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

Abstract

Arithmetic functions are used in many important computer programs such as computer algebra systems and cryptographic software. The latter are critical applications whose correct implementation deserves to be formally guaranteed. They are also computation-intensive applications, so that programmers often resort to low-level assembly code to implement arithmetic functions. We propose an approach for the construction of a library of formally verified low-level arithmetic functions. To build our library, we first introduce a formalization of data structures for signed multi-precision arithmetic in low-level programs. We use this formalization to verify the implementation of several primitive arithmetic functions using Separation logic, an extension of Hoare logic to deal with pointers. Since 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 style of verification is illustrated with a concrete implementation of the binary extended gcd algorithm.