## Faster computation of the Tate pairing.(English)Zbl 1222.14069

The paper proposes some improvements in the computation of the Tate pairing on elliptic curves $$E$$, both in Weierstrass form and in Edwards form, curves defined over a non-binary finite field $$F_q$$ and with even embedding degree (for a prime $$n|\sharp(E)$$ the embedding degree $$k$$, with respect to $$n$$, is the multiplicative degree of $$q$$ modulo $$n$$).
For $$E$$ in Weierstrass form, Miller’s algorithm computes efficiently the Tate pairing, using the chord-and-tangent method for the addition and doubling of points. Section 3 presents new formulas for the addition and doubling steps in Miller’s algorithm. Those formulas use a representation of the points of $$E$$ in Jacobian coordinates $$(X:Y:Z:T)$$, $$T^2=Z$$, and the paper gives its cost in term of the costs $$m,s,M,S$$ of multiplication and squaring in $$\mathbb F_q$$ and $$\mathbb F_{q^k}$$.
A twisted Edwards curve, introduced by D. J. Bernstein et al. [in: AFRICACRYPT 2008. Casablanca, Morocco, 2008. Lect. Notes Comput. Sci. 5023, 389–405 (2008; Zbl 1142.94332)], is a curve giving by an equation: $$E_{\text{ad}}: ax^2+y^2=1+dx^2y^2$$, whose (affine) points have efficient addition formulas. Since the equation of $$E_{\text{ad}}$$ has degree four the chord-and-tangent geometric interpretation of the addition is not more valid, but section 4 of the paper gives (theorem 2) a new geometric interpretation of the addition law for $$E_{\text{ad}}$$, and with this tool section 5 shows how to compute Tate pairing on twisted Edwards curves.
Section 6 gives the comparison of the proposed formulas with others in the literature, as the paper of S. Ionica and A. Joux [in: INDOCRYPT 2008. Kharagpur, India, 2008. Lect. Notes Comput. Sci. 5365, 400–413 (2008; Zbl 1203.94104)], concluding that “ ... our new formulas for Edwards curves solidly beat all previous formulas published for Tate computation on Edwards curves” and “ Our new formulas for pairings on arbitrary Edwards curves are faster than all formulas previously known for Weierstrass curves except for the very special curves with $$a_4=0$$.”.
Finally, sections 7 and 8 present construction and numerical examples (with embedding degree $$k=6,8,10,22$$) of pairing-friendly Edwards curves, examples covering the most common security levels.

### MSC:

 11Y16 Number-theoretic algorithms; complexity 11G20 Curves over finite and local fields 14G50 Applications to coding theory and cryptography of arithmetic geometry 94A60 Cryptography

### Citations:

Zbl 1142.94332; Zbl 1203.94104

EFD; SageMath
Full Text:

### References:

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