203 research outputs found
The Q-curve construction for endomorphism-accelerated elliptic curves
We give a detailed account of the use of -curve reductions to
construct elliptic curves over 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 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 equipped with efficient
endomorphisms for every p \textgreater{} 3, and exhibit examples of
twist-secure curves over for the efficient Mersenne prime
.Comment: To appear in the Journal of Cryptology. arXiv admin note: text
overlap with arXiv:1305.540
Computing cardinalities of Q-curve reductions over finite fields
We present a specialized point-counting algorithm for a class of elliptic
curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo
inert primes and, more generally, any elliptic curve over F\_{p^2} with a
low-degree isogeny to its Galois conjugate curve. These curves have interesting
cryptographic applications. Our algorithm is a variant of the
Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree
endomorphism in place of Frobenius. While it has the same asymptotic asymptotic
complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of
Drew Sutherlan
Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians
The first step in elliptic curve scalar multiplication algorithms based on
scalar decompositions using efficient endomorphisms-including
Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as
well as higher-dimensional and higher-genus constructions-is to produce a short
basis of a certain integer lattice involving the eigenvalues of the
endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar
coefficients, and the faster the resulting scalar multiplication. Typically,
knowledge of the eigenvalues allows us to write down a long basis, which we
then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more
specialized algorithm. In this work, we use elementary facts about quadratic
rings to immediately write down a short basis of the lattice for the GLV, GLS,
GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real
multiplication constructions. We do not pretend that this represents a
significant optimization in scalar multiplication, since the lattice reduction
step is always an offline precomputation---but it does give a better insight
into the structure of scalar decompositions. In any case, it is always more
convenient to use a ready-made short basis than it is to compute a new one
Point compression for the trace zero subgroup over a small degree extension field
Using Semaev's summation polynomials, we derive a new equation for the
-rational points of the trace zero variety of an elliptic curve
defined over . Using this equation, we produce an optimal-size
representation for such points. Our representation is compatible with scalar
multiplication. We give a point compression algorithm to compute the
representation and a decompression algorithm to recover the original point (up
to some small ambiguity). The algorithms are efficient for trace zero varieties
coming from small degree extension fields. We give explicit equations and
discuss in detail the practically relevant cases of cubic and quintic field
extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph
Isogeny-based post-quantum key exchange protocols
The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented
Counting Points on Genus 2 Curves with Real Multiplication
We present an accelerated Schoof-type point-counting algorithm for curves of
genus 2 equipped with an efficiently computable real multiplication
endomorphism. Our new algorithm reduces the complexity of genus 2 point
counting over a finite field (\F_{q}) of large characteristic from
(\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we
compute a 256-bit prime-order Jacobian, suitable for cryptographic
applications, and also the order of a 1024-bit Jacobian
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