8,395 research outputs found
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
Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians
International audienceThe 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
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
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
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
A Non-commutative Cryptosystem Based on Quaternion Algebras
We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion
algebras. This cryptosystem uses bivariate polynomials as the underling ring.
The multiplication operation in our cryptosystem can be performed with high
speed using quaternions algebras over finite rings. As a consequence, the key
generation and encryption process of our cryptosystem is faster than NTRU in
comparable parameters. Typically using Strassen's method, the key generation
and encryption process is approximately times faster than NTRU for an
equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure
that makes inefficient standard lattice attacks on the private key. This
entails a higher computational complexity for attackers providing the
opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is
more resistant than NTRU against known attacks at an equivalent parameter set.
Moreover, message protection is feasible through larger polynomials and this
allows us to obtain the same security level as other NTRU-like cryptosystems
but using lower dimensions.Comment: Submitted for possible publicatio
A Survey on Homomorphic Encryption Schemes: Theory and Implementation
Legacy encryption systems depend on sharing a key (public or private) among
the peers involved in exchanging an encrypted message. However, this approach
poses privacy concerns. Especially with popular cloud services, the control
over the privacy of the sensitive data is lost. Even when the keys are not
shared, the encrypted material is shared with a third party that does not
necessarily need to access the content. Moreover, untrusted servers, providers,
and cloud operators can keep identifying elements of users long after users end
the relationship with the services. Indeed, Homomorphic Encryption (HE), a
special kind of encryption scheme, can address these concerns as it allows any
third party to operate on the encrypted data without decrypting it in advance.
Although this extremely useful feature of the HE scheme has been known for over
30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE)
scheme, which allows any computable function to perform on the encrypted data,
was introduced by Craig Gentry in 2009. Even though this was a major
achievement, different implementations so far demonstrated that FHE still needs
to be improved significantly to be practical on every platform. First, we
present the basics of HE and the details of the well-known Partially
Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which
are important pillars of achieving FHE. Then, the main FHE families, which have
become the base for the other follow-up FHE schemes are presented. Furthermore,
the implementations and recent improvements in Gentry-type FHE schemes are also
surveyed. Finally, further research directions are discussed. This survey is
intended to give a clear knowledge and foundation to researchers and
practitioners interested in knowing, applying, as well as extending the state
of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the
survey that is being submitted to ACM CSUR and has been uploaded to arXiv for
feedback from stakeholder
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