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Abstract. Since most public key cryptosystem primitives require the computation of modular exponentiation as their main building block, the problem of performing modular exponentiation efficiently has received considerable attention over the years. It is known that optimal (shortest) addition chains are the key mathematical concept for accomplishing modular exponentiations optimally. However, finding an optimal addition chain of length r is an NP-hard problem whose search space size is comparable to r!. In this contribution we explore the usage of a Genetic Algorithm (GA) approach for the problem of finding optimal (shortest) addition chains. We explain our GA strategy in detail reporting several promising experimental results that suggest that evolutionary algorithms may be a viable alternative to solve this illustrious problem in a quasi optimal fashion.

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In this contribution, we derive a novel parallel formulation of the standard Itoh-Tsujii algorithm for multiplicative inverse computation over GF(2 m ). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best performance when using a special class of irreducible trinomials, namely, P (X) = X m + X k + 1, with m and k odd numbers and when implemented in hardware platforms. Under these conditions, our experimental results show that our parallel version of the Itoh-Tsujii algorithm yields a speedup of about 30% when compared with the standard version of it. Implemented in a Virtex 3200E FPGA device, our design is able to compute multiplicative inversion over GF(2 193 ) after 20 clock cycles in about 0.94µS.

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