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Specific Congruence Classes of Integer Parameters for Generating BLS Curves for Fast Pairings

Pairings are widely used for innovative protocols such as ID-based encryption and group signature authentication. According to the recent works, the Barreto-Lynn-Scott (BLS) family of pairing-friendly elliptic curves are suggested for the pairings at the various security levels. The BLS family has specific polynomial parameters in terms of an integer x0 for generating the pairing-friendly elliptic curves with various embedding degrees k, which are called BLS curves. The important fact is that one can find congruence classes of x0 which give rise to the BLS curves having a benefit of an efficient performing field arithmetics. However, except for the BLS curves with k = 24, such the practical usable congruence classes of x0 have not been provided at this time. In this manuscript, the authors provide the specific congruence classes generating the practical subfamilies of the BLS curves with k = 2i • 3 and 3j with arbitrary positive integers i and j.