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49-526: The following is a simplified description of EdDSA, ignoring details of encoding integers and curve points as bit strings; the full details are in the papers and RFC. An EdDSA signature scheme is a choice: These parameters are common to all users of the EdDSA signature scheme. The security of the EdDSA signature scheme depends critically on the choices of parameters, except for the arbitrary choice of base point—for example, Pollard's rho algorithm for logarithms

98-408: A {\displaystyle a} , b {\displaystyle b} , A {\displaystyle A} , and B {\displaystyle B} such that α a β b = α A β B {\displaystyle \alpha ^{a}\beta ^{b}=\alpha ^{A}\beta ^{B}} . If the underlying group

147-552: A {\displaystyle a} , if x i ∈ S 2 {\displaystyle x_{i}\in S_{2}} then increment b {\displaystyle b} . Let G {\displaystyle G} be a cyclic group of order n {\displaystyle n} , and given α , β ∈ G {\displaystyle \alpha ,\beta \in G} , and

196-473: A collision , requires on average only 2 evaluations using a birthday attack . Some of the applications that use cryptographic hashes, such as password storage, are only minimally affected by a collision attack . Constructing a password that works for a given account requires a preimage attack, as well as access to the hash of the original password (typically in the shadow file) which may or may not be trivial. Reversing password encryption (e.g., to obtain

245-439: A hash function . If x i {\displaystyle x_{i}} is in S 0 {\displaystyle S_{0}} then double both a {\displaystyle a} and b {\displaystyle b} ; if x i ∈ S 1 {\displaystyle x_{i}\in S_{1}} then increment

294-446: A U.S. federal standard. The SHA-2 family of algorithms are patented in the U.S. The United States has released the patent under a royalty-free license. As of 2011, the best public attacks break preimage resistance for 52 out of 64 rounds of SHA-256 or 57 out of 80 rounds of SHA-512, and collision resistance for 46 out of 64 rounds of SHA-256. With the publication of FIPS PUB 180-2, NIST added three additional hash functions in

343-546: A hash security lower than 112 bits after 2013. The previous revision from 2007 specified the cutoff to be the end of 2010. In August 2012, NIST revised SP800-107 in the same manner. The NIST hash function competition selected a new hash function, SHA-3 , in 2012. The SHA-3 algorithm is not derived from SHA-2. The SHA-2 hash function is implemented in some widely used security applications and protocols, including TLS and SSL , PGP , SSH , S/MIME , and IPsec . The inherent computational demand of SHA-2 algorithms has driven

392-430: A length in bits not a multiple of eight while supporting both variants. Hash values of an empty string (i.e., a zero-length input text). Even a small change in the message will (with overwhelming probability) result in a different hash, due to the avalanche effect . For example, adding a period to the end of the following sentence changes approximately half (111 out of 224) of the bits in the hash, equivalent to picking

441-514: A method for generating initial values for truncated versions of SHA-512. Additionally, a restriction on padding the input data prior to hash calculation was removed, allowing hash data to be calculated simultaneously with content generation, such as a real-time video or audio feed. Padding the final data block must still occur prior to hash output. In July 2012, NIST revised SP800-57, which provides guidance for cryptographic key management. The publication disallowed creation of digital signatures with

490-495: A new hash at random: 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. The computation of the ch and maj values can be optimized the same way as described for SHA-1 . SHA-224 is identical to SHA-256, except that: SHA-512 is identical in structure to SHA-256, but: SHA-384 is identical to SHA-512, except that: SHA-512/t

539-590: A partition G = S 0 ∪ S 1 ∪ S 2 {\displaystyle G=S_{0}\cup S_{1}\cup S_{2}} , let f : G → G {\displaystyle f:G\to G} be the map and define maps g : G × Z → Z {\displaystyle g:G\times \mathbb {Z} \to \mathbb {Z} } and h : G × Z → Z {\displaystyle h:G\times \mathbb {Z} \to \mathbb {Z} } by Consider, for example,

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588-419: A password to try against a user's account elsewhere) is not made possible by the attacks. (However, even a secure password hash cannot prevent brute-force attacks on weak passwords .) In the case of document signing, an attacker could not simply fake a signature from an existing document—the attacker would have to produce a pair of documents, one innocuous and one damaging, and get the private key holder to sign

637-426: A period from late 2014 and early 2015. Similarly, Microsoft announced that Internet Explorer and Edge would stop honoring public SHA-1-signed TLS certificates from February 2017. Mozilla disabled SHA-1 in early January 2016, but had to re-enable it temporarily via a Firefox update, after problems with web-based user interfaces of some router models and security appliances . For a hash function for which L

686-465: A secret value called a nonce unique to each signature. In the signature schemes DSA and ECDSA , this nonce is traditionally generated randomly for each signature—and if the random number generator is ever broken and predictable when making a signature, the signature can leak the private key, as happened with the Sony PlayStation 3 firmware update signing key. In contrast, EdDSA chooses

735-1000: Is a set of cryptographic hash functions designed by the United States National Security Agency (NSA) and first published in 2001. They are built using the Merkle–Damgård construction , from a one-way compression function itself built using the Davies–Meyer structure from a specialized block cipher. SHA-2 includes significant changes from its predecessor, SHA-1 . The SHA-2 family consists of six hash functions with digests (hash values) that are 224, 256, 384 or 512 bits: SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224, SHA-512/256 . SHA-256 and SHA-512 are novel hash functions whose digests are eight 32-bit and 64-bit words, respectively. They use different shift amounts and additive constants, but their structures are otherwise virtually identical, differing only in

784-423: Is a solution as expected. As n = 1018 {\displaystyle n=1018} is not prime , there is another solution γ 2 = 519 {\displaystyle \gamma _{2}=519} , for which 2 519 = 1014 = − 5 ( mod 1019 ) {\displaystyle 2^{519}=1014=-5{\pmod {1019}}} holds. The running time

833-630: Is approximately O ( n ) {\displaystyle {\mathcal {O}}({\sqrt {n}})} . If used together with the Pohlig–Hellman algorithm , the running time of the combined algorithm is O ( p ) {\displaystyle {\mathcal {O}}({\sqrt {p}})} , where p {\displaystyle p} is the largest prime factor of n {\displaystyle n} . SHA-512 Pseudo-collision attack against up to 46 rounds of SHA-256. SHA-2 ( Secure Hash Algorithm 2 )

882-698: Is assumed to be random-looking and thus is likely to enter into a loop of approximate length π n 8 {\displaystyle {\sqrt {\frac {\pi n}{8}}}} after π n 8 {\displaystyle {\sqrt {\frac {\pi n}{8}}}} steps. One way to define such a function is to use the following rules: Partition G {\displaystyle G} into three disjoint subsets S 0 {\displaystyle S_{0}} , S 1 {\displaystyle S_{1}} , and S 2 {\displaystyle S_{2}} of approximately equal size using

931-486: Is cyclic of order n {\displaystyle n} , by substituting β {\displaystyle \beta } as α γ {\displaystyle {\alpha }^{\gamma }} and noting that two powers are equal if and only if the exponents are equivalent modulo the order of the base, in this case modulo n {\displaystyle n} , we get that γ {\displaystyle \gamma }

980-413: Is expected to take approximately ℓ π / 4 {\displaystyle {\sqrt {\ell \pi /4}}} curve additions before it can compute a discrete logarithm, so ℓ {\displaystyle \ell } must be large enough for this to be infeasible, and is typically taken to exceed 2 . The choice of ℓ {\displaystyle \ell }

1029-573: Is identical to SHA-512 except that: The SHA-512/t IV generation function evaluates a modified SHA-512 on the ASCII string "SHA-512/ t ", substituted with the decimal representation of t . The modified SHA-512 is the same as SHA-512 except its initial values h0 through h7 have each been XORed with the hexadecimal constant 0xa5a5a5a5a5a5a5a5 . Sample C implementation for SHA-2 family of hash functions can be found in RFC ; 6234 . In

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1078-467: Is known as edwards25519 , and is birationally equivalent to the Montgomery curve known as Curve25519 . The equivalence is x = u v − 486664 , y = u − 1 u + 1 . {\displaystyle x={\frac {u}{v}}{\sqrt {-486664}},\quad y={\frac {u-1}{u+1}}.} The original team has optimized Ed25519 for

1127-477: Is limited by the choice of q {\displaystyle q} , since by Hasse's theorem , # E ( F q ) = 2 c ℓ {\displaystyle \#E(\mathbb {F} _{q})=2^{c}\ell } cannot differ from q + 1 {\displaystyle q+1} by more than 2 q {\displaystyle 2{\sqrt {q}}} . The hash function H {\displaystyle H}

1176-1010: Is normally modelled as a random oracle in formal analyses of EdDSA's security. Within an EdDSA signature scheme, 2 c S B = 2 c R + 2 c H ( R ∥ A ∥ M ) A . {\displaystyle 2^{c}SB=2^{c}R+2^{c}H(R\parallel A\parallel M)A.} 2 c S B = 2 c ( r + H ( R ∥ A ∥ M ) s ) B = 2 c r B + 2 c H ( R ∥ A ∥ M ) s B = 2 c R + 2 c H ( R ∥ A ∥ M ) A . {\displaystyle {\begin{aligned}2^{c}SB&=2^{c}(r+H(R\parallel A\parallel M)s)B\\&=2^{c}rB+2^{c}H(R\parallel A\parallel M)sB\\&=2^{c}R+2^{c}H(R\parallel A\parallel M)A.\end{aligned}}} Ed25519

1225-517: Is one of the solutions of the equation ( B − b ) γ = ( a − A ) ( mod n ) {\displaystyle (B-b)\gamma =(a-A){\pmod {n}}} . Solutions to this equation are easily obtained using the extended Euclidean algorithm . To find the needed a {\displaystyle a} , b {\displaystyle b} , A {\displaystyle A} , and B {\displaystyle B}

1274-459: Is the EdDSA signature scheme using SHA-512 (SHA-2) and an elliptic curve related to Curve25519 where − x 2 + y 2 = 1 − 121665 121666 x 2 y 2 , {\displaystyle -x^{2}+y^{2}=1-{\frac {121665}{121666}}x^{2}y^{2},} The twisted Edwards curve E / F q {\displaystyle E/\mathbb {F} _{q}}

1323-402: Is the number of bits in the message digest , finding a message that corresponds to a given message digest can always be done using a brute force search in 2 evaluations. This is called a preimage attack and may or may not be practical depending on L and the particular computing environment. The second criterion, finding two different messages that produce the same message digest, known as

1372-845: 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 the formal CMVP validation, which is required by law for certain applications. As of December 2013, there are over 1300 validated implementations of SHA-256 and over 900 of SHA-512, with only 5 of them being capable of handling messages with

1421-494: The integer factorization problem. The goal is to compute γ {\displaystyle \gamma } such that α γ = β {\displaystyle \alpha ^{\gamma }=\beta } , where β {\displaystyle \beta } belongs to a cyclic group G {\displaystyle G} generated by α {\displaystyle \alpha } . The algorithm computes integers

1470-533: The x86-64 Nehalem / Westmere processor family. Verification can be performed in batches of 64 signatures for even greater throughput. Ed25519 is intended to provide attack resistance comparable to quality 128-bit symmetric ciphers . Public keys are 256 bits long and signatures are 512 bits long. Ed25519 is designed to avoid implementations that use branch conditions or array indices that depend on secret data, in order to mitigate side-channel attacks . As with other discrete-log-based signature schemes, EdDSA uses

1519-427: The 'x86-64' numbers are native 64-bit code. While SHA-256 is designed for 32-bit calculations, it does benefit from code optimized for 64-bit processors on the x86 architecture. 32-bit implementations of SHA-512 are significantly slower than their 64-bit counterparts. Variants of both algorithms with different output sizes will perform similarly, since the message expansion and compression functions are identical, and only

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1568-530: The SHA family. The algorithms are collectively known as SHA-2, named after their digest lengths (in bits): SHA-256, SHA-384, and SHA-512. The algorithms were first published in 2001 in the draft FIPS PUB 180-2, at which time public review and comments were accepted. In August 2002, FIPS PUB 180-2 became the new Secure Hash Standard , replacing FIPS PUB 180-1, which was released in April 1995. The updated standard included

1617-454: The U.S. National Institute of Standards and Technology says, "Federal agencies should stop using SHA-1 for...applications that require collision resistance as soon as practical, and must use the SHA-2 family of hash functions for these applications after 2010" (emphasis in original). NIST's directive that U.S. government agencies ought to, but not explicitly must, stop uses of SHA-1 after 2010

1666-405: The algorithm uses Floyd's cycle-finding algorithm to find a cycle in the sequence x i = α a i β b i {\displaystyle x_{i}=\alpha ^{a_{i}}\beta ^{b_{i}}} , where the function f : x i ↦ x i + 1 {\displaystyle f:x_{i}\mapsto x_{i+1}}

1715-855: The group generated by 2 modulo N = 1019 {\displaystyle N=1019} (the order of the group is n = 1018 {\displaystyle n=1018} , 2 generates the group of units modulo 1019). The algorithm is implemented by the following C++ program: The results are as follows (edited): That is 2 681 5 378 = 1010 = 2 301 5 416 ( mod 1019 ) {\displaystyle 2^{681}5^{378}=1010=2^{301}5^{416}{\pmod {1019}}} and so ( 416 − 378 ) γ = 681 − 301 ( mod 1018 ) {\displaystyle (416-378)\gamma =681-301{\pmod {1018}}} , for which γ 1 = 10 {\displaystyle \gamma _{1}=10}

1764-555: The hash function SHAKE256 and the elliptic curve edwards448 , an (untwisted) Edwards curve related to Curve448 in RFC   7748 . Ed448 has also been approved in the final version of the FIPS 186-5 standard. Pollard%27s rho algorithm for logarithms Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve

1813-515: The initial hash values and output sizes are different. The best implementations of MD5 and SHA-1 perform between 4.5 and 6 cycles per byte on modern processors. Testing was performed by the University of Illinois at Chicago on their hydra8 system running an Intel Xeon E3-1275 V2 at a clock speed of 3.5 GHz, and on their hydra9 system running an AMD A10-5800K APU at a clock speed of 3.8 GHz. The referenced cycles per byte speeds above are

1862-452: The innocuous document. There are practical circumstances in which this is possible; until the end of 2008, it was possible to create forged SSL certificates using an MD5 collision which would be accepted by widely used web browsers. Increased interest in cryptographic hash analysis during the SHA-3 competition produced several new attacks on the SHA-2 family, the best of which are given in

1911-501: The nonce deterministically as the hash of a part of the private key and the message. Thus, once a private key is generated, EdDSA has no further need for a random number generator in order to make signatures, and there is no danger that a broken random number generator used to make a signature will reveal the private key. Note that there are two standardization efforts for EdDSA, one from IETF, an informational RFC   8032 and one from NIST as part of FIPS 186-5. The differences between

1960-529: The number of rounds. SHA-224 and SHA-384 are truncated versions of SHA-256 and SHA-512 respectively, computed with different initial values. SHA-512/224 and SHA-512/256 are also truncated versions of SHA-512, but the initial values are generated using the method described in Federal Information Processing Standards (FIPS) PUB 180-4. SHA-2 was first published by the National Institute of Standards and Technology (NIST) as

2009-511: The original SHA-1 algorithm, with updated technical notation consistent with that describing the inner workings of the SHA-2 family. 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 . In October 2008, the standard was updated in FIPS PUB 180-3, including SHA-224 from the change notice, but otherwise making no fundamental changes to

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2058-967: The proposal of more efficient solutions, such as those based on application-specific integrated circuits (ASICs) hardware accelerators. SHA-256 is used for authenticating Debian software packages and in the DKIM message signing standard; SHA-512 is part of a system to authenticate archival video from the International Criminal Tribunal of the Rwandan genocide . SHA-256 and SHA-512 are proposed for use in DNSSEC . Unix and Linux vendors are moving to using 256- and 512-bit SHA-2 for secure password hashing. Several cryptocurrencies , including Bitcoin , use SHA-256 for verifying transactions and calculating proof of work or proof of stake . The rise of ASIC SHA-2 accelerator chips has led to

2107-400: The standard. The primary motivation for updating the standard was relocating security information about the hash algorithms and recommendations for their use to Special Publications 800-107 and 800-57. Detailed test data and example message digests were also removed from the standard, and provided as separate documents. In January 2011, NIST published SP800-131A, which specified a move from

2156-549: The standards have been analyzed, and test vectors are available. Notable uses of Ed25519 include OpenSSH , GnuPG and various alternatives, and the signify tool by OpenBSD . Usage of Ed25519 (and Ed448) in the SSH protocol has been standardized. In 2023 the final version of the FIPS 186-5 standard included deterministic Ed25519 as an approved signature scheme. Ed448 is the EdDSA signature scheme defined in RFC   8032 using

2205-509: The table below, internal state means the "internal hash sum" after each compression of a data block. In the bitwise operations column, "Rot" stands for rotate no carry , and "Shr" stands for right logical shift . All of these algorithms employ modular addition in some fashion except for SHA-3. More detailed performance measurements on modern processor architectures are given in the table below. The performance numbers labeled 'x86' were running using 32-bit code on 64-bit processors, whereas

2254-445: The table below. Only the collision attacks are of practical complexity; none of the attacks extend to the full round hash function. At FSE 2012, researchers at Sony gave a presentation suggesting pseudo-collision attacks could be extended to 52 rounds on SHA-256 and 57 rounds on SHA-512 by building upon the biclique pseudo-preimage attack. Implementations of all FIPS-approved security functions can be officially validated through

2303-432: The then-current minimum of 80-bit security (provided by SHA-1) allowable for federal government use until the end of 2013, to 112-bit security (provided by SHA-2) being both the minimum requirement (starting in 2014) and the recommended security level (starting from the publication date in 2011). In March 2012, the standard was updated in FIPS PUB 180-4, adding the hash functions SHA-512/224 and SHA-512/256, and describing

2352-512: The use of scrypt -based proof-of-work schemes. SHA-1 and SHA-2 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. SHA-1 is being retired for most government uses;

2401-445: Was hoped to accelerate migration away from SHA-1. The SHA-2 functions were not quickly adopted initially, despite better security than SHA-1. Reasons might include lack of support for SHA-2 on systems running Windows XP SP2 or older and a lack of perceived urgency since SHA-1 collisions had not yet been found. The Google Chrome team announced a plan to make their web browser gradually stop honoring SHA-1-dependent TLS certificates over

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