## Generic Groups, Collision Resistance, and ECDSA (2002)

Venue: | Designs, Codes and Cryptography |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Brown02genericgroups,,

author = {Daniel R. L. Brown},

title = {Generic Groups, Collision Resistance, and ECDSA},

journal = {Designs, Codes and Cryptography},

year = {2002},

volume = {35},

pages = {119--152}

}

### OpenURL

### Abstract

Proved here is the sufficiency of certain conditions to ensure the Elliptic Curve Digital Signature Algorithm (ECDSA) existentially unforgeable by adaptive chosen-message attacks. The sufficient conditions include (i) a uniformity property and collision-resistance for the underlying hash function, (ii) pseudo-randomness in the private key space for the ephemeral private key generator, (iii) generic treatment of the underlying group, and (iv) a further condition on how the ephemeral public keys are mapped into the private key space. For completeness, a brief survey of necessary security conditions is also given. Some of the necessary conditions are weaker than the corresponding sufficient conditions used in the security proofs here, but others are identical.

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Citation Context ...es not do, and moreover, assume a condition on the hash function stronger than thesrst condition above. This work seems to be thesrst advance in the provable security of ECDSA. 1 Introduction Koblitz =-=[32]-=- and Miller [40] independently proposed elliptic curve cryptography in 1985. The National Institute of Standards and Technology (NIST) proposed the Digital Signature Algorithm (DSA) in 1991 (see [10])... |

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Citation Context ...oreover, assume a condition on the hash function stronger than thesrst condition above. This work seems to be thesrst advance in the provable security of ECDSA. 1 Introduction Koblitz [32] and Miller =-=[40]-=- independently proposed elliptic curve cryptography in 1985. The National Institute of Standards and Technology (NIST) proposed the Digital Signature Algorithm (DSA) in 1991 (see [10]). Vanstone [53] ... |

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Citation Context ...; y) : x; y 2 F q ; y 2 = x 3 +ax+bg if q is an odd prime or the set f(x; y) : x; y 2 F q ; y 2 + xy = x 3 + ax 2 + bg if q is a power of two. The group operations involve severalseld operations, see =-=[6, 31, 33, 38, 40, 47-=-] for example. Specic choices for the parameters q; a; b; n; t and G are recommended in [21, 47]. These recommendations help achieve eciency, security and interoperability. The ECDSA Conversion Functi... |

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Citation Context ...erty and an additional minor property. Our condition for the group is stronger than [44]: we use more than just the intractability of the elliptic curve discrete logarithm; instead we model the group =-=[48]-=- generically. On balance, our conditions are roughly equal 3 to [44], since one condition is weaker and the other ? Supported in part by a National Science and Engineering Research Council of Canada I... |

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Citation Context ...onfusion will result because our analysis will not focus on the DSA group specically. The DSA Conversion Function The DSA conversion function is f : hGi ! Zn : a 7! (a mod n). Nguyen and Sharplinksi [=-=42]-=- give partial results showing towards showing the DSA conversion function is almost-bijective. Experiments with small values of n show that the DSA conversion function is as almost-bijective of the sa... |

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Citation Context ...ns. For a more thorough treatment of the security issues surrounding the design of hash functions, see Damgard [17, 18], Menezes, van Oorschot and Vanstone [39], Preneel [45], Simon [50] and Stinson [=-=51, 5-=-2]. Some of our denitions are somewhat simplied to re ect the common practice of using asxed hash function in a signature scheme rather than the goal of designing a family of secure hash functions. Fo... |

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Citation Context ...; y) : x; y 2 F q ; y 2 = x 3 +ax+bg if q is an odd prime or the set f(x; y) : x; y 2 F q ; y 2 + xy = x 3 + ax 2 + bg if q is a power of two. The group operations involve severalseld operations, see =-=[6, 31, 33, 38, 40, 47-=-] for example. Specic choices for the parameters q; a; b; n; t and G are recommended in [21, 47]. These recommendations help achieve eciency, security and interoperability. The ECDSA Conversion Functi... |

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Citation Context ...ns. For a more thorough treatment of the security issues surrounding the design of hash functions, see Damgard [17, 18], Menezes, van Oorschot and Vanstone [39], Preneel [45], Simon [50] and Stinson [=-=51, 5-=-2]. Some of our denitions are somewhat simplied to re ect the common practice of using asxed hash function in a signature scheme rather than the goal of designing a family of secure hash functions. Fo... |

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Citation Context ...onsidered One can dene non-standard adversaries such as those thatsnd: an additional signature for an already signed message, an additional public-private key pair for a given message-signature pair [=-=7, 30, 3-=-7], attacks in the multi-user setting, and attacks depending on nonstandard modied key generation. We will not consider such adversaries here. 5.1 A Signature Scheme Generalizing ECDSA and DSA We dene... |

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Citation Context ...success probability 1 have runningtime about 2 80 . Thus it is reasonable to say that SHA-1 is collision-resistant of strength roughly (1; 2 80 ). For the purposes of ECDSA and DSA, however, Vaudenay =-=[54]-=- noted that collisions in the hash function h n;SHA-1 are what matters. Some measures are therefore needed to help ensure that the domain parameter n was not selected n = SHA-1(M 1 ) SHA-1(M 2 ) by a ... |

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Citation Context ... [40] independently proposed elliptic curve cryptography in 1985. The National Institute of Standards and Technology (NIST) proposed the Digital Signature Algorithm (DSA) in 1991 (see [10]). Vanstone =-=[53]-=- proposed an elliptic curve analogue of DSA, the Elliptic Curve Digital Signature Algorithm (ECDSA), in 1992. The current form of ECDSA was proposed by the IEEE P1363 working group in 1995. Koblitz de... |

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Citation Context ...0] give an excellent survey on ECDSA. Detailed specications of ECDSA are given in several approved standards: ISO 14888-3 [28], IEEE Std 1363-2000 [27], ANSI X9.62 [2], and FIPS 186-2 [21] (see also [=-=47]-=-). Applications of ECDSA are proposed in several other standards such as WAP WTLS, IETF S/MIME, IETF TLS, and IETF IPSEC IKE. A proof of conditional security ensures that a cryptographic technique mee... |

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Citation Context ...needed and D as large as needed. The pseudorandomness aspect of the above denition of uniformity is similar to a property called \entropysmoothing " by Shoup [49, page 20]. Schweinberger and Sho=-=up [46-=-] also use entropy-smoothing hash functions. 3.6 Relationships Between Hash Security Properties If the range of the hash-function is innite, then a collision-resistant hash function can fail to be one... |