ((not e) and g)
t1 := h + s1 + ch + k[i] + w[i]

h := g
g := f
f := e
e := d + t1
d := c
c := b
b := a
a := t1 + t2

Add this chunk's hash to result so far:
h0 := h0 + a
h1 := h1 + b
h2 := h2 + c
h3 := h3 + d
h4 := h4 + e
h5 := h5 + f
h6 := h6 + g
h7 := h7 + h

Produce the final hash value (big-endian):
digest = hash = h0 append h1 append h2 append h3 append h4 append h5 append h6 append h7
The
ch
and
maj
functions can be optimized the same way as described for SHA-1.

SHA-224 is identical to SHA-256, except that:

• the initial variable values
h0
through
h7
are different, and
• the output is constructed by omitting
h7
.

SHA-512 is identical in structure, but:

• all numbers are 64 bits long,
• there are 80 rounds instead of 64,
• the initial values and additive constants are extended to 64 bits, and
• the shift and rotate amounts used are different.

SHA-384 is identical to SHA-512, except that:

• the initial values
h0
through
h7
are different, and
• the output is constructed by omitting
h6
and
h7
.

## References

• Florent Chabaud, Antoine Joux: Differential Collisions in SHA-0. CRYPTO 1998. pp56–71
• Eli Biham, Rafi Chen, Near-Collisions of SHA-0, Cryptology ePrint Archive, Report 2004/146, 2004 (appeared on CRYPTO 2004) [1]
• Joux, Carribault, Lemuet, Jalby: Collision for the full SHA-0 algorithm, CRYPTO 2004 [2]
• Xiaoyun Wang, Hongbo Yu and Yiqun Lisa Yin, Efficient Collision Search Attacks on SHA-0, CRYPTO 2005 [3]
• Xiaoyun Wang, Yiqun Lisa Yin and Hongbo Yu, Finding Collisions in the Full SHA-1, CRYPTO 2005 [4]
• Henri Gilbert, Helena Handschuh: Security Analysis of SHA-256 and Sisters. Selected Areas in Cryptography 2003: pp175–193
• {{{author}}}, Federal Register, Proposed Revision of Federal Information Processing Standard (FIPS) 180, Secure Hash Standard

### Implementations

• The OpenSSL Project – The widely used OpenSSL
crypto
library includes free, open-source implementations of SHA-1, SHA-224, SHA-256, SHA-384, and SHA-512
• Crypto++ Crypto++ Library is a free C++ class library of cryptographic schemes.

### Test Vectors

The NESSIE project test vectors for SHA-1, SHA-256, SHA-384, and SHA-512.

### SHA1 reverse lookup databases

• SHA1search - reverses MD5 and SHA1 hashes (330 000+ strings)