Jaime Gutierrez

Bio: Jaime Gutierrez is an academic researcher from University of Cantabria. The author has contributed to research in topics: Pseudorandom number generator & Rational function. The author has an hindex of 20, co-authored 97 publications receiving 1126 citations.

Predicting nonlinear pseudorandom number generators

Abstract: Let p be a prime and let a and b be elements of the finite field Fp of p elements. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation u n+1 ≡ au -1 n +b mod p. We show that if sufficiently many of the most significant bits of several consecutive values u n of the ICG are given, one can recover the initial value u 0 (even in the case where the coefficients a and b are not known). We also obtain similar results for the quadratic congruential generator (QCG), v n+1 ≡ f(v n ) mod p, where f ∈ F p [X]. This suggests that for cryptographic applications ICG and QCG should be used with great care. Our results are somewhat similar to those known for the linear congruential generator (LCG), x n+1 ≡ ax n + b mod p, but they apply only to much longer bit strings. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for LCG.

Hyperelliptic Curves with Extra Involutions

Abstract: The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus Lg of such genus-g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves Hg. The authors present a birational parameterization of Lg via dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈ Hg with |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈ Lg such that the Klein 4-group is embedded in the reduced automorphism group of p. Further, for g = 3, they show that for every moduli point p ∈ H3 such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

Hyperelliptic curves with extra involutions

Abstract: The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of $\L_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in \L_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in \L_g$ such that the Klein 4-group is embedded in the reduced automorphism group of $\p$. Further, for $g=3$ we show that for every moduli point $\p \in \H_3$ such that $| \Aut (\p) | > 4$, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.

Multivariate Polynomial Decomposition

Abstract: In this paper, we discuss several notions of decomposition for multivariate polynomials, focussing on the relation with Luroth's theorem in field theory, and the finiteness and uniqueness of decompositions. We also present two polynomial time algorithms for decomposing (sparse) multivariate polynomials over an arbitrary field.

Modern computer algebra

Abstract: 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.

Gröbner Bases와 응용

Abstract: 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

Discrete math = 離散数学

Abstract: What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof