Residue Class Ring with Identical Representation Function

The universal binary neurons are a central point of this Chapter. The mathematical models of the universal binary neuron over the field of the complex numbers, the residue class ring and the finite field are considered. A notion of the P-realizable Boolean function, which may be considered as a generalization of a notion of the threshold Boolean function, is introduced. It is shown that the implementation with a single neuron the input/output mapping described by non-threshold Boolean functions is possible, if weights are complex, and activation function of the neuron is a function of the argument of the weighted sum (similar to multi-valued neuron). It is also possible to define an activation function on the residue class ring and the finite field in such a way that the implementation of non-threshold Boolean functions will be possible on the single neuron with weights from these sets. P-realization of multiple-valued function over finite algebras and residue class ring in particular is also considered. The general features of P-realizable Boolean functions are considered, also as the synthesis of universal binary neuron.

Decomposition of linear sequential circuits over residue class rings

We consider linear parametric descriptions of logic functions over residue class rings and present minimal descriptions for boolean functions of two, three, and four variables.

The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{v}$</tex></formula> can be divided into classical cyclotomy or generalized cyclotomy, depending on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$v$</tex> </formula> prime or composite. In this paper, we introduce a generalized cyclotomy of order <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex></formula> over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots, p_{n}^{e_{n}}}$</tex></formula> , which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p_{1},p_{2},\ldots,p_{n}$</tex></formula> are pairwise distinct odd primes satisfying <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d\vert (p_{i}-1)$</tex></formula> for all <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1\leq i\leq n$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$e_{1},e_{2},\ldots,e_{n}$</tex></formula> are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{p_{1}p_{2}}$</tex> </formula> . Besides, we also construct an infinite series of near-optimal codebooks over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\BBZ}_{p_{1}p_{2}}$</tex></formula> , as well as some infinite series of asymptotically optimal difference systems of sets over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots,p_{n}^{e_{n}}}$</tex> </formula> .

In this chapter, we show how to compute in residue class rings and their multiplicative groups. We also discuss algorithms for finite abelian groups. These techniques are of great importance in cryptographic algorithms.KeywordsPrime NumberFinite FieldCommutative RingUnit GroupMultiplicative GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The K-vector space structure on residue class rings of polynomial rings has already been used in Section 6.3 in connection with zero-dimensional ideals. An important result was that an ideal I is zero-dimensional if and only if the residue class ring modulo I is finite-dimensional as a K-vector space. In this chapter we discuss a number of important algorithms that use linear algebra in connection with Grobner bases. The focus is on zero-dimensional ideals.

In this paper, we propose a Combined Public Key Cryptosystem based on Conic Curves (CPK-CCC) over the residue class ring without trusted third parties, assume the existence of trusted Key Management Center (KMC). The security of this scheme relies on the problem of factorizing large integer and computing discrete logarithm on conic over the residue class ring. The implementation principles of CPK-CCC scheme and its designing structure are described in details. We employs seeded secret technique combine technique for generation of userpsilas Private/Secret Key (SK) and Public Key (PK) pairs for requested users. Moreover a precise definition of the CPK-CCC cryptosystem is given. Property analysis of the new scheme is also made in comparison with that of the other cryptosystems, which leads to the conclusion that CPK-CCC cryptosystem may be applied as an optimized approach towards efficient and effective network authentication.

On Gröbner bases and Krull dimension of residue class rings of polynomial rings over integral domains

An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring $\mathbb{Z}_N$, N=pq for two distinct odd primes p and q. The result is a generalization of an estimate of exponential sums on rational point groups of elliptic curves over finite fields. The bound is applied to showing the pseudorandomness of a large family of binary sequences constructed by using elliptic curves over $\mathbb{Z}_N$.

Let us recall some classes of rings. A ring R is said to be k-nil-clean if each element can be written as a sum of a nilpotent and k idempotents. A ring R is said to be fine if each non-zero element can be written as a sum of a unit and a nilpotent. A ring R is called nil-good if every element is a nilpotent or a sum of a nilpotent and a unit. And, finally, ring R is called nil-good clean if every element is a sum of a nilpotent, an idempotent, and a unit. In this paper, we continue our work on additive problems in formal matrix rings over residue class rings. We have found necessary and sufficient conditions for the nilpotency of a formal matrix over residue class rings. After that we have shown that a ring of such matrices is (p –1)-nil-clean and nil-good clean. Also, answering the question posed in the previous article of the second co-author, we prove that a ring of formal matrices over residue rings is never nil-good, and, therefore, not fine.

The work is devoted to some arithmetic applications to the theory of symmetric groups.Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group 𝑆_𝑛 of degree 𝑛 of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisned 𝑘, 𝜙(𝑘), and 𝑘𝜙(𝑘), where 𝑘 ≤ 𝑛, 𝜙 – Euler function, those representations jf grups (Z/𝑘Z, +), (Z/𝑘Z)* and theorem product in the form of degree substitutions 𝑘. In this case isomorphic embeddings of these groups are constructed following the proof of Cayley’s theorem,but along with this, a linear binomial is used Z/𝑘Z residue class rings, where gcd (𝑎, 𝑘) = 1.In addition, the result concerning the isomorphic embedding of a group (Z/𝑘Z)* in to a group (Z/𝑘Z)* in to a group 𝑆_𝑘 extends to an alternating group 𝐴_𝑘 for odd 𝑘.The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group 𝑆_𝑛. In particular, applying the Euler-Maclaurin summation formula and bounds for the 𝑘 in prime, a lower bound for maximum number of prime divisors of cyclic orders in the symmetric group 𝑆_𝑛.

Let \(n\ge 1, e\ge 1, k\ge 2\) and c be integers. An integer u is called a unit in the ring \({\mathbb {Z}}_n\) of residue classes modulo n if \(\gcd (u, n)=1\). A unit u is called an exceptional unit in the ring \({\mathbb {Z}}_n\) if \(\gcd (1-u,n)=1\). We denote by \({\mathcal {N}}_{k,c,e}(n)\) the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^e+\cdots +x_k^e\equiv c \pmod n\) with all \(x_i\) being exceptional units in the ring \({\mathbb {Z}}_n\). In 2017, Mollahajiaghaei presented a formula for the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^2+\cdots +x_k^2\equiv c\pmod n\) with all \(x_i\) being the units in the ring \({\mathbb {Z}}_n\). Meanwhile, Yang and Zhao gave an exact formula for \({\mathcal {N}}_{k,c,1}(n)\). In this paper, by using Hensel’s lemma, exponential sums and quadratic Gauss sums, we derive an explicit formula for the number \({\mathcal {N}}_{k,c,2}(n)\). Our result extends Mollahajiaghaei’s theorem and that of Yang and Zhao.

Let G k be the de Bruijn graph with 2k vertices. When reproduction labels from a 4-letter alphabet A 4 are assigned linearly to the edges of G k using a three row binary generating matrix, a 2k-state trellis code results which is suitable for encoding a source with alphabet A 4 at a rate of one bit per source sample with respect to the Hamming fidelity criterion. There is a special type of code which results in this way called systematic code. A systematic code belongs to three sets of codes, each set isomorphic to a residue class ring of polynomials over GF(2). A systematic code sweeps out a cycle in each of its three residue class rings via repeated multiplication by the residue class of polynomial x. The local coordinates of a systematic code are non-negative integer parameters which describe how the code is positioned in each of its three cycles. Our main result is that a systematic code is uniquely determined by its local coordinates, except in rare instances. The paper concludes with some preliminary work indicating how a code with good distortion performance in coding a memoryless source with alphabet A 4 can be found in relatively small sets of systematic codes whose local coordinates are of a certain type.

Reduced Gröbner bases and Macaulay–Buchberger Basis Theorem over Noetherian rings

Sums of exceptional units in residue class rings