Papers › The SEA algorithm for endomorphisms of supersingular elliptic curves
The SEA algorithm for endomorphisms of supersingular elliptic curves
Travis Morrison, Lorenz Panny, Jana Sotáková, Michael Wills
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For a prime p > 3 and a supersingular elliptic curve E defined over 𝔽_(p²) with j(E)∉{0,1728}, consider an endomorphism α of E represented as a composition of L isogenies of degree at most d. We prove that the trace of α may be computed in O(n⁴(logn)² + dLn³) bit operations, where n = log(p), using a generalization of the SEA algorithm for computing the trace of the Frobenius endomorphism of an ordinary elliptic curve. When L∈O(logp) and d∈O(1), this complexity matches the heuristic complexity of the SEA algorithm. Our theorem is unconditional, unlike the complexity analysis of the SEA algorithm, since the kernel of an arbitrary isogeny of a supersingular elliptic curve is defined over an extension of constant degree, independent of p. We also provide practical speedups, including a fast algorithm to compute the trace of α modulo p.
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