Unique Prime Cartesian Factorization of Graphs Over Finite Fields

Implementing an improved ID‐based cryptographic mechanism is the main objective of the proposed work. In this article, an ID‐based cryptographic (IBC) technique using generalized discrete logarithm problem (GDLP) and the integer factorization problem (IFP) presented by Meshram et al have improved. Meshram et al have given IBC technique without using the bilinear pair and also, reveal that their technique can attain data protection and security objectives. Besides, their technique will deter the adversary from eavesdropping the encrypted information or the secret key of the user. However, it has found that their system carries a deadlock problem. Encryption process, as expected by the user, is not guaranteed to be secure. It is because the user may require private information about key authentication center (KAC), which has kept secret from users. Pang et al have proposed an improved technique that overcomes the deadlock problem. It has found that Pang et al have not discussed the analysis and proofs of security. In this article, generalized discrete logarithms in multiplicative group over finite fields and IFP have used to improve the technique and also a key distribution system has discussed. It has analyzed that the proposed strategy is safer than the technique described by Pang et al. Also, it has found that the proposed technique addresses the deadlock problem.

<!-- *** Custom HTML *** --> This chapter is mostly a review, discussing unique factorization of positive integers, unique factorization of polynomials whose coefficients are rational or real or complex, signs of permutations, and matrix algebra. Sections 1–2 concern unique factorization of positive integers. Section 1 proves the division and Euclidean algorithms, used to compute greatest common divisors. Section 2 establishes unique factorization as a consequence and gives several number-theoretic consequences, including the Chinese Remainder Theorem and the evaluation of the Euler $\varphi$ function. Section 3 develops unique factorization of rational and real and complex polynomials in one indeterminate completely analogously, and it derives the complete factorization of complex polynomials from the Fundamental Theorem of Algebra. The proof of the fundamental theorem is postponed to Chapter IX. Section 4 discusses permutations of a finite set, establishing the decomposition of each permutation as a disjoint product of cycles. The sign of a permutation is introduced, and it is proved that the sign of a product is the product of the signs. Sections 5–6 concern matrix algebra. Section 5 reviews row reduction and its role in the solution of simultaneous linear equations. Section 6 defines the arithmetic operations of addition, scalar multiplication, and multiplication of matrices. The process of matrix inversion is related to the method of row reduction, and it is shown that a square matrix with a one-sided inverse automatically has a two-sided inverse that is computable via row reduction.

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply‐vertex‐transitive Γ‐decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply‐vertex‐transitive Γ‐decomposition of Km × e for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α‐labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T5 or the wheel W6. We also show that sometimes strong difference families lead to regular Γ‐decompositions of a complete graph. We construct, for instance, a regular cube‐decomposition of K16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008

In this paper, we introduce a new method to solve the elliptic curve discrete logarithm probelm (ECDLP) over a finite field by using the elliptic curve lifting problem. Moreover, we propose to find a non-trivial point in E1(Q) in order to get a lifted elliptic curve with rank smaller than the number of lifted points. By this method, we conclude that finding a non-trivial point in E1(Q) implies solving the ECDLP, the discrete logarithm problem (DLP) and the integer factorization problem (IFP). Finally, we find that the minimum of canonical height of a point in E1(Q) is almost O(|Ẽ(Fp)|), which means that it is too large to be found by the brute force search.

Discrete Fourier transforms (DFTs) over finite fields have widespread applications in various communication and storage systems. Hence reducing the computational complexities of DFTs is of great significance. Recently proposed cyclo-tomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexities. Unfortunately, they have very high additive complexities. Techniques such as common subexpression elimination (CSE) can be used to reduce the additive complexities of CFFTs, but their effectiveness for long DFTs is limited by their complexities. In this paper, we propose prime factor cyclotomic Fourier transforms (PFCFTs), which use CFFTs as sub-DFTs via the prime factor algorithm. When the length has co-prime factors, the short lengths of the sub-DFTs allow us to use CSE to significantly reduce their additive complexities. In comparison to previously proposed fast Fourier transforms, our PFCFTs achieve reduced overall complexities when the lengths of DFTs are at least 255, and the improvement significantly increases as the length grows. This approach enables us to propose the first efficient DFTs with very long length (e.g., 4095-point) in the literature. Finally, our PFCFTs are also advantageous for hardware implementation due to their regular structure.

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if n has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs.We present a normal form for collisions of any number of decompositions with any number of components and describe exactly the (non)uniqueness of the parameters in the tame case. This yields an exact formula for the number of such collisions over a finite field. We conclude with a fast algorithm for the exact number of decomposable polynomials of degree n over a finite field Fq of characteristic coprime to n.

Summary form only given. The reliability assessment of future distribution networks is an important subject due to the increasing demand for more reliable service with a lower frequency and duration of interruption. The connection of future distribution networks may not be simple radial circuits and analyzing the reliability of such a network is usually complicated and time consuming. In this paper, a technical approach for evaluating the reliability of a networked distribution system is described. The proposed algorithm is based on the identification of circuit minimal tie sets using the concept of Petri nets. Prime number encoding and unique prime factorization are then applied to classify the remaining combinations as tie sets, cut sets, or minimal cut sets. A Markov model is used to compute the availability and failure frequency of the network. A well-known test bed [the Roy Billinton Test System (RBTS)] is used to illustrate the analysis and calculate different load and system reliability indices, including SAIFI and SAIDI. The method shown is algorithmic and appears suitable for offline comparisons of alternative secondary distribution system designs on the basis of their reliability.

The reliability assessment of future distribution networks is an important subject due to the increasing demand for more reliable service with a lower frequency and duration of interruption. The connection of future distribution networks may not be simple radial circuits and analyzing the reliability of such a network is usually complicated and time consuming. In this paper, a technical approach for evaluating the reliability of a networked distribution system is described. The proposed algorithm is based on the identification of circuit minimal tie sets using the concept of Petri nets. Prime number encoding and unique prime factorization are then applied to classify the remaining combinations as tie sets, cut sets, or minimal cut sets. A Markov model is used to compute the availability and failure frequency of the network. A well-known test bed [the Roy Billinton Test System (RBTS)] is used to illustrate the analysis and calculate different load and system reliability indices, including SAIFI and SAIDI. The method shown is algorithmic and appears suitable for offline comparisons of alternative secondary distribution system designs on the basis of their reliability.

We have seen that number rings are not always unique factorization domains: Elements may not factor uniquely into irreducibles. (See exercise 29, chapter 1, and exercise 15, chapter 2 for examples of non-unique factorization.) However we will prove that the nonzero ideals in a number ring always factor uniquely into prune ideals. This can “be regarded as a generalization of unique factorization in ℤ, where the ideals are just the principal ideals (n) and the prime ideals are the ideals (p), where p is a prime integer.

The rapid development of quantum computing constitutes a significant threat to modern Public-Key Cryptography (PKC). key, signature and Key Encapsulation Mechanism (KEM) sysThe use of Shor's algorithm [1] with potential powerful tem based on a double exponentiation with matrix exponents. quantum computers could easily break the two most widely In this paper, a high-throughput pipelined architecture of used public key cryptosystems, namely, RSA and Elliptic DME-(3,2,48), with $m =3 , n =2$, and $q = 2 ^{e}\,= 2 ^{48}$, has been Curve Cryptography (ECC), based on integer factorization and presented and hardware implementations over Xilinx FPGAs discrete logarithm problems. For this reason, Post-Quantum Artix-7 have been performed. These settings correspond with Cryptography (PQC) [2] based on alternative mathematical the reference implementation given in the NIST proposal [7], features has become a fundamental research topic due to its where the finite field $GF( 2 ^{48})$ is generated by the type I resistance against quantum computers. The National Institute of Standards and Technology (NIST) has even opened a call for proposals of quantum-resistant PKC algorithms in order to standardize one or more PQC algorithms. Cryptographic systems that appear to be extremely difficult to break with large quantum computers are hash-based cryptography, lattice-based cryptography, code-based cryptography, and multivariate-quadratic cryptography [3]. Furthermore, efficient hardware implementations are highly required for these alternative quantum-resistant cryptosystems [4].

Group based transactions are becoming common via handhelds. Single key based systems may not be able to meet various security requirements. In this paper, we propose a threshold signature scheme based on Pedersen distributed key generation principle which is suitable for handheld devices and ad-hoc networks. Existing distributed key generation protocols use either cryptosystems based on the hardness of discrete logarithm over a finite field or integer factorization. Elliptic curve cryptosystems provide a promising alternative with efficiency which is suitable for low-power devices in terms of memory and processing overhead. In the proposed scheme, the public key from the key generation protocol follows a uniform distribution in the elliptic curve additive group, and the signature can be generated and verified efficiently. We evaluated the proposed key generation protocol and signature scheme using PARI/GP, and the key generation time takes a fraction of a second and the signature signing and verifying can be finished in a few milliseconds on the LINUX Intel PXA 255 processor.

Unique Prime Factorization and Lattice Points

Unique Prime Factorization and Lattice Points

‘Conjectures and theorems’ investigates a number of topics, such as the distribution of prime numbers, and two unsolved problems, Goldbach’s conjecture and the twin prime conjecture. The factorization of positive integers into primes is unique, but this does not hold for certain other systems of numbers. A more in-depth look at unique factorization gives deeper results, including a proposed result of Gauss. Mathematicians in the 1950s and 1960s confirmed that he was correct, as shown in the so-called ‘Baker-Heegner-Stark theorem’.

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n &gt; 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.