Residue Ring Modulo Research Articles

ИССЛЕДОВАНИЕ АЛГОРИТМОВ СИНТЕЗА САМОПРОВЕРЯЕМЫХ ЦИФРОВЫХ УСТРОЙСТВ НА ОСНОВЕ ЛОГИЧЕСКОЙ КОРРЕКЦИИ СИГНАЛОВ С ПРИМЕНЕНИЕМ ВЗВЕШЕННЫХ КОДОВ БОУЗА – ЛИНА

When synthesizing self-checking digital devices based on Boolean correction of signals, it is proposed to use weight-based Bose – Lin codes, the construction principles of which imply preliminary weighting of data symbols by natural numbers. Two “basic” structures are proposed for the synthesis of built-in control circuits for groups of six outputs of the diagnostic object. The structures are based on weight-based Bose – Lin codes with summation in the residue ring modulo M=4. There are 15 such noise-protected codes with the number of data symbols m=4, which allows to select the best option as a base code in the builtin control circuit according to various criteria, including achieving self-checking properties even in cases where this cannot be achieved using traditional approaches, including duplication. Two algorithms for the synthesis of built-in control circuits based on Boolean signal correction have been developed, allowing the use of correction of only two of the six functions in the basic structure. For basic structures, there are 720 ways to construct an integrated control circuit based on Boolean correction of signals using each weight-based Bose – Lin code, which makes it possible to choose the best way to implement a self-checking device, considering various indicators (structural redundancy, testability, etc.). The operation of the algorithms is demonstrated on simple examples. The results of experiments with test digital circuits from the MCNC Benchmars set confirming the efficiency of the developed algorithms are given. It is shown that with a large number of outputs, there is an astronomical number of ways to organize built-in control circuits, which makes it possible to build self-checking devices with various characteristics. The use of Boolean correction of signals using weight-based Bose – Lin codes can be used in the development and design of self-checking digital devices on various element bases.

Синтез самопроверяемых цифровых устройств на основе взвешенных кодов с суммированием, проверочные символы которых описываются самодвойственными булевыми функциями

The proposed article considers sum code in a residue ring over a given modulo M of the data vector weighted bits. Test symbols are described by self-dual Boolean functions. It is proven that such codes can be constructed only under the condition M  {2, 4, 8, …} for certain arrays of weighting coefficients. The division into special classes of the weight-based sum code special case in the residue ring modulo M=4 is given. Cases with the number of data bits m=4 are considered. It has been determined that only two options for weighing with weight coefficients from arrays [1, 1, 2, 3] and [1, 2, 2, 2], without considering permutations of weights, give codes with the function selfduality property that describe check symbols. The concurrent error-detection circuit structure based on weight-based sum code in a residue ring by modulo M=4 with the m=4 data bits is given.

Two-Module Weight-Based Sum Code in Residue Ring Modulo M=4

The paper describes research results of features of error detection in data vectors by sum codes. The task is relevant in this setting, first of all, for the use of sum codes in the implementation of the checkable discrete systems and the technical means for the diagnosis of their components. Methods for sum codes constructing are described. A brief overview in the field of methods for sum codes constructing is provided. The article highlights codes for which the values of all data bits are taken into account once by the operations of summing their values or the values of the weight coefficients of the bits during the formation of the check vector. The paper also highlights codes that are formed when the data vectors are initially divided into subsets, in particular, into two subsets. An extension of the sum code class obtained by isolating two independent parts in the data vectors, as well as weighting the bits of the data vectors at the stage of code construction, is proposed.
 The paper provides a generalized algorithm for two-module weighted codes construction, and describes their features obtained by weighing with non-ones weight coefficients for one of data bits in each of the subvectors, according to which the total weight is calculated. Particular attention is paid to the two-module weight-based sum code, for which the total weight of the data vector in the residue ring modulo M = 4 is determined. It is shown that the purpose of the inequality between the bits of the data vector in some cases gives improvements in the error detection characteristics compared to the well-known two-module codes. Some modifications of the proposed two-module weighted codes are described. A method for calculating the total number of undetectable errors in the two-module sum codes in the residue ring modulo M = 4 with one weighted bit in each of the subsets is proposed. Detailed characteristics of error detection by the considered codes both by the multiplicities of undetectable errors and by their types (unidirectional, symmetrical and asymmetrical errors) are given. The proposed codes are compared with known codes. A method for the synthesis of two-module sum encoders on a standard element base of the single signals adders is proposed. The classification of two-module sum codes is presented.

Application of sequential method for constructing a statistical attack on the LPN-C cipher system over residue ring modulo 2N

LPN-C is one of the first post-quantum symmetric cipher systems, which security relies on the complexity of solving the LPN problem. The original version of the cipher system is defined over the field of two elements, nevertheless it is naturally generalized to the case of an arbitrary finite ring. Usually such generalization associated with the complication of the algebraic structure of underlain object used for construction of a cipher system increases its security to known attacks, however, as shown below, the LPN-C cipher system over a residue ring modulo 2Nrepresents an exception to this rule. In this article, an attack on the LPN-C cipher system over the residue ring modulo 2Nis proposed. The attack is based on recovering the key by sequential solving N systems of linear equations corrupted by noise over the field of order two. It is shown that the proposed attack is significantly more effective in comparison with traditional attacks of the same type based on direct solving these systems using the generalized BKW algorithm. The obtained results indicate that the residue rings modulo 2N, N≥2, are not expedient for constructing LPN-C cipher systems, since this does not lead to significant increasing the security in comparison with 1N= case.

Injectivity on distribution of elements in the compressed sequences derived from primitive sequences over ℤ p e $\mathbb {Z}_{p^{e}}$

Let p ≥ 3 be a prime, e ≥ 2 an integer and $\mathbb {Z}_{p^{e}}$ the residue ring modulo pe. Let σ(x) be a primitive polynomial of degree n over $\mathbb {Z}_{p^{e}}$ and let G′(σ(x), pe) be the set of primitive linear recurring sequences over $\mathbb {Z}_{p^{e}}$ generated by σ(x). A compressing mapping $\varphi :\mathbb {Z}_{p^{e}}\rightarrow \mathscr {A}$ naturally induces a mapping $\widehat {\varphi }$ on G ′(σ(x), pe), i.e., $\widehat {\varphi }$ maps a sequence (…,si− 1,si,si+ 1,… ) to (…,φ(si− 1),φ(si),φ(si+ 1),… ). For any pair of sequences in $\{\widehat {\varphi }(\underline {s}):\underline {s}\in G^{\prime }_{~}(\sigma (x),p^{e})\}$ , it is desirable to determine whether (at least) one element of $\mathscr {A}$ is distributed differently in them. For $\emptyset \neq D\subseteq \mathscr {A}$ , $\widehat {\varphi }$ is said to be injective on G ′(σ(x), pe) w.r.t. D-uniformity if for any two distinct sequences $\underline {u},\underline {v}\in G^{\prime }_{~}(\sigma (x),p^{e})$ , the distribution of at least one element of D in $\widehat {\varphi }(\underline {u})$ differs from that in $\widehat {\varphi }(\underline {v})$ . A sufficient condition on φ is given to ensure that $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. If $\left (\left ((x^{p^{n}-1}-1)^{2}\bmod \sigma (x)\right ) \bmod p^{3}\right ) \notin p^{2}\mathbb {Z}_{p}$ , then an equivalent condition on φ is obtained to decide whether $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. Furthermore, quantitative estimation suggests that almost all mappings on $\mathbb {Z}_{p^{e}}$ induce injective mappings on G ′(σ(x), pe) as p and e increase.

Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over ${\bf Z}/(p^{e})$

Let p be an odd prime number, e an integer greater than 1, and Z/(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> ) the integer residue ring modulo p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> . In this paper, we obtain an improved result of the previous paper (IEEE Trans. Inf. Theory, 56(1) (2010) 555-563) on distribution properties of compressing sequences derived from primitive sequences over Z/(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> ). It is shown that two primitive sequences α and b generated by a strongly primitive polynomial f(x) over Z/(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> ) are the same, if there exist s∈ Z/(p) and k∈ Z(p)* such that the distribution of in their compressing sequences a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-1</sub> +η(a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,⋯.a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> ) and b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-1</sub> +η(b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,⋯,b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> ) is coincident at the positions t with α(t)=k, where η(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,⋯,x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> ) is an (e-)-variable polynomial over Z/(p) with the coefficient of x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p-1</sup> ⋯x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p-1</sup> x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p-1</sup> not equal to (-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> · (p+1)/2 and α is an m-sequence over Z/(p)determined by f(x) and a. Compared with the previous result, this gives a more precise characterization on the positions of a compressing sequence, i.e., of the form a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-1</sub> +η(a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> ), derived from a primitive sequence a over Z/(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> ) that completely determines a. In particular, the result is also true for the highest level sequence ae <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sub> by taking η(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,⋯,x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e-2</sub> )=0.

Further Results on the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

This paper studies the distinctness of primitive sequences over Z/(M) modulo 2, where M is an odd integer that is composite and square-free, and Z/(M) is the integer residue ring modulo M. A new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f(x) over Z/(M) are pairwise distinct modulo 2. Such result improves a recent result obtained in our previous paper, and consequently, the set of primitive sequences over Z/(M) that can be proven to be distinct modulo 2 is greatly enlarged.

On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> be a square-free odd integer and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</b> /( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) the integer residue ring modulo <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> . This paper studies the distinctness of primitive sequences over <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</b> /( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) modulo 2. Recently, for the case of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pq</i> , a product of two distinct prime numbers <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> , the problem has been almost completely solved. As for the case that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">'</sup> +1 over <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</b> /( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) is distinct modulo 2, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">'</sup> is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</b> /( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ), which are related closely to our main results.

On the distinctness of modular reductions of primitive sequences modulo square-free odd integers

Let M be a square-free odd integer with at least two different prime factors and Z/(M) the integer residue ring modulo M. In this paper, it is shown that for two primitive sequences a̲=(a(t))t⩾0 and b̲=(b(t))t⩾0 generated by a primitive polynomial of degree n over Z/(M), a̲=b̲ if and only if a(t)≡b(t)modH for all t⩾0, where H>2 is an integer divisible by a prime number coprime with M. This result is obtained basing on the assumption that every element in Z/(M) occurs in a primitive sequence of order n over Z/(M), which is known to be valid for most Mʼs if n>6.

A new result on the distinctness of primitive sequences over [formula omitted] modulo 2

Let Z / ( p q ) be the integer residue ring modulo pq with odd prime numbers p and q. This paper studies the distinctness problem of modulo 2 reductions of two primitive sequences over Z / ( p q ) , which has been studied by H.J. Chen and W.F. Qi in 2009. First, it is shown that almost every element in Z / ( p q ) occurs in a primitive sequence of order n > 2 over Z / ( p q ) . Then based on this element distribution property of primitive sequences over Z / ( p q ) , previous results are greatly improved and the set of primitive sequences over Z / ( p q ) that are known to be distinct modulo 2 is further enlarged.

Solvability of the problem of taking the discrete logarithm

It is proved that consideration of the solvability problem for taking the discrete logarithm in a residue ring modulo composite M can be reduced to consideration of a similar problem in residue rings modulo pq, where p and q are prime divisors of M. For moduli of form pq, necessary and sufficient conditions for solvability checking are obtained in some cases. In addition, the problem of raising a solution of an exponential comparison in a residue ring modulo p α is solved.

Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

We show that, for sufficiently large integersm and X, for almost all a = 1,...,m the ratios a/x and the products ax, where |x| 6 X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums

A Note on Low-Correlation Zone Signal Sets

In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size, integer residue ring modulo , and the subset in the complex field which consists of powers of a primitive th root of unity. We show a connection between these sequence designs and ldquocompletely noncyclicrdquo Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction.

On the regular Sylow p-subgroups of Chevalley groups over $$\mathbb{Z}_{p^m } $$

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo pm is regular for n2 < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring modulo M for an ?almost squarefree? integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over we obtain a stronger bound ?on average? over all initial values. This bound is new even in the case when M = p is prime.

Distribution of exponential functions with squarefull exponent in residue rings

For a real x ≥ 1 we denote by S[ x] the set of squarefull integers n ≤ x, that is, the set of positive integers n ≤ such that l 2| n for any prime divisor l| n. We estimate exponential sums of the form T a (m,x) = ∑ n∈S[x] exp(2πiaϑ n/m) where ν is a fixed integer with gcd( ν, m) = 1, and apply them to studying the distribution of the powers ν n , n ϵ S[ x], in the residue ring modulo m ≥ 1.

Double exponential sums over thin sets

We estimate double exponential sums of the form Sa(X, Y) = E E exp(27riaI0xY/p), xGE X yG Y where V is of multiplicative order t modulo the prime p and X and Y are arbitrary subsets of the residue ring modulo t. In the special case t = p 1, our bound is nontrivial for I XI > I Yl > p15/16+6 with any fixed 6 > 0, while if in addition we have Xl > p1-6/4 it is nontrivial for I YI ? p3/4+6. Let p be a prime and let Fp be a finite field of p elements. For an integer m > 1 we denote by Z,m = {0, . .. , m 1} the residue ring modulo m. We also identify Fp with the set {o,... ,p-1}. Finally we define e(z) = exp(27ri/p) and use log z for the natural logarithm of z. Throughout the paper the implied constants in symbols 'O', ' ' may occasionally, where obvious, depend on the small positive parameter E and are absolute otherwise (we recall that A A are equivalent to A 0 (B)). We fix an element ) E IFp of multiplicative order t, that is, 79s 7 1, 1 pl5/16+6 with any fixed 6 > 0. Further examples are given below. Our results rely on the following estimate for certain double exponential sums from [1]; see the proof of Theorem 8 of that paper. Let A E Fp be of multiplicative order T. For any a, b (E IF we have the bound 4 (1) E| e(a\v +bAuv) 1 satisfies (2) T(m) < m(1). 2 Our main estimate is the following. Theorem. For any sets X, Y C Zt, the bound max ISa( X, Y)| << I x 1/21 yI5/6tl/2pl/8+E aEF* p holds. Proof. For a divisor dlt we denote by Y(d) the subset of y Y Y with gcd(y, t) d. Then ISa(X, Y)I ? Z ud| dlt