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Verified Indifferentiable Hashing into Elliptic Curves
"... Abstract. Many cryptographic systems based on elliptic curves are proven secure in the Random Oracle Model, assuming there exist probabilistic functions that map elements in some domain (e.g. bitstrings) onto uniformly and independently distributed points in a curve. When implementing such systems, ..."
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Abstract. Many cryptographic systems based on elliptic curves are proven secure in the Random Oracle Model, assuming there exist probabilistic functions that map elements in some domain (e.g. bitstrings) onto uniformly and independently distributed points in a curve. When implementing such systems, and in order for the proof to carry over to the implementation, those mappings must be instantiated with concrete constructions whose behavior does not deviate significantly from random oracles. In contrast to other approaches to publickey cryptography, where candidates to instantiate random oracles have been known for some time, the first generic construction for hashing into ordinary elliptic curves indifferentiable from a random oracle was put forward only recently by Brier et al. We present a machinechecked proof of this construction. The proof is based on an extension of the CertiCrypt framework with logics and mechanized tools for reasoning about approximate forms of observational equivalence, and integrates mathematical libraries of group theory and elliptic curves. 1
ProjectTeam CACAO Curves, Algebra, Computer Arithmetic, and so On
"... c t i v it y e p o r t 2009 Table of contents ..."
5. New Results.............................................................................. 3
"... c t i v it y e p o r t 2007 Table of contents ..."
A language of patterns for subterm selection
, 2012
"... Abstract. This paper describes the language of patterns that equips the SSReflect proof shell extension for the Coq system. Patterns are used to focus proof commands on subexpressions of the conjecture under analysis in a declarative manner. They are designed to ease the writing of proof scripts and ..."
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Abstract. This paper describes the language of patterns that equips the SSReflect proof shell extension for the Coq system. Patterns are used to focus proof commands on subexpressions of the conjecture under analysis in a declarative manner. They are designed to ease the writing of proof scripts and to increase their readability and maintainability. A pattern can identify the subexpression of interest approximating the subexpression itself, or its enclosing context or both. The user is free to choose the most convenient option. Patterns are matched following an extremely precise and predictable discipline, that is carefully designed to admit an efficient implementation. In this paper we report on the language of patterns, its matching algorithm and its usage in the formal library developed by the Mathematical Components team to support the verification of the Odd Order Theorem. 1
Certificates Based on Hensel’s Lifting
, 2011
"... If it is quite easy to check a given integer is a root of a given polynomial with integer coefficients, verifying we know all the integral roots of a polynomial requires a different approach. In both univariate and bivariate cases, we introduce a type of integral roots certificates and the correspon ..."
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If it is quite easy to check a given integer is a root of a given polynomial with integer coefficients, verifying we know all the integral roots of a polynomial requires a different approach. In both univariate and bivariate cases, we introduce a type of integral roots certificates and the corresponding checker specification, based on Hensel’s lifting. We provide a formalization of this iterative algorithm from which we deduce a formal proof of the correctness of the checkers, with the help of the Coq proof assistant along with the SSReflect extension. The ultimate goal of this work is to provide a component that will be involved in a complete certification chain for solving the Table Maker’s Dilemma in an exact way.