On construction of a library of formally verified low-level arithmetic functions

Abstract

Most information security infrastructures rely on cryptography, which is usually implemented with low-level arithmetic functions. The formal verification of these functions therefore becomes a prerequisite to firmly assess any security property. We propose an approach for the construction of a library of formally verified low-level arithmetic functions that can be used to implement realistic cryptographic schemes in a trustful way. For that purpose, we introduce a formalization of data structures for signed multi-precision arithmetic and we experiment it with formal verification of basic functions, using Separation logic. Because this direct style of formal verification leads to technically involved specifications, we also propose for larger functions to show a formal simulation relation between pseudo-code and assembly. This is illustrated with the binary extended gcd algorithm.

How to Get an Efficient yet Verified Arbitrary-Precision Integer Library

“…Affeldt verified a binary GCD algorithm and the functions it depends on [2]. Neither multiplications nor divisions are present.…”

WhyMP, a formally verified arbitrary-precision integer library

“…Affeldt used Coq to verify a binary extended GCD algorithm implemented in a variant of MIPS assembly [1], as well as the basic arithmetic functions the algorithm depends on. The work uses GMP's number representation and a memory model based on separation logic.…”

“…As such, GMP has had its share of bugs. 1 We advocate using formal verification to ensure memory safety and the absence of correctness bugs for all inputs. GMP features several layers, each one handling different kinds of numbers.…”

Proof Pearl: A Verified Bignum Implementation in x86-64 Machine Code

“…The most closely related work on verified implementation of arithmetic functions is that of Affeldt [2], Fischer [6], Berghofer [4] and Moore [11]. We will also compare with the first author's early poster on this topic [13], and reflect on recent trends in programming logics for assembly verification.…”