The Q-curve construction for endomorphism-accelerated elliptic curves

We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}\_{p^2}$ 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 $p$ 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 $\mathbb{F}\_{p^2}$ equipped with efficient endomorphisms for every p \textgreater{} 3, and exhibit examples of twist-secure curves over $\mathbb{F}\_{p^2}$ for the efficient Mersenne prime $p = 2^{127}-1$.Comment: To appear in the Journal of Cryptology. arXiv admin note: text overlap with arXiv:1305.540

Families of fast elliptic curves from Q-curves

We construct new families of elliptic curves over \FF_{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. Our construction is based on reducing \QQ-curves-curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates-modulo inert primes. As a first application of the general theory we construct, for every $p > 3$, two one-parameter families of elliptic curves over \FF_{p^2} equipped with endomorphisms that are faster than doubling. Like GLS (which appears as 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 $p$ is fixed. Unlike GLS, we also offer the possibility of constructing twist-secure curves. Among our examples are prime-order curves equipped with fast endomorphisms, with almost-prime-order twists, over \FF_{p^2} for $p = 2^{127}-1$ and $p = 2^{255}-19$

Faster computation of the Tate pairing

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 pairing-friendly Edwards curves.Comment: 15 pages, 2 figures. Final version accepted for publication in Journal of Number Theor

FourQ: four-dimensional decompositions on a Q-curve over the Mersenne prime

We introduce FourQ, a high-security, high-performance elliptic curve that targets the 128-bit security level. At the highest arithmetic level, cryptographic scalar multiplications on FourQ can use a four-dimensional Gallant-Lambert-Vanstone decomposition to minimize the total number of elliptic curve group operations. At the group arithmetic level, FourQ admits the use of extended twisted Edwards coordinates and can therefore exploit the fastest known elliptic curve addition formulas over large prime characteristic fields. Finally, at the finite field level, arithmetic is performed modulo the extremely fast Mersenne prime $p=2^{127}-1$. We show that this powerful combination facilitates scalar multiplications that are significantly faster than all prior works. On Intel\u27s Broadwell, Haswell, Ivy Bridge and Sandy Bridge architectures, our software computes a variable-base scalar multiplication in 50,000, 56,000, 69,000 cycles and 72,000 cycles, respectively; and, on the same platforms, our software computes a Diffie-Hellman shared secret in 80,000, 88,000, 104,000 cycles and 112,000 cycles, respectively. These results show that, in practice, FourQ is around four to five times faster than the original NIST P-256 curve and between two and three times faster than curves that are currently under consideration as NIST alternatives, such as Curve25519

Machine-Level Software Optimization of Cryptographic Protocols

This work explores two methods for practical cryptography on mobile devices. The first method is a quantum-resistant key-exchange protocol proposed by Jao et al.. As the use of mobile devices increases, the deployment of practical cryptographic protocols designed for use on these devices is of increasing importance. Furthermore, we are faced with the possible development of a large-scale quantum computer in the near future and must take steps to prepare for this possibility. We describe the key-exchange protocol of Jao et al. and discuss their original implementation. We then describe our modifications to their scheme that make it suitable for use in mobile devices. Our code is between 18-26% faster (depending on the security level). The second is an highly optimized implementation of Miller's algorithm that efficiently computes the Optimal Ate pairing over Barreto-Naehrig curves proposed by Grewal et al.. We give an introduction to cryptographic pairings and describe the Tate pairing and its variants. We then proceed to describe Grewal et al.'s implementation of Miller's algorithm, along with their optimizations. We describe our use of hand-optimized assembly code to increase the performance of their implementation. For the Optimal Ate pairing over the BN-446 curve, our code is between 7-8% faster depending on whether the pairing uses affine or projective coordinates

Families of fast elliptic curves from Q-curves

We construct new families of elliptic curves over \FF_{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. Our construction is based on reducing \QQ-curves---curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates---modulo inert primes. As a first application of the general theory we construct, for every $p > 3$, two one-parameter families of elliptic curves over \FF_{p^2} equipped with endomorphisms that are faster than doubling. Like GLS (which appears as 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 $p$ is fixed. Unlike GLS, we also offer the possibility of constructing twist-secure curves. Among our examples are prime-order curves equipped with fast endomorphisms, with almost-prime-order twists, over \FF_{p^2} for $p = 2^{127}-1$ and $p = 2^{255}-19$

Endomorphisms for faster elliptic curve cryptography on a large class of curves

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant-Lambert-Vanstone (GLV) method. We extend results of Iijima, Matsuo, Chao and Tsujii which give such homomorphisms for a large class of elliptic curves by working over quadratic extensions and demonstrate that these results can be applied to the GLV method. Our implementation runs in between 0.70 and 0.84 the time of the previous best methods for elliptic curve point multiplication on curves without small class number complex multiplication. Further speedups are possible when using more special curves