1,699 research outputs found
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
On the quaternion -isogeny path problem
Let \cO be a maximal order in a definite quaternion algebra over
of prime discriminant , and a small prime. We describe a
probabilistic algorithm, which for a given left -ideal, computes a
representative in its left ideal class of -power norm. In practice the
algorithm is efficient, and subject to heuristics on expected distributions of
primes, runs in expected polynomial time. This breaks the underlying problem
for a quaternion analog of the Charles-Goren-Lauter hash function, and has
security implications for the original CGL construction in terms of
supersingular elliptic curves.Comment: To appear in the LMS Journal of Computation and Mathematics, as a
special issue for ANTS (Algorithmic Number Theory Symposium) conferenc
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