2017/07/28 by Christian J. Berghoff, Berghoff, Christian J.
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic #math.NT #msc:11Y40
paper · pdf · doi:10.48550/arxiv.1707.09330
13 pages, follow-up to arXiv:1707.08075.pdf and https://arxiv.org/pdf/1707.08610.pdf
arxiv created 2017/07/28 · arxiv updated 2018/01/22
This work builds on earlier results. We define universal elliptic Gauß sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable representation in terms of the j-invariant and two other modular functions. We analyse the necessary computations in detail and derive an alternative approach for determining the trace of the Frobenius homomorphism for Atkin primes using these pre-computations. A rough run-time analysis shows, however, that this new method is not competitive with existing ones.