Papers › An Elementary Method For Fast Modular Exponentiation With Factored Modulus
An Elementary Method For Fast Modular Exponentiation With Factored Modulus
Anay Aggarwal, Manu Isaacs
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We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let a,n,m be positive integers and suppose m factors canonically as ∏ᵢ₌₁ᵏ pᵢ^(eᵢ). Choose integer parameters tᵢ∈[1, eᵢ] for 1≤i≤k. Then we can compute the modular exponentiation aⁿm in O(max(eᵢ/tᵢ)+∑ᵢ₌₁ᵏ tᵢlogpᵢ) steps (i.e., modular operations). We go on to analyze this algorithm mathematically and programmatically, showing significant asymptotic improvement in specific cases. Specifically, for an infinite family of m we achieve a complexity of O(√(logm)) steps, much faster than the Repeated Squaring Algorithm, which has complexity O(logm). Additionally, we extend our algorithm to matrices and hence general linear recurrences. The complexity is similar; with the same setup we can exponentiate matrices in GL_d(ℤ/mℤ) in less than O(max(eᵢ/tᵢ)+d²∑ᵢ₌₁ᵏ tᵢlogpᵢ) steps. This improves Fiduccia's algorithm and the results of Bostan and Mori in the case of ℤ/mℤ. We prove analogous results for ℤ/pᵏℤ ring extensions.
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