A New Method for Differential Cryptanalysis on Addition on Residue Class Ring Z/2nZ

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.

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.

Synthesis of multifunction and universal logic elements over residue class rings. I

Given a positive integer [see formula in PDF] and the residue class ring [see formula in PDF], we set [see formula in PDF] to be the group of units in[see formula in PDF], i.e., [see formula in PDF]. Let [see formula in PDF] be the number of solutions of[see formula in PDF] with [see formula in PDF]. In this note, we determine an explicit expression of [see formula in PDF]. This extends the results of Sun and Yang in 2014.

Several modules M over algebras with straightening law A have a structure which is similar to the structure of A itself: M has a system of generators endowed with a natural partial order, a standard basis over the ring B of coefficients, and the multiplication A × M → A satisfies a “straightening law”. We call them modules with straightening law, briefly MSLs. In section 1 we recall the notion of an algebra with straightening law together with those examples which will be important in the sequel. Section 2 contains the basic results on MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue class rings with respect to them, (ii) “generic” modules defined by generic, alternating or symmetric matrices of indeterminates, (iii) certain modules related to differentials and derivations of determinantal rings. The essential homological invariant of a module is its depth. We discuss how to compute the depth of an MSL in section 4. The main tool are filtrations related to the MSL structure. The last section contains a natural strengthening of the MSL axioms which under certain circumstances leads to a straightening law on the symmetric algebra. The main examples of such modules are the “generic” modules defined by generic and alternating matrices. The notion of an MSL was introduced by the author in [Br.3] and discussed extensively during the workshop. The main differences of this survey to [Br.3] are the more detailed study of examples and the treatment of the depth of MSLs which is almost entirely missing in [Br.3]

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.

On additive properties of two sets in a residue class ring

Residue Class Rings

Generalized bilinear forms graphs and MRD codes over a residue class ring

For a given polynomial f and an ideal I of a polynomial ring K[X] over a field K, we give a necessary and sufficient condition for I to have a smallest ideal extension J such that f is invertible in the residue class ring K[X]/J. If the condition holds, J is shown to be the saturation ideal I:∞. We also show primary decompositions of the ideals I:∞ and I+9fm;:, where m is a natural number such that I:∞ = I:fm, all together forms a primary decomposition of I with no modification. These observations are especially useful for polynomial rings with parameters.

<p style='text-indent:20px;'>Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> prime or composite, cyclotomy on a residue class ring <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{Z}}_{v} $\end{document}</tex-math></inline-formula> can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [<xref ref-type="bibr" rid="b40">40</xref>], we introduce a generalized cyclotomy of order <inline-formula><tex-math id="M4">\begin{document}$ e $\end{document}</tex-math></inline-formula> on the ring <inline-formula><tex-math id="M5">\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q_j $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ i\neq j $\end{document}</tex-math></inline-formula>) may not be co-prime, which includes classical cyclotomy as a special case. Here, <inline-formula><tex-math id="M9">\begin{document}$ q_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ q_k $\end{document}</tex-math></inline-formula> are powers of primes with an integer <inline-formula><tex-math id="M13">\begin{document}$ e|(q_i-1) $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M14">\begin{document}$ 1\leq i\leq k $\end{document}</tex-math></inline-formula>. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and <inline-formula><tex-math id="M15">\begin{document}$ d $\end{document}</tex-math></inline-formula>-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.</p>

Letbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

Permutation polynomials on matrices

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

We shall consider partially ordered rings as a generalization of ordered rings. We give properties or related matters of partially ordered rings, using convex ideals or non-negative semicones. Also, we consider order-preserving isomorphisms between residue class rings which are partially ordered rings.