Vladimir Sergeevich Anashin

Lomonosov Moscow State University | MSU · Department of Mathematical Cybernetics

In this paper, it is rigorously proven that since observational data (i.e., numerical values of physical quantities) are rational numbers only due to inevitably nonzero measurements errors, the conclusion about whether Nature at the smallest scales is discrete or continuous, random and chaotic, or strictly deterministic, solely depends on experimen...

Title of the talk: A look at quantum systems via p-adic 1-Lipschitz maps. Author: Vladimir Anashin Affiliations: 1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University. 2. Federal Research Center `Information and Control’ , Russian Academy of Sciences Abstract: In the talk, it will be argued that basic notio...

Abstract: In the talk we describe causal functions that are consistent with Volovich postulates which read: (1) only rational numbers can be observed; irrational numbers cannot; (2) distances smaller than Planck length cannot be measured; (3) fundamental physical laws should be invariant with respect to change of number field. Causal functions over...

The paper announces results which may lead to a novel (basically, information-theoretic) interpretation of quantum mechanics under assumptions of causality and existence of the smallest temporal interval. The main mathematical tool of the interpretation is [Formula: see text]-adic 1-Lipschitz maps. The latter maps are exactly the maps produced by l...

ABSTRACT: During the course, by the automaton we mean a transducer, i.e., a sequential machine which maps symbols of a finite input alphabet to symbols of a finite output alphabet so that any output symbol depends on corresponding input symbol and of current state of the machine, whereas any input symbol changes current state of the machine. An aut...

The paper announces results which may lead to a novel (basically, information-theoretic) interpretation of quantum mechanics under assumptions of causality and existence of the smallest temporal interval. The main mathematical tool of the interpretation are p-adic 1-Lipschitz maps. The latter maps are exactly the maps produced by letter-to-letter t...

The problem of making a correct diagnosis of a patient's disease is always is up to date. In this paper, we consider the problem classification of patients by type of mental illness based on analysis of EEG dendrograms of the brain. Shor O., Glik A., Yaniv-Rosenfeld A., Valevski A., Weizman A., Khrennikov A., Benninger F. proposed a method for anal...

The talk given at the conference dedicated to 75-th anniversary of I.V.Volovich. http://www.mathnet.ru/php/conference.phtml?eventID=1&confid=1971&if_videolibrary=1&option_lang=eng

This is mostly a survey paper on the p-adic (and wider, an ulrametric) theory of automata functions, though sketch proofs are given in a few cases. In the paper, by the automaton we mostly mean a transducer, i.e., a sequential machine which maps symbols of a finite input alphabet to symbols of a finite output alphabet so that any output symbol depe...

In the talk, we develop an ultrametric (to be more specific, a p- adic) approach to causality and time. The goal is to finally construct a math- ematical theory of functions which agrees with the current physical models of causality and time both at macro- and Planck scales. Common physical models are mostly based on Archimedean time and space whic...

Physically unclonable functions (PUFs) are currently attracting great interest in information security. In [1] the PUFs are described as follows: A physically unclonable function or PUF is best described as “an expression of an inherent and unclonable instance-specific feature of a physical object”. This means that through physical reasoning it is...

ERRATA to the `Ergodic transformations in the space of p-adic integers’

This paper contains a brief review of a very diverse and vast scientific work of Andrei Yurievich Khrennikov on the occasion of his 60th birthday.

Every automaton (a letter-to-letter transducer) whose input/output alphabets consist of p symbols produces a 1-Lipschitz map from p-adic integers to p-adic integers; and vice versa, every that map can be performed by a suitable automaton. Every that map can be regarded as a discrete causal map. By further developing of ideas of our earlier works «Q...

Invited talk given at the meeting of the Chinese Association for Cryptologic Research.

Talk given at the conference `New Trends in Mathematical and Theoretical Physics’ (October 03–07, 2016, MIAN, Gubkina, 8, Moscow) . Video of the talk available at http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=14908

In the paper, we show that matter waves can be derived from discrete-ness and causality. Namely we show that matter waves can naturally be ascribed to finite discrete causal systems, the Mealy automata having binary input/output which are bit sequences. If assign real numerical values ('measured quantities') to bit sequences, the waves arise as a c...

Abstract—It is shown that the class of all C 2-smooth real functions that can be computed (in a new, but natural sense precisely defined below) on Mealy machines (letter-to-letter transducers or, briefly, transducers) consists of affine functions only. Moreover, it turns out that all these functions can be naturally associated with wave functions o...

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are Fp = {0, 1,..., p - 1} produces a 1-Lipschitz map f A from the space Zp of p-adic integers to Zp . The map fA can naturally be plotted in a unit real square I2 ⊂ R2: To an m-letter non-empty word v = γm-1γm-2... γ0 there corresponds a number 0.v ∈ R with bas...

A textbook on the p-adic ergodic theory and its applications to computer science and cryptology. Based on a lecture course given at the Graduate University of Chinese Academy of Sciences for MS- and PhD-students.

We present two low time-cost methods to evaluate arbitrary T-function on k-bit words; both methods use only fast computer instructions (integer addition and/or bitwise logical instructions) and calls to memory. The methods can be applied in a design of T-function-based stream ciphers for fast encryption software in heavy-traffic networks.

We find linear (as well as quadratic) relations in a very large class of T-functions. The relations may be used in analysis of T-function-based stream ciphers.

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $<f;\mathbf S_{2^{-r}}(a)>$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\...

In the paper we develop the $p$-adic theory of discrete automata. Every automaton $\mathfrak A$ (transducer) whose input/output alphabets consist of $p$ symbols can be associated to a continuous (in fact, 1-Lipschitz) map from $p$-adic integers to $p$ integers, the automaton function $f_\mathfrak A$. The $p$-adic theory (in particular, the $p$-adic...

In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior of the system w.r.t. variety of word transformations performed by the system: We call a system completely transitive if, given arbitrary pair $a,b$ of finite words that have equal lengths,...

We find linear (as well as quadratic) relations in a very large class of T-functions. The relations may be used in analysis of T-function-based stream ciphers.

The paper presents new criteria for bijectivity/transitivity of T-functions and fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz $p$-adic functions and to study measure-preservation/ergodicity of these.

Characterization of ergodicity of p-adic dynamical systems by using the van der Put Basis is presented. Results for p-adic maps that provide the opportunity to characterize their important properties, including ergodicity and the preservation of the Haar measure, in terms of coefficients with respect to the van der Put basis, are obtained. Put basi...

In order to determine transitive polynomials on finite solvable groups, we develop ergodic theory for polynomial transformations on profinite groups with operators.

A mixed identity in variables over a group is a word (where the coefficients lie in , , and ) taking the value 1 for any values of the variables in . The concept of a mixed variety of groups is introduced as an object corresponding to a certain set of mixed identities and generalizing the concept of a variety of groups; an analogue of Birkhoff's th...

The paper develops techniques in order to construct computer programs, pseudorandom number generators (PRNG), that produce uniformly distributed sequences. The paper exploits an approach that treats standard processor instructions (arithmetic and bitwise logical ones) as continuous functions on the space of 2-adic integers. Within this approach, a...

These are lecture notes of a 20-hour course at the International Summer School \emph{Mathematical Methods and Technologies in Computer Security} at Lomonosov Moscow State University, July 9--23, 2006. Loosely speaking, a $T$-function is a map of $n$-bit words into $n$-bit words such that each $i$-th bit of image depends only on low-order bits $0,.....

Let L1 be the set of all mappings f : Zp --> Zp of the space of all p-adic integers Zp into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping f ∈ L1 is ergodic with respect to the normalized Haar measure on Zp if and only if f induces a single cycle permutation on each residue ring Z/pkZ modulo pk, for all k = 1,...

The paper develops a novel approach to stream cipher design: Both the state update function and the output function of the corresponding pseudorandom generators are compositions of arithmetic and bitwise logical operations, which are standard instructions of modern microprocessors. Moreover, both the state update function and the output function ar...

ABC is a synchronous stream cipher submitted to ECRYPT Stream Cipher Project [2]. The changes proposed in this paper increase ABC keystream period to 2 32 ·(2 127 − 1) words and the size of ABC internal state to 1287 bits while keeping all the guaranteed properties of the keystream without a considerable overhead.

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 paper study counter-dependent pseudorandom generators; the latter are generators such that their state transition function (and output function) is being modified dynamically while working: For such a generator the recurrence sequence of states satisfies a congruence $x_{i+1}\equiv f_i(x_i)\pmod{2^n}$, while its output sequence is of the form $...

The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In s...

Such an approach enables us to establish relations between "discrete" and "continuous" properties of some classes of functions. For example, from this point of view the functions known as determinate in the theory of automata turn out to be exactly the functions that satisfy the Lipschitz condition with coefficient 1. There also exists the relation...

A group G is called functionally complete if for an arbitrary natural number n every mapping f: Gn ? G can be realized by a “polynomial” in at most n variables over the group G. We know that a group G is functionally complete if and only if it is either trivial or a finite simple non-Abelian group [Ref. Zh. Mat. 9A174 (1975)]. In this article the “...

ABC is a synchronous stream cipher submitted to eSTREAM [4]. In the previous paper [3] we suggested minor tweaks that increase the period and the secret state of ABC. Here we describe how these tweaks make ABC v.2 [5] resistant to certain attacks, including the ones pre-sented in [6] and [7]. We also note that the paper [8] contains multiple errors...

Loosely speaking, a T-function is a mapping from k-bit words into r-bit words such that each i-th bit of image depends only on low-order bits 0,⋯,i of the pre-image. For example, all arithmetic operations (addition, multiplication) are T-functions, all bitwise logical operations (XOR, AND, etc.) are T-functions. Any composition of T-functions is a...