CONTENTS - Emerald

7 Computational Number Theory
041701 `The Rabin-Miller primality test: composite numbers which pass
it'
F Arnault, Mathematics of Computation v 64 no 209 (Jan 95) pp 355{361
The author provides a new technique, based on biquadratic reciprocity, for generating
strong pseudoprimes with respect to a given set of bases. He exhibits an integer
which is a strong pseudoprime with respect to the rst forty six prime numbers.
041702 `The Magic Words Are Squeamish Ossifrage'
DAtkins, M Gra , AK Lenstra, PC Leyland, Asiacrypt 94 pp 219{229
The authors describe how they factored the 129-digit RSA challenge which was
published in 1977 in Scienti c American. They used the double large prime variation
of the multiple polynomial quadratic sieve value, and were assisted by the spare cycles
of 1600 machines on the Internet. It was discovered that the results of the computation
can be approximated by a quartic function of the number of relations received, rather
than as a quadratic function as expected. They conclude that 512-bit RSA moduli are
vulnerable to any organisation willing to spend a few million dollars and wait a few
months.
041703 `On the squared unsymmetric Lanczos method'
AT Chronopoulos, Journal of Computational and Applied Mathematics v 54 (94) pp
65{78
The author discusses an optimisation of the Lanczos method for nding eigenvalues
of nonsymmetric matrices. Tests on a system of 40,000 equations show it to be
competitive and economical of memory.
041704 `Small Zeros of Quadratic Congruences modulo pq, II'
TCochrane, Journal of Number Theoryv50no2(Feb 95) pp 299{308
For n 4, the quadratic congruence Q(x1;x2; :::; xn) 0 (mod pq) has a nonzero
solution with maxjxij
p pq, and this is the best possible such result.
041705 `Computing (x), M(x) and (x)'
M Deleglise, J Rivat, ANTS 94 p 264
The authors report a slight improvement inthe Lagarias-Miller-Odlyzko method
for calculating (x) | their variant runs a multiple of O(log x) faster at the cost of a
similar increase in memory size.
041706 `On Orders of Optimal Normal Basis Generators'
SH Gao, SA Vanstone, ANTS 94 p 220
The authors did a numerical investigation of a large number of optimal normal
bases for elds of characteristic 2. Where is primitive and = + ,1 generates
the basis, e can be computed in O(n:w(e)) operations, where w(e) is the Hamming
weight of e. If were an arbitrary value, the cost would be O(n log n log log n log e)
computations.
041707 `The complexity of greatest common divisor computations'
BS Majewski, G Havas, ANTS 94 pp 184{193
The authors consider the complexity of expressing the GCD of n>2numbers as
a linear combination of them, and shows that this problem is NP-complete. However,
the largest multiplier cannot exceed half the largest input number.
041708 `Parallel decomposition of modular exponentiation for RSA cryptosystem'
S Shimonaka, N Takeda, H Nagase, ISITA 94 pp 269{271
The authors propose calculating M e by decomposing M into factors and raising
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