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Faster Computation of Tate Pairings
, 904
"... Abstract. This paper proposes new explicit formulas for the doubling and addition step in Miller’s algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by pr ..."
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Cited by 4 (0 self)
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Abstract. This paper proposes new explicit formulas for the doubling and addition step in Miller’s algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in the addition and doubling. Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also speed up pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairingfriendly Edwards curves.
Faster Computation of the Tate Pairing
"... This paper proposes new explicit formulas for the doubling and addition steps in Miller’s algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpreta ..."
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Cited by 1 (1 self)
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This paper proposes new explicit formulas for the doubling and addition steps in Miller’s algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller’s algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairingfriendly Edwards curves.
A remark on an article of S. Müller
, 2009
"... Proposition 1 Let E: y 2 = F(x) = x 3 + ax + b be an elliptic curve defined over Fp having three rational 2torsion points. If p ≡ 3 mod 4, there are no rational 4torsion points on E. Remember Swan’s theorem [3]. Let () a p denote the Legendre’s symbol. Theorem 2 Let f(X) be a squarefree polynomia ..."
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Proposition 1 Let E: y 2 = F(x) = x 3 + ax + b be an elliptic curve defined over Fp having three rational 2torsion points. If p ≡ 3 mod 4, there are no rational 4torsion points on E. Remember Swan’s theorem [3]. Let () a p denote the Legendre’s symbol. Theorem 2 Let f(X) be a squarefree polynomial of degree d and n its number of irreducible factors modulo p. Then