Efficient Computation of Singular Moduli with Application in Cryptography
We present an implementation that turns out to be most efficient in practice to compute singular moduli within a fixed floating point precision. First, we show how to efficiently determine the Fourier coefficients of the modular function j and related functions γ2, f2, and η. Comparing several alternative methods for computing singular moduli, we show that in practice the computation via the η-function turns out to be the most efficient one. An important application with respect to cryptography is that we can speed up the generation of cryptographically strong elliptic curves using the Complex Multiplication Approach.
Keywordsclass group complex multiplication cryptography elliptic curve Fourier series modular function ring class polynomial singular modulus
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