70 research outputs found
Optimal TNFS-secure pairings on elliptic curves with composite embedding degree
In this paper we present a comprehensive comparison between pairing-friendly elliptic curves, considering di erent curve forms and twists where possible. We de ne an additional measure of the e- ciency of a parametrized pairing-friendly family that takes into account the number eld sieve (NFS) attacks (unlike the -value). This measure includes an approximation of the security of the discrete logarithm problem in F pk , computed via the method of Barbulescu and Duquesne [4]. We compute the security of the families presented by Fotiadis and Konstantinou in [14], compute some new families, and compare the eciency of both of these with the (adjusted) BLS, KSS, and BN families, and with the new families of [20]. Finally, we recommend pairing-friendly elliptic curves for security levels 128 and 192
Pairing Computation on Elliptic Curves of Jacobi Quartic Form
This paper proposes explicit formulae for the addition step and
doubling step in Miller\u27s algorithm to compute Tate pairing on
Jacobi quartic curves.
We present a geometric interpretation of the group law on Jacobi
quartic curves, %and our formulae for Miller\u27s
%algorithm come from this interpretation.
which leads to formulae for Miller\u27s algorithm. The doubling step
formula is competitive with that for Weierstrass curves and Edwards
curves. Moreover, by carefully choosing the coefficients, there
exist quartic twists of Jacobi quartic curves from which pairing
computation can benefit a lot. Finally, we provide some examples of
supersingular and ordinary pairing friendly Jacobi quartic curves
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Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction
Pairing computation on Edwards curves with high-degree twists
In this paper, we propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a littler faster than that proposed by Arene et.al.. Finally, to improve the efficiency of pairing computation we present twists of degree 4 and 6 on twisted Edwards curves
The Pairing Computation on Edwards Curves
We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric
interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves
Archimedean local height differences on elliptic curves
To compute generators for the Mordell-Weil group of an elliptic curve over a
number field, one needs to bound the difference between the naive and the
canonical height from above. We give an elementary and fast method to compute
an upper bound for the local contribution to this difference at an archimedean
place, which sometimes gives better results than previous algorithms.Comment: 10 pages, comments welcom
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