Analysis of Modular Arithmetic Markus Müller-Olm1 and Helmut Seidl2

Abstract. We consider integer arithmetic modulo a power of 2 as pro-vided by mainstream programming languages like Java or standard im-plementations of C. The difficulty here is that the ring Zm of integers modulo m = 2w, w> 1, has zero divisors and thus cannot be embedded into a field. Not withstanding that, we present intra- and inter-procedural algorithms for inferring for every program point u, affine relations be-tween program variables valid at u. Our algorithms are not only sound but also complete in that they detect all valid affine relations. Moreover, they run in time linear in the program size and polynomial in the number of program variables and can be implemented by using the same modular integer arithmetic as the target language to be analyzed.