Solving systems of differential equations of addition

Mixing addition modulo 2(n) (+) and exclusive-or (E)) have a host of applications in symmetric cryptography as the operations are fast and nonlinear over GF(2). We deal with a frequently encountered equation (x + y) circle plus ((x circle plus alpha) + (y circle plus beta)) = gamma. The difficulty of solving an arbitrary system of such equations - named differential equations of addition (DEA) - is an important consideration in the evaluation of the security of many ciphers against differential attacks. This paper shows that the satisfiability of an arbitrary set of DEA - which has so far been assumed hard for large n - is in the complexity class P. We also design an efficient algorithm to obtain all solutions to an arbitrary system of DEA with running time linear in the number of solutions. Our second contribution is solving DEA in an adaptive query model where an equation is formed by a query (alpha, beta) and oracle output gamma. The challenge is to optimize the number of queries to solve (x + y) circle plus ((x circle plus alpha) + (y circle plus beta)) = gamma. Our algorithm solves this equation with only 3 queries in the worst case. Another algorithm solves the equation (x+y) circle plus (x + (y circle plus beta)) = gamma with (n - t - 1) queries in the worst case (t is the position of the least significant '1' of x), and thus, outperforms the previous best known algorithm by Muller - presented at FSE '04 - which required 3(n - 1) queries. Most importantly, we show that the upper bounds, for our algorithms, on the number of queries match worst case lower bounds. This, essentially, closes further research in this direction as our lower bounds are optimal. Finally we describe applications of our results in differential cryptanalysis.