Domingo Gómez-Pérez

Bio: Domingo Gómez-Pérez is an academic researcher from University of Cantabria. The author has contributed to research in topics: Finite field & Pseudorandom number generator. The author has an hindex of 12, co-authored 58 publications receiving 414 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.

Predicting the Inversive Generator

Abstract: Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation \(U_{n+1}\equiv au{^{-1}_{n}}+b {\rm mod} p\).We show that if b and sufficiently many of the most significant bits of three consecutive values u n of the ICG are given, one can recover in polynomial time the initial value u 0 (even in the case where the coefficient a is unknown) provided that the initial value u 0 does not lie in a certain small subset of exceptional values.

Predicting the Inversive Generator

Abstract: Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation \(U_{n+1}\equiv au{^{-1}_{n}}+b {\rm mod} p\).We show that if b and sufficiently many of the most significant bits of three consecutive values u n of the ICG are given, one can recover in polynomial time the initial value u 0 (even in the case where the coefficient a is unknown) provided that the initial value u 0 does not lie in a certain small subset of exceptional values.

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.

Visual learning analytics of educational data: A systematic literature review and research agenda

Abstract: We present a systematic literature review of the emerging field of visual learning analytics. We review existing work in this field from two perspectives: First, we analyze existing approaches, audiences, purposes, contexts, and data sources—both individually and in relation to one another—that designers and researchers have used to visualize educational data. Second, we examine how established literature in the fields of information visualization and education has been used to inform the design of visual learning analytics tools and to discuss research findings. We characterize the reviewed literature based on three dimensions: (a) connection with visualization background; (b) connection with educational theory; and (c) sophistication of visualization(s). The results from this systematic review suggest that: (1) little work has been done to bring visual learning analytics tools into classroom settings; (2) few studies consider background information from the students, such as demographics or prior performance; (3) traditional statistical visualization techniques, such as bar plots and scatter plots, are still the most commonly used in learning analytics contexts, while more advanced or novel techniques are rarely used; (4) while some studies employ sophisticated visualizations, and some engage deeply with educational theories, there is a lack of studies that both employ sophisticated visualizations and engage deeply with educational theories. Finally, we present a brief research agenda for the field of visual learning analytics based on the findings of our literature review.

The Large Sieve

Abstract: Let A be a thin set in Z n , and A N the intersection of A with the ball of diameter N centred at the origin. When n = 1 we have seen in §9.7 (as a consequence of Siegel’s theorem) that |A N | = O(N 1/2) when N → ∞. To prove a similar result when n ≥ 2 one needs a different method, based on the large sieve inequality (cf. [Co]).

Teaching and Learning Analytics to Support Teacher Inquiry: A Systematic Literature Review

Abstract: Teacher inquiry is identified as a key global need for driving the continuous improvement of the teaching and learning conditions for learners. However, specific barriers (mainly related to teachers’ data literacy competences), can defer teachers from engaging with inquiry to improve their teaching practice. To alleviate these barriers and support teacher inquiry, the concept of Teaching and Learning Analytics (TLA) has been proposed, as a complementing synergy between Teaching Analytics and Learning Analytics. Teaching and Learning Analytics aims to provide a framework in which the insights generated by Learning Analytics methods and tools can become meaningfully translated for driving teachers’ inquiry to improve their teaching practice, captured through Teaching Analytics methods and tools. In this context, TLA have been identified as a research challenge with significant practical impact potential. This chapter contributes the first systematic literature review in the emerging research field of Teaching and Learning Analytics. The insights gained from the systematic literature review aim to (a) transparently outline the existing state-of-the-art following a structured analysis methodology, as well as (b) elicit insights and shortcomings which could inform future work in the Teaching and Learning Analytics research field.

Topics in Geometry, Coding Theory and Cryptography

Abstract: The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.