The field of high-assurance cryptography is quickly maturing, yet a unified foundational framework for end-to-end formal verification of efficient cryptographic implementations is still missing. To address this gap, we use the Coq proof assistant to formally connect three existing tools: (1) the Hacspec emergent cryptographic specification language; (2) the Jasmin language for efficient, high-assurance cryptographic implementations; and (3) the SSProve foundational verification framework for modular cryptographic proofs. We first connect Hacspec with SSProve by devising a new translation from Hacspec specifications to imperative SSProve code. We validate this translation by considering a second, more standard translation from Hacspec to purely functional Coq code and generate a proof of the equivalence between the code produced by the two translations. We further define a translation from Jasmin to SSProve, which allows us to formally reason in SSProve about efficient cryptographic implementations in Jasmin. We prove this translation correct in Coq with respect to Jasmin's operational semantics. Finally, we demonstrate the usefulness of our approach by giving a foundational end-to-end Coq proof of an efficient AES implementation. For this case study, we start from an existing Jasmin implementation of AES that makes use of hardware acceleration and prove that it conforms to a specification of the AES standard written in Hacspec. We use SSProve to formalize the security of the encryption scheme based on the Jasmin implementation of AES.

We generalize the Bernstein-Yang (BY) algorithm [11] for constant-time modular inversion to compute the Kronecker symbol, of which the Jacobi and Legendre symbols are special cases. We first develop a basic and easy-to-implement algorithm, defined with full-precision division steps. We then describe an optimized version due to Hamburg [21] over word-sized inputs, and formally verify its correctness. Along the way, we introduce a number of optimizations for implementing both versions in constant time. The resulting algorithms are particularly suitable for computing the Legendre symbol with dense prime p, where no efficient addition chain is known for exponentiating to p-1 over 2, as it is often the case in pairing-friendly elliptic curves. Our high-speed implementation for a range of parameters shows that the new algorithm is up to 40 times faster than exponentiation, and up to 25.7% faster than the previous state of the art. We illustrate our techniques with hashing to elliptic curves using the SwiftEC algorithm [17], with savings of 14.7%-48.1%, and to accelerating the CTIDH isogeny-based key exchange [7], with savings of 3.5-13.5%.

The security of modern cryptography depends on multiple factors, from sound hardness assumptions to correct implementations that resist side-channel cryptanalysis. Curve-based cryptography is not different in this regard, and substantial progress in the last few decades has been achieved in both selecting parameters and devising secure implementation strategies. In this context, the security of implementations of field inversion is sometimes overlooked in the research literature, because (i) the approach based on Fermat's Little Theorem (FLT) suffices performance-wise for many parameters used in practice; (ii) it is typically invoked only at the very end of a cryptographic computation, with a small impact on performance; (iii) it is challenging to implement securely for general parameters without a significant performance penalty. However, field inversion can process sensitive information and must be protected with side-channel countermeasures like any other cryptographic operation, as illustrated by recent attacks [1]–[3]. In this work, we focus on implementing field inversion for primes of cryptographic interest with security against timing attacks, irrespective of whether the FLT-based inversion can be efficiently implemented. We extend the Fiat-Crypto framework, which synthesizes provably correct-by-construction implementations, to implement the Bernstein-Yang inversion algorithm as a step towards this goal. This allows a correct implementation of prime field inversion to be synthesized for any prime. We benchmark the implementations across a range of primes for curve-based cryptography and they outperform traditional FLT-based approaches in most cases, with observed speedups up to 2 for the largest parameters. Our work is already used in production in the MirageOS unikernel operating system, zig programming language, and the ECCKiila framework [4]