A Formal Library for Elliptic Curves in the Coq Proof Assistant

International audienceA preliminary step towards the verification of elliptic curve cryptographic algorithms is the development of formal libraries with the corresponding mathematical theory. In this paper we present a formaliza-tion of elliptic curves theory, in the SSReflect extension of the Coq proof assistant. Our central contribution is a library containing many of the objects and core properties related to elliptic curve theory. We demonstrate the applicability of our library by formally proving a non-trivial property of elliptic curves: the existence of an isomorphism between a curve and its Picard group of divisors

Jasmin: high-assurance and high-speed cryptography

Jasmin is a framework for developing high-speed and high-assurance cryptographic software. The framework is structured around the Jasmin programming language and its compiler. The language is designed for enhancing portability of programs and for simplifying verification tasks. The compiler is designed to achieve predictability and effciency of the output code (currently limited to x64 platforms), and is formally verified in the Coq proof assistant. Using the supercop framework, we evaluate the Jasmin compiler on representative cryptographic routines and conclude that the code generated by the compiler is as efficient as fast, hand-crafted, implementations. Moreover, the framework includes highly automated tools for proving memory safety and constant-time security (for protecting against cache-based timing attacks). We also demonstrate the effectiveness of the verification tools on a large set of cryptographic routines.TEC4Growth - Pervasive Intelligence, Enhancers and Proofs of Concept with Industrial Impact/NORTE- 01-0145-FEDER- 000020info:eu-repo/semantics/publishedVersio

Computer-aided proofs for multiparty computation with active security

Secure multi-party computation (MPC) is a general cryptographic technique that allows distrusting parties to compute a function of their individual inputs, while only revealing the output of the function. It has found applications in areas such as auctioning, email filtering, and secure teleconference. Given its importance, it is crucial that the protocols are specified and implemented correctly. In the programming language community it has become good practice to use computer proof assistants to verify correctness proofs. In the field of cryptography, EasyCrypt is the state of the art proof assistant. It provides an embedded language for probabilistic programming, together with a specialized logic, embedded into an ambient general purpose higher-order logic. It allows us to conveniently express cryptographic properties. EasyCrypt has been used successfully on many applications, including public-key encryption, signatures, garbled circuits and differential privacy. Here we show for the first time that it can also be used to prove security of MPC against a malicious adversary. We formalize additive and replicated secret sharing schemes and apply them to Maurer's MPC protocol for secure addition and multiplication. Our method extends to general polynomial functions. We follow the insights from EasyCrypt that security proofs can be often be reduced to proofs about program equivalence, a topic that is well understood in the verification of programming languages. In particular, we show that in the passive case the non-interference-based definition is equivalent to a standard game-based security definition. For the active case we provide a new NI definition, which we call input independence

An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in Any Characteristic

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group