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

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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.

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Authors and Affiliations

  1. National Institute of Advanced Industrial Science and Technology, Central 2, 1-1-1, Umezono, Tsukuba, Ibaraki, Japan

    Reynald Affeldt

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  1. Reynald Affeldt
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Correspondence to Reynald Affeldt.

Additional information

A preliminary version of this work appeared in the proceedings of the 27th ACM SIGAPP Symposium On Applied Computing (SAC 2012), Software Verification and Testing Track [3].

Appendix: Additional assembly code

Appendix: Additional assembly code

This section provides for the sake of completeness assembly code that is explicitly referred to in the body of this paper. See [5] for other assembly code or formal proofs (see Figs. 18, 19, 20).

Fig. 18
figure 18

In-place signed–unsigned subtraction (appears in Fig. 5)

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Fig. 19
figure 19

Unsigned–unsigned addition (appears in Figs. 4 and 18)

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Fig. 20
figure 20

Assembly code for the main function of the binary extended gcd algorithm (see Fig. 10 for the corresponding pseudo-code)

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Affeldt, R. On construction of a library of formally verified low-level arithmetic functions. Innovations Syst Softw Eng 9, 59–77 (2013). https://doi.org/10.1007/s11334-013-0195-x

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