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Title:
Barreto-Naehrig Curve With Fixed Coefficient - Efficiently Constructing Pairing-Friendly Curves -
Authors: Masaaki Shirase
Abstract:This paper describes a method for constructing Barreto-Naehrig (BN) curves and twists of BN curves that are pairing-friendly and have the embedding degree 12 by using just primality tests without a complex multiplication (CM) method. Specifically, this paper explains that the number of points of elliptic curves y^2=x^3\pm 16 and y^2=x^3 \pm 2 over \Fp is given by 6 polynomials in z, n_0(z),\cdots, n_5(z), two of which are irreducible, classified by the value of z\bmod{12} for a prime p(z)=36z^4+36z^3+24z^2+6z+1 with z an integer. For example, elliptic curve y^2=x^3+2 over \Fp always becomes a BN curve for any z with z \equiv 2,11\!\!\!\pmod{12}. Let n_i(z) be irreducible. Then, to construct a pairing-friendly elliptic curve, it is enough to find an integer z of appropriate size such that p(z) and n_i(z) are primes.
ePrint: https://eprint.iacr.org/2010/134
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