S. B. Gashkov

Lomonosov Moscow State University | MSU · Division of Mathematics

LXXVIII Moscow mathematical Olympics Problems and solutions

Доказана алгоритмическая неразрешимость нескольких задач, связанных с кусочно-полиномиальными функциями одной действительной переменной, имеющими бесконечное число узлов.

Yuri Valentinovich Nesterenko (to the 75th anniversary)

The paper devoted to the 70-th anniversary of V.N.Chubaricov and its scientic byography

В работе предпринимается обзор современного состояния теории быстрых алгоритмов умножения чисел и многочленов. Рассматривается процесс эволюции методов умножения от первых блочных алгоритмов Карацубы и Тоома 1960-х гг. к методам 1970-х гг., опирающимся на дискретное преобразование Фурье (ДПФ), и далее к новейшим методам, разработанным в 2007–2019 г...

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It is proved that for an arbitrary polynomial \(f\left( x \right) \in {\mathbb{Z}_{{p^n}}}\left[ X \right]\) of degree d the Boolean complexity of calculation of one its root (if it exists) equals O(dM(nλ(p))) for a fixed prime p and growing n, where λ(p) = ⌈log2p⌉, and M(n) is the Boolean complexity of multiplication of two binary n-bit numbers. G...

It was proved that the complexity of square root computation in the Galois field GF(3s), s = 2kr, is equal to O(M(2k)M(r)k + M(r) log2r) + 2kkr1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3s) is equal to O(M(2k)M(r))...

The chapter introduces a comparative analysis of the complexity of the Tate pairing operation on a supersingular elliptic curve and the complexity of the final exponentiation in the tripartite key agreement cryptographic protocol. The analysis takes into account a possibility of using different bases of finite fields in combination. Operations of m...

Для линейного положительного оператора Коровкина $$ f(x)\to t_n(f;x)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)E(t) dt, $$ где $E(x)$ - многочлен Эгервари-Сасса, и соответствующего ему интерполяционного среднего $$ t_{n,N}(f;x)=\frac{1}{N}\sum_{k=-N}^{N-1} E_n(x-\frac{\pi k}{N})f(\frac{\pi k}{N}), $$ доказаны при $N > n/2$ неравенства типа Джексона $$ \|...

For the linear positive Korovkin operator f(x)→tn(f;x)=1π∫−ππf(x+t)E(t)dt, where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean tn,N(f;x)=1N∑k=−NN−1En(x−πkN)f(πkN), the Jackson-type inequalities ‖tn,N(f;x)−f(x)‖≤(1+π)ωf(1n),‖tn,N(f;x)−f(x)‖≤2ωf(πn+1), where ωf (x) denotes the modulus of continuity, are proved for N >...

В работе рассматривается аддитивная сложность матриц, составленных из натуральных степеней наибольших общих делителей и наименьших общих кратных номеров строк и столбцов. Показано, что сложность (n×n)-матрицы, составленной из чисел НОДr(i,k), над базисом {x+y} асимптотически равна rnlog2n при n→∞, а сложность (n×n)-матрицы, составленной из чисел НО...

In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the n × n matrix formed by the numbers GCD^r(i, k) over the basis {x + y} is asymptotically equal to rn log_2n as n→∞, an...

In this paper, we generalize an approach of switching between different bases of a finite field to efficiently implement distinct stages of algebraic algorithms. We consider seven bases of finite fields supporting optimal normal bases of types 2 and 3: polynomial, optimal normal, permuted, redundant, reduced, doubled polynomial, and doubled reduced...

Quadratic and superquadratic estimates are obtained for the computational complexity of some linear transforms by circuits over the base {x + y} ∪ {ax: |a| ≤ C} consisting of addition and scalar multiplications by bounded constants. Upper bounds of order O(n log n) of the computational complexity are also proved for the linear basis {ax + by: a, b...

For the linear Stirling transforms of both kinds, which are well-known in combinatorics, we obtain close to optimal estimates of the complexity of computation by vector addition chains and non-branching programs composed of arithmetic operations over real numbers. A relation between these problems and the Lagrange and Newton interpolation is discus...

We obtain order-sharp quadratic and slightly higher estimates of the computational complexity of certain linear transformations (binomial, Stirling, Lah, Gauss, Serpiński, Sylvester) in the basis {x + y} ⋃ {ax : |a| ≤ C} consisting of the operations of addition and inner multiplication by a bounded constant as well as upper bounds O(n log n) for th...

In this chapter the idea of using optimal normal bases (o.n.b.) of second and third types in combination with polynomial basis of field F(q n ) is detailed using a new modification of o.n.b. called reduced optimal normal basis –1, β 1, …, β n − 1 corresponding to a permutated o.n.b. β 1, …, β n − 1 Operations of multiplication, rising to power q i...

It is proved that the product of n complex variables can be represented as a sum of m = 2 n−1 n-powers of linear forms of n variables and for any m < 2 n−1 there is no such identity with m summands being nth powers of linear forms.

For the complexity of multiplication in a standard basis of the field GF(2 n ), where n=2·3 k , the upper bound 5nlog 3 nlog 2 log 3 n+O(nlogn) for multiplication complexity and an asymptotically 2.5 times greater bound for inversion complexity are obtained. As a consequence, for the complexity of multiplication of binary polynomials the upper boun...

We give a review of some works on the complexity of implementation of arithmetic operations in finite fields by Boolean circuits.

This work suggests a method for deriving lower bounds for the complexity of polynomials with positive real coefficients implemented by circuits of functional elements over the monotone arithmetic basis . Using this method, several new results are obtained. In particular, we construct examples of polynomials of degree in each of the variables with c...

For an arbitrary Boolean function of n variables, we show how to construct formulas of complexity O(2n/2) in the bases $$\left\{ {x - y,xy,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right], } \left\{ {x - y,x*y,2x,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right],}$$ , where x * y = max(−1, min(1, x))max(−1, min(1, y)). The obtained e...

We show that any function satisfying the Lipschitz condition on a given closed interval can be approximately computed by a scheme (nonbranching program) in the basis composed of functions $$x - y, \left| x \right|, x*y = \min (\max (x,0),1)\min (\max (y,0),1),$$ and all constants from the closed interval [0, 1]; here the complexity of the scheme...

Under consideration is the problem of constructing a square Booleanmatrix A of order n without “rectangles” (it is a matrix whose every submatrix of the elements that are in any two rows and two columns does not consist of 1s). A linear transformation modulo two defined by A has complexity o(ν(A) − n) in the base {⊕}, where ν(A) is the weight of A,...

We prove some improvements for well-known upper bound of complexity of testing irreducibility of polynomials over finite fields. Also the fast modification of well-known probabilistic algorithm finding a normal bases in special finite fields is presented.

We study scheme (hardware) and program (software) methods of multiplication of polynomials over fields of characteristic 7 in order to apply them to pairing-based cryptographic protocols on hyperelliptic curves of genus three. We consider hardware and software implementations of arithmetic in GF(7), GF(72), GF(7n ), GF(77n ), and GF(714n ) and esti...

Let n = (p − 1) · p k , where p is a prime number such that 2 is a primitive root modulo p, and 2p−1 − 1 is not a multiple of p 2. For a standard basis of the field GF(2n ), a multiplier of complexity O(log log p)n log n log logp n and an inverter of complexity O(log p log log p)n log n log logp n are constructed. In particular, in the case p = 3 t...

We suggest a method of realisation of inversion over the standard bases of finite fields GF(p^n) by means of circuits over GF(p) of complexity and depth , where ε > 0, and w < 1.667 is the exponent of multiplication of n^1/2 × n^1/2 and n^1/2 × n matrices. Inversion over Gaussian normal bases is realised by a circuit of complexity O(ε^(-b) n^(1+cε...

The attainability of the lower bound by order is proved for the complexity of a problem of approximation of Lipschitz functions by diagrams of functional elements in bases consisting of a finite number of Lipschitz functions and a continuum of constants from a bounded set.

Some methods are considered for the circuit design of n-digit adders of small depth. The efficiency of synthesis methods is determined by the depth of circuits for n < 1000.

Upper complexity estimates are proved for implementation of Boolean functions by formulas in bases consisting of a finite number of continuous real functions and a continuum of constants. For some bases upper complexity estimates coincide with lower ones.

A survey of methods for constructing bit-parallel circuits for arithmetic operations in finite fields.

It is shown that Lozhkin’s method (1981) for minimization of the depth of formulas with a bounded number of changing types of elements in paths from input to output and Hoover-Klawe-Pippenger’s method (technical report in 1981, journal publication in 1984) for minimization of the depth of circuits with unbounded branching by insertion of trees from...

Рассматриваются методы построения схем n-разрядных суммато-ров малой глубины (эффективность методов синтеза определяетсяглубиной схем при n <1000)

We obtain estimates of complexity and depth of Boolean inverter circuits in normal and polynomial bases of finite fields. In particular, we show that it is possible to construct a Boolean inverter circuit in the normal basis of the field GF(2n) whose complexity is at most (λ(n -1) + (n+ o(1))λ(n)/λ(λ(n)))M(n) and the depth is at most (λ(n - 1) + 2)...

Получены оценки сложности и глубины булевых схем для инвертирования в нор­мальных и полиномиальных базисах конечных полей. В частности, показано, что для инвертирования в нормальном базисе поля GF(2^n) можно построить булеву схему со сложностью не более (l(n — 1) + (1 + о(1))l(n)/l(l(n)))М(n) и глубиной не более (l(n — 1)+ 2)D(n), где М(n), D(n) —...

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with re...

We obtain bounds for the complexity of circuit realisation of the system of differentials of orders from one to k of an arbitrary elementary function in terms of the circuit complexity of this function. Similar bounds are obtained for the complexities of realisation of the Jacobian and Hessian matrices. We point out some applications to deduction o...

Modifications of classical algorithms for multiplication and division of polynomials, which are efficient for a small number of nonzero coefficients of one factor or of the divisor, respectively, are considered. A hybrid algorithm for multiplication of polynomials in GF(2n) that involves a modification of the classical algorithm or of the Karatsuba...

We estimate the complexity of transition from normal bases to standard ones and discuss the related problems of effective realization of arithmetic operations in finite fields of high dimensionality.

Introduction. The algortithm of integration of rationl fractions is one of the most significant and most often used algorithms of analysis. It is well known that the integral of a rational fraction can be represented as the sum of a rational and a logarithmic part, and that the rational part can be separated only by means of arithmetic operations....

We give a fast algorithm for multiplication of polynomials with real-valued coefficients without resort to complex numbers and the fast Fourier transformation. The efficiency of this algorithm is compared with the multiplication algorithm based on the discrete Hartley transformation. We demonstrate that the complexity of the Hartley transformation...

An original scheme for multiplying polynomials based on Karatsuba's method and a modification of the method based on the Discrete Fourier Transform (DFT) are presented. Based on theoretical estimates of the algorithm complexity it is shown that fast variants of the classical algorithm of polynomial multiplication have advantages in runtime for poly...

Let m be a positive integer and Z*m be the set of all positive integeres which are no greater than m and relatively prime to m. A number s ∈ Z*m is called a witness of primality of m if the sequence s(m-1)2-1 (mod m), i = 0,1,...,r, m - 1 = 2rt, where t is odd, consists only of ones, or begins with ones and continues by minus one and, may be, then...

For a number of rational, polynomial and linear bases asymptotic estimates of the complexity and depth of ε-approximation of almost all numbers by schemes in these bases are obtained and a continual analogue of the Shannon effect from the complexity theory of Boolean functions is established. For some monotone linear bases the so-called Shannon sem...

For any function L(ε) satisfying some natural conditions we prove the existence of a ∊ R whose complexity of ε -approximation by schemes in the basis {x±y, xy, x/y, 1} is of order L(ε) and under some additional restrictions on L(ε) the asymptotic equivalence of the complexity and L(ε) is established. A connection between hardly realizable Boolean f...

Without Abstract

We find upper and lower estimates for the complexity of approximate calculation of real analytic functions of one real variable by means of networks of functional elements and formulas.

In this paper we interpret the Berlekamp-Massey algorithm (BMA) for synthesis of linear feedback shift register (LFSR) as an algorithm computing Pade approximants for Laurent series over arbitrary field. This interpretation of the BMA is based on a iterated procedure for computing of the sequence of polynomials orthogonal to some sequence of polyno...

The R. Bellman’s problem on computation of a monomial x 1 n 1 ⋯x m n m and the D. Knuth’s problem on computation of a set of powers x n 1 ,dots,x n m are considered. It is shown that log 2 max 1≤i≤m n i + (1+o(1)) log 2 ∏ i=1 m n i / log 2 log 2 ∏ i=1 m n i + O (m) operations of multiplication is sufficient for the solution of these problems. N. Pi...