5,905 research outputs found
Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes
We give a general framework for uniform, constant-time one-and
two-dimensional scalar multiplication algorithms for elliptic curves and
Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer
surface, where we can exploit faster and more uniform pseudomultiplication,
before recovering the proper "signed" output back on the curve or Jacobian.
This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and
Joye to genus 2, and also to two-dimensional scalar multiplication. Our results
show that many existing fast pseudomultiplication implementations (hitherto
limited to applications in Diffie--Hellman key exchange) can be wrapped with
simple and efficient pre-and post-computations to yield competitive full scalar
multiplication algorithms, ready for use in more general discrete
logarithm-based cryptosystems, including signature schemes. This is especially
interesting for genus 2, where Kummer surfaces can outperform comparable
elliptic curve systems. As an example, we construct an instance of the Schnorr
signature scheme driven by Kummer surface arithmetic
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
A Generic Approach to Searching for Jacobians
We consider the problem of finding cryptographically suitable Jacobians. By
applying a probabilistic generic algorithm to compute the zeta functions of low
genus curves drawn from an arbitrary family, we can search for Jacobians
containing a large subgroup of prime order. For a suitable distribution of
curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus
3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime
fields with group orders over 180 bits in size, improving previous results. Our
approach is particularly effective over low-degree extension fields, where in
genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3}
with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average
time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio
Edwards curves and CM curves
Edwards curves are a particular form of elliptic curves that admit a fast,
unified and complete addition law. Relations between Edwards curves and
Montgomery curves have already been described. Our work takes the view of
parameterizing elliptic curves given by their j-invariant, a problematic that
arises from using curves with complex multiplication, for instance. We add to
the catalogue the links with Kubert parameterizations of X0(2) and X0(4). We
classify CM curves that admit an Edwards or Montgomery form over a finite
field, and justify the use of isogenous curves when needed
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
, 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 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 and
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
Millipeds from the eastern Dakotas and western Minnesota, USA, with an account of Pseudopolydesmus serratus (Say, 1821) (Polydesmida: Polydesmidae); first published records from six states and the District of Columbia
The diplopod orders Callipodida and Polydesmida, and their respective families Abacionidae and
Xystodesmidae, are initially recorded from South Dakota as is Polydesmidae from North Dakota. Other new records of
indigenous taxa include Abacion Rafinesque, 1820/A. texense (Loomis, 1937) and Pleuroloma/P. flavipes, both by
Rafinesque, 1820, from South Dakota, and Pseudopolydesmus Attems, 1898/P. serratus (Say, 1821) from Alabama,
Connecticut, Delaware, New Hampshire, North Dakota, South Carolina, and the District of Columbia. New records of
Aniulus garius Chamberlin, 1912, A. (Hakiulus) d. diversifrons (Wood, 1867), and Oriulus venustus (Wood, 1864)
(Julida: Parajulidae) are provided for western Minnesota and/or eastern North Dakota. Published records from these
states are summarized, and the introduced taxa, Julidae/Cylindroiulus Verhoeff, 1894/C. caeruleocinctus (Wood, 1864)
and Paradoxosomatidae/Oxidus Cook, 1911/O. gracilis (C. L. Koch, 1847), are newly recorded from the Dakotas. The
distribution of P. serratus, which extends from Maine to South Carolina and the Florida panhandle, west to Texas, and
north to Fargo, North Dakota is described and discussed. This distribution exhibits a prominent southeastern lacuna
which we hypothesize suggests replacement by younger, more successful species, as postulated for a similar distributional
gap in Scytonotus granulatus (Say, 1821)
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
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