Algorithms for Exponentiation in Finite Fields

Algorithms for Exponentiation in Finite Fields

and infinitely many integers n, multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglogn), division with O(n log n loglogn) operations in Fq , and exponentiation of an arbitrary element in Fqn with O(n2 loglogn) operations in Fq . We also prove that using a polynomial basis exponentiation in F2n can be done with the same number of operations in F2 for all n. The pre...

Exponentiation Using Addition Chains for Finite Fields

Algorithms for Multi-exponentiation

This paper compares different approaches for computing power products ∏ 1≤i≤k g ei i in commutative groups. We look at the conventional simultaneous exponentiation approach and present an alternative strategy, interleaving exponentiation. Our comparison shows that in general groups, sometimes the conventional method and sometimes interleaving exponentiation is more efficient. In groups where in...

Exponentiation in Power Series Fields

We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a nonsurjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through t...

Self-Randomized Exponentiation Algorithms

Exponentiation is a central process in many public-key cryptosystems such as RSA and DH. This paper introduces the concept of self-randomized exponentiation as an efficient means for preventing DPAtype attacks. Self-randomized exponentiation features several interesting properties: – it is fully generic in the sense that it is not restricted to a particular exponentiation algorithm; – it is par...