Jacobi Quartic Curves Research Articles

A special scalar multiplication algorithm on Jacobi quartic curves

A special scalar multiplication algorithm on Jacobi quartic curves

Simplified isogeny formulas on twisted Jacobi quartic curves

Isogenies between elliptic curves play a very important role in elliptic curve related cryptosystems and cryptanalysis. It is widely known that different models of elliptic curves would induce different computational costs of elliptic curve arithmetic, and several works have been devoted to accelerate the isogeny computation on various curve models. This paper studies the case of the Jacobi quartic model, which is a classic form of elliptic curves. A new w-coordinate system on extended Jacobi quartic curves is introduced for Montgomery-like group arithmetic. Explicit formulas for 2-isogenies and odd ℓ-isogenies on the specific curves are presented, and based on the w-coordinate system, the computation of such isogenies could be further simplified.

How to Compute an Isogeny on the Extended Jacobi Quartic Curves?

How to Compute an Isogeny on the Extended Jacobi Quartic Curves?

An optimal Tate pairing computation using Jacobi quartic elliptic curves

This research paper proposes new explicit formulas to compute the Tate pairing on Jacobi quartic elliptic curves. We state the first geometric interpretation of the group law on Jacobi quartic curves by presenting the functions which arise in the addition and doubling. We draw together the best possible optimization that can be used to efficiently evaluate the Tate pairing using Jacobi quartic curves. They are competitive with all published formulas for Tate pairing computation using Short Weierstrass or Twisted Edwards curves. Finally we present several examples of pairing-friendly Jacobi quartic elliptic curves which provide optimal Tate pairing.

Edwards Curves and Extended Jacobi Quartic-Curves for Efficient Support of Elliptic-Curve Cryptosystems in Embedded Systems

The efficient support of cryptographic protocols based on elliptic curves is crucial when embedded processors are adopted as the target hardware platforms. The implementation of Elliptic Curve Cryptography (ECC) offers a variety of design options, mostly covering the specific family of curves and the related coordinate system. At the same time, theory shows that a limited set of solutions can actually lead to efficient computational implementations. This paper analyzes these configurations from an applicative perspective, and mainly addresses two families of curves, namely, Edwards curves and extended Jacobi quartic curves, under a variety of coordinate systems. The experimental results prove that the ECC schemes based on either Edwards curves or extended Jacobi quartic curves can attain remarkable performances in terms of computational efficiency on low-cost, lowresources processors. Edwards curves, in particular, yield best performances when expressed in inverted coordinates.

Efficient computation of pairings on Jacobi quartic elliptic curves

Abstract This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y 2 = d X 4 + Z 4 $Y^{2}=dX^{4}+Z^{4}$ . We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing, we obtain a theoretical gain up to 27 % $27\%$ and 39 % $39\%$ , depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to 27 % $27\%$ more efficient compared to the case of Weierstrass curves with quartic twists.

Another Approach to Pairing Computation on Jacobi Quartic Curves

Another Approach to Pairing Computation on Jacobi Quartic Curves

Legendre polynomials and the elliptic genus

The famous Legendre polynomials are intricately connected with the theory of elliptic integrals and with the formal groups associated to Jacobi quartic curves Y2 = 1 26X2 + &X4. For instance, Honda [4] has obtained new congruences for these polynomials using his theory of formal groups. Congruences modulo p2 involving Legendre polynomials are at the root of the construction of elliptic cohomology [S]. On the other hand, Legendre polynomials may be interpreted as matrix coefficients for finite-dimensional representations of the algebraic group X(2) [3]. In this article, we use well-known facts about the reduction modulo p of such representations to find a representation-theoretic statement which implies the Schur congruences for the Legendre polynomials (see Section 1). We use similar methods to obtain a new derivation for some of Honda’s congruences. In Section 3, we prove a statement about the reduction modulo a large prime number p of a certain representation of an orthogonal group to obtain a congruence (Corollary 3.2) for Gegenbauer polynomials. We also give a congruence for Tchebycheff polynomials (Proposition 3.4). The expression of the Legendre polynomials as matrix coefficients allows us to write down an intriguing formula (Proposition 2.1) for the logarithm of the formal group associated to a Jacobi quartic. This formula shows that this formal power series is an “expectation value” (in the sense of quantum theory) for the representation of GL(2) in the space of formal power series in two variables X and Y. The role of the vacuum vector is played by the formal power series exy. The parameter t for the formal power series appears as the entry of a scalar (2, 2)-matrix. This suggests that some