37,732 research outputs found
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
Riemann zeta function and quantum chaos
A brief review of recent developments in the theory of the Riemann zeta
function inspired by ideas and methods of quantum chaos is given.Comment: Lecture given at International Conference on Quantum Mechanics and
Chaos, Osaka, September 200
Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields
We introduce a new algorithm, which is faster and requires less computing
resources than the ones previously known, to compute the Euler-Kronecker
constants for the prime cyclotomic fields
, where is an odd prime and is a primitive
-root of unity. With such a new algorithm we evaluated and
, where is the Euler-Kronecker constant of
the maximal real subfield of , for some very large primes
thus obtaining two new negative values of :
and We also evaluated and for
every odd prime , thus enlarging the size of the previously known
range for and . Our method also reveals that
difference can be computed in a much
simpler way than both its summands, see Section 3.4. Moreover, as a by-product,
we also computed
for every odd prime , where are the Dirichlet
-functions, run over the non trivial Dirichlet characters mod and
is the trivial Dirichlet character mod . As another by-product of
our computations, we will also provide more data on the generalised Euler
constants in arithmetic progressions. The programs used to performed the
computations here described and the numerical results obtained are available at
the following web address:
\url{http://www.math.unipd.it/~languasc/EK-comput.html}.Comment: 25 pages, 6 tables, 4 figures. Third known example of negative values
for Ek(q) inserted. Complete set of computation of Ek(q) and Ek(q)^+ for
every prime up to 10^6; computation of max|L'/L(1,chi)| for the same primes
inserted. Two references added, typos correcte
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