The p-adic ergodic theory and applications

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2001

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

Linear finite transducers underlie a series of schemes for Public Key Criptography (PKC) proposed in the 90s of the last century. The uninspiring and arid language then used, condemned these works to oblivion. Although some of these schemes were after shown to be insecure, the promise of a new system of PKC relying in diferent complexity assumptions is still quite exciting. The algorithms there used depend heavilly on the results of invertibility of linear transducers. In this paper we introduce the notion of post-initial linear tranducer, which is an extension of the notion of linear finite tranducer with memory, and for which the previous fundamental results on invertibility hold. Using this notion, we give a necessary and sufficient condition for left invertibility with fixed delay of finite transducers, as well as a new explicit method to obtain a left inverse.

Doklady Mathematics, 2012

Finite automata public-key cryptosystems rely upon characterizations of some types of invertible finite automata, and methods of obtain them as well as their respective inverses.

Annales UMCS, Informatica, 2012

Let K be a finite commutative ring and f = f (n) a bijective polynomial map f (n) of the Cartesian power K n onto itself of a small degree c and of a large order. Let f y be a multiple composition of f with itself in the group of all polynomial automorphisms, of free module K n. The discrete logarithm problem with the "pseudorandom" base f (n) (solve f y = b for y) is a hard task if n is "sufficiently large". We will use families of algebraic graphs defined over K and corresponding dynamical systems for the explicit constructions of such maps f (n) of a large order with c =2such that all nonidentical powers f y are quadratic polynomial maps. The above mentioned result is used in the cryptographical algorithms based on the maps f (n)-in the symbolic key exchange protocols and public keys algorithms.

Mathematics

Cryptology, since its advent as an art, art of secret writing, has slowly evolved and changed, above all since the middle of the last century. It has gone on to obtain a more solid rank as an applied mathematical science. We want to propose some annotations in this regard in this paper. To do this, and after reviewing the broad spectrum of methods and systems throughout history, and from the traditional classification, we offer a reordering in a more compact and complete way by placing the cryptographic diversity from the algebraic binary relations. This foundation of cryptological operations from the principles of algebra is enriched by adding what we call pre-cryptological operations which we show as a necessary complement to the entire structure of cryptology. From this framework, we believe that it is improved the diversity of questions related to the meaning, the fundamentals, the statute itself, and the possibilities of cryptological science.

Corr, 2004

The paper study counter-dependent pseudorandom number generators based on $m$-variate ($m>1$) ergodic mappings of the space of 2-adic integers $\Z_2$. The sequence of internal states of these generators is defined by the recurrence law $\mathbf x_{i+1}= H^B_i(\mathbf x_i)\bmod{2^n}$, whereas their output sequence is %while its output sequence is of the $\mathbf z_{i}=F^B_i(\mathbf x_i)\mod 2^n$; here $\mathbf x_j, \mathbf z_j$ are $m$-dimensional vectors over $\Z_2$. It is shown how the results obtained for a univariate case could be extended to a multivariate case.

Finite Fields and Their Applications, 2014

A recursive construction of complete mappings over finite fields is provided in this work. These permutation polynomials, characterized by the property that both f (x) ∈ F q [x] and its associated mapping f (x) + x are permutations, have an important application in cryptography in the construction of bent-negabent functions which actually leads to some new classes of these functions. Furthermore, we also provide a recursive construction of mappings over finite fields of odd characteristic, having an interesting property that both f (x) and f (x + c) + f (x) are permutations for every c ∈ F q. Both the multivariate and univariate representations are treated and some results concerning fixed points and the cycle structure of these permutations are given. Finally, we utilize our main result for the construction of so-called negabent functions and bent functions over finite fields.

Cryptographic Applications of Analytic Number Theory, 2003

International Journal of Computer Theory and Engineering, 2012

International Journal of Mathematics and Mathematical Sciences, 2012

Bernoulli numbers, Bernoulli polynomials, and Euler numbers, Euler polynomials were studied by many authors. Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials possess many interesting properties and arise in many areas of mathematics and physics. These numbers are still in the center of the advanced mathematical research. Especially, in number theory and quantum theory, they have many applications. p-Adic analysis with q-analysis includes several domains in mathematics and physics, including the number theory, algebraic geometry, algebraic topology, mathematical analysis, mathematical physics, string theory, field theory, stochastic differential equations, quantum groups, and other parts of the natural sciences. The intent of this special issue was to survey major interesting results and current trends in the theory of p-adic analysis associated with q-analogs of zeta functions, Hurwitz zeta functions, Dirichlet series, L-series, special values, q-analogs of Bernoulli, Euler, and Genocchi numbers and polynomials, q-integers, q-integral, q-identities, q-special functions, qcontinued fractions, gamma functions, sums of powers, q-analogs of multiple zeta functions, Barnes multiple zeta functions, multiple L-series, and computational and numerical aspects of q-series and q-analysis. The Guest Editors and Referees of this special issue are well-known mathematicians that work in this field of interest. Thus, we got the best articles to be included in this issue.

2020

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational throug...

2015

In this work, we study the elliptic curve over the ring ; ; where d is a positive integer. More precisely in cryptography applications, we will give many various explicit formulas describing the binary operations calculus in . The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic Curve Cryptosystems. Keywords— Elliptic Curves, Finite Ring, Cryptography..

Tit, 2007

In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2-adic complexity arise naturally to measure the (non)randomness of a given string. Here, we define an isometry K on which is the precise equivalent to Euclid's algorithm over the reals to calculate the continued fraction expansion of a formal power series over. K allows us to deduce the linear and jump complexity profiles of the input sequence and since K is an isometry, the resulting-sequence is independent and identically distributed (i.i.d.) for i.i.d. input. Hence the linear and jump complexity profiles may be modeled via Bernoulli experiments. We thus can apply the precise bounds as collected by Révész, among others the Law of the Iterated Logarithm The second topic is approximation by FCSR and AFSR registers, as defined by Goresky, Klapper, and Xu. For the 2-adic span, we derive an isometry A on. The corresponding jump complexity behaves on average exactly like coin tossing. Also, we give a general procedure to obtain an isometry from any approximation algorithm like LFSR, FCSR, or AFSR register synthesis. Focusing on the behavior of the result after applying the isometries instead of the complexities itself, permits us to apply the known bounds for Bernoulli experiments and to compare different complexities in a unified way, considering only their induced isometries. In this way, we can give sharp bounds on what behavior of the linear, jump, and 2-adic complexities is permitted for pseudorandom sequences, and what should be rejected.

De Gruyter eBooks, 2013