Extended Euclidean Algorithm and CRT Algorithm

In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].This work was supported by JSPS KAKENHI 21240001 and 22300285Okazaki Hiroyuki - Shinshu University, Nagano, JapanAoki Yosiki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.NZMATH development Group. http://tnt.math.se.tmu.ac.jp/nzmath/.Donald E. Knuth. Art of Computer Programming. Volume 2: Seminumerical Algorithms, 3rd Edition, Addison-Wesley Professional, 1997.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Bound-intersection detection for multiple-symbol differential unitary space-time modulation

This paper considers multiple-symbol differential detection (MSD) of differential unitary space-time modulation (DUSTM) over multiple-antenna systems. We derive a novel exact maximum-likelihood (ML) detector, called the bound-intersection detector (BID), using the extended Euclidean algorithm for single-symbol detection of diagonal constellations. While the ML search complexity is exponential in the number of transmit antennas and the data rate, our algorithm, particularly in high signal-to-noise ratio, achieves significant computational savings over the naive ML algorithm and the previous detector based on lattice reduction. We also develop four BID variants for MSD. The first two are ML and use branch-and-bound, the third one is suboptimal, which first uses BID to generate a candidate subset and then exhaustively searches over the reduced space, and the last one generalizes decision-feedback differential detection. Simulation results show that the BID and its MSD variants perform nearly ML, but do so with significantly reduced complexity

Performance Increasing Approaches For Binary Field Inversion

Authors propose several approaches for increasing performance of multiplicative inversion algorithm in binary fields based on Extended Euclidean Algorithm (EEA). First approach is based on Extended Euclidean Algorithm specificity: either invariant polynomial u remains intact or swaps with invariant polynomial v. It makes it possible to avoid necessity of polynomial v degree computing. The second approach is based on searching the next matching index when calculating the degree of the polynomial, since degree polynomial invariant u at least decreases by 1, then it is possible to use current value while further calculation the degree of the polynomial