2,967 research outputs found
A Modular Integer GCD Algorithm
This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC \u2793: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U-bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (-p/2,p/2) such that b=UV ^-1(mod p) . When the algorithm is executed on a bit common CRCW PRAM with O(n log n log log log n) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/log n) time on the same number of processors
On the calculation of the linear complexity of periodic sequences
Based on a result of Hao Chen in 2006 we present a general procedure how to reduce the determination of the linear complexity of a sequence over a finite field \F_q of period to the determination of the linear complexities of sequences over \F_q of period . We apply this procedure to some classes of
periodic sequences over a finite field \F_q obtaining efficient algorithms to determine the linear complexity
Decoding Cyclic Codes up to a New Bound on the Minimum Distance
A new lower bound on the minimum distance of q-ary cyclic codes is proposed.
This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for
some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are
special cases of our bound. For some classes of codes the bound on the minimum
distance is refined. Furthermore, a quadratic-time decoding algorithm up to
this new bound is developed. The determination of the error locations is based
on the Euclidean Algorithm and a modified Chien search. The error evaluation is
done by solving a generalization of Forney's formula
GCD Computation of n Integers
Greatest Common Divisor (GCD) computation is one of the most important
operation of algorithmic number theory. In this paper we present the algorithms
for GCD computation of integers. We extend the Euclid's algorithm and
binary GCD algorithm to compute the GCD of more than two integers.Comment: RAECS 201
A Discrete Logarithm-based Approach to Compute Low-Weight Multiples of Binary Polynomials
Being able to compute efficiently a low-weight multiple of a given binary
polynomial is often a key ingredient of correlation attacks to LFSR-based
stream ciphers. The best known general purpose algorithm is based on the
generalized birthday problem. We describe an alternative approach which is
based on discrete logarithms and has much lower memory complexity requirements
with a comparable time complexity.Comment: 12 page
Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance
The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its
generalization by Hartmann and Tzeng are lower bounds on the minimum distance
of simple-root cyclic codes. We generalize these two bounds to the case of
repeated-root cyclic codes and present a syndrome-based burst error decoding
algorithm with guaranteed decoding radius based on an associated folded cyclic
code. Furthermore, we present a third technique for bounding the minimum
Hamming distance based on the embedding of a given repeated-root cyclic code
into a repeated-root cyclic product code. A second quadratic-time probabilistic
burst error decoding procedure based on the third bound is outlined. Index
Terms Bound on the minimum distance, burst error, efficient decoding, folded
code, repeated-root cyclic code, repeated-root cyclic product cod
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