1,303 research outputs found

    Efficient algorithms for pairing-based cryptosystems

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    We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics.We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairing-based cryptography

    On the existence of distortion maps on ordinary elliptic curves

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    Distortion maps allow one to solve the Decision Diffie-Hellman problem on subgroups of points on the elliptic curve. In the case of ordinary elliptic curves over finite fields, it is known that in most cases there are no distortion maps. In this article we characterize the existence of distortion maps in the remaining cases.Comment: 3 Pages (Updated version corrects an error in the previous version

    Computing cardinalities of Q-curve reductions over finite fields

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    We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of Drew Sutherlan

    The Q-curve construction for endomorphism-accelerated elliptic curves

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    We give a detailed account of the use of Q\mathbb{Q}-curve reductions to construct elliptic curves over F_p2\mathbb{F}\_{p^2} with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when pp is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over F_p2\mathbb{F}\_{p^2} equipped with efficient endomorphisms for every p \textgreater{} 3, and exhibit examples of twist-secure curves over F_p2\mathbb{F}\_{p^2} for the efficient Mersenne prime p=21271p = 2^{127}-1.Comment: To appear in the Journal of Cryptology. arXiv admin note: text overlap with arXiv:1305.540

    On Using Expansions to the Base of -2

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    This short note investigates the effects of using expansions to the base of -2. The main applications we have in mind are cryptographic protocols, where the crucial operation is computation of scalar multiples. For the recently proposed groups arising from Picard curves this leads to a saving of at least 7% for the computation of an m-fold. For more general non-hyperelliptic genus 3 curves we expect a larger speed-up.Comment: 5 page
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