Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

1 S-polynomials eliminating the leading term Buchberger's criterion and algorithm 2 Wavelet Design construct wavelet filters 3 Proof of the Buchberger Criterion two lemmas proof of the Buchberger criterion termination and elimination

Gröbner Bases와 응용

In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian integers, the subset of the complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.

Modeling Toroidal Networks with the Gaussian Integers

Circulant graphs have been extensively investigated over the past 30 years because of their broad application to different fields of theory and practice. Two known surveys on circulant networks including a survey on undirected circulants have been published: by Bermond et al. [Distributed loop computer networks: A survey, J. Parallel Distributed Comput.24 (1995) 2–10] and by Hwang [A survey on multi-loop networks, Theoret. Comput. Sci.299 (2003) 107–121]. The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks. We focus on the survey connected with study of structural and communicative properties of circulant networks.

A survey on undirected circulant graphs

Character Sums with Exponential Functions and their Applications: Multiplicative Translations of Arbitrary Sets Modulo p

Let p be a prime and let c be an integer modulo p. The Pollard generator is a sequence (un) of pseudorandom numbers defined by the relation un+1equivun 2+c mod p. It is shown that if c and 9/14 of the most significant bits of two consecutive values un,un+1 of the Pollard generator are given, one can recover in polynomial time the initial value u0 with a probabilistic algorithm. This result is an improvement of a theorem in a recent paper which requires that 2/3 of the most significant bits be known

Attacking the Pollard Generator

https://personales.unican.es/gomezd/Articles/Ibeas/TheoreticalComputerScience.pdf

Optimal routing in double loop networks

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

/pdf/common-factors-of-resultants-modulo-p-1q4yjifphw.pdf

COMMON FACTORS OF RESULTANTS MODULO p

Let p be a prime and let be an elliptic curve defined over the finite field of p elements For a given point the linear congruential genarator on elliptic curves (EC-LCG) is a sequence (Un) of pseudorandom numbers defined by the relation: where denote the group operation in and is the initial value or seed We show that if G and sufficiently many of the most significants bits of two consecutive values U n, Un+1 of the EC-LCG are given, one can recover the seed U0 (even in the case where the elliptic curve is private) provided that the former value U n does not lie in a certain small subset of exceptional values We also estimate limits of a heuristic approach for the case where G is also unknown This suggests that for cryptographic applications EC-LCG should be used with great care Our results are somewhat similar to those known for the linear and non-linear pseudorandom number congruential generator

Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits

Let p be a prime and let E(IFp) be an elliptic curve defined over the finite field IFp of p elements. For a given point G ∈ E(IFp) the linear congruential genarator on elliptic curves (EC-LCG) is a sequence (Un) of pseudorandom numbers defined by the relation Un = Un−1 ⊕G = nG⊕ U0, n = 1, 2, . . . , where ⊕ denote the group operation in E(IFp) and U0 ∈ E(IFp) is the initial value or seed. We show that if G and sufficiently many of the most significants bits of two consecutive values Un, Un+1 of the EC-LCG are given, one can recover the seed U0 (even in the case where the elliptic curve is private) provided that the former value Un does not lie in a certain small subset of exceptional values. We also estimate limits of a heuristic approach for the case where G is also unknown. This suggests that for cryptographic applications EC-LCG should be used with great care. Our results are somewhat similar to those known for the linear and non-linear pseudorandom number congruential generator.

/pdf/inferring-sequences-produced-by-a-linear-congruential-5eov3mnlxh.pdf

Álvar Ibeas

Papers

Attacking the Pollard Generator

Optimal routing in double loop networks

COMMON FACTORS OF RESULTANTS MODULO p

Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits

Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits.