Search Results (11)

It is demonstrated that neuromorphic materials designed for computational tasks can be effectively implemented by drawing an analogy with trigger-based systems built upon classical binary elements. Among the most promising approaches in this context are systems that perform computations based on the Residue Number System (RNS). A specific implementation of a trigger-based adder employing the proposed methodology is presented and tested through simulation modeling. This adder utilizes the representation of natural numbers as elements of a subtraction ring modulo P, where P is the product of Mersenne prime numbers. This configuration enables component-wise, independent execution of arithmetic operations. It is further shown that analogous trigger-based systems can be realized using recurrent neural network analogs, particularly those implemented with neuromorphic materials. The study emphasizes that it is possible to construct a neural network, especially one based on neuromorphic substrates, that can perform logical operations equivalent to those executed by conventional binary circuitry. A key challenge in the proposed approach lies in implementing an operation analogous to the carry mechanism employed in classical binary adders. An algorithm addressing this issue is proposed, based on the transition from computations modulo P to computations modulo 2P. Full article

This paper introduces a novel method for selecting parameters of finite fields formed by 2 × 2 matrices over a finite field of integers modulo a prime p. The method aims to simultaneously determine both the field parameters and primitive elements, thereby optimizing the construction of cryptographic algorithms. The proposed approach leverages the properties of quadratic residues and non-residues, simplifying the process of finding matrix field parameters while maintaining computational efficiency. The method is particularly effective when the prime number p is either a Mersenne prime or (p + 1)/2 is also a prime. This study demonstrates that the resulting matrix fields can be practically computed, offering a high degree of flexibility for cryptographic protocols such as key agreement and secure data transmission. Compared to previous methods, the new method reduces the parameter search space and provides a structured way to identify primitive elements without the need for a separate search procedure. The findings have significant implications for the development of efficient cryptographic systems using matrix-based finite fields. Full article

Mersenne prime numbers, expressed in the form (2ⁿ − 1), have long captivated researchers due to their unique properties. The presented work aims to develop a symmetric cryptographic algorithm using a novel technique based on the logical properties of Mersenne primes. Existing encryption algorithms exhibit certain challenges, such as scalability and design complexity. The proposed novel modular multiplicative inverse property over Mersenne primes simplifies the encryption/decryption process. The simplification is achieved by computing the multiplicative inverse using cyclic bit shift operation. The proposed image encryption/decryption scheme involves a series of exor, complement, bit shift, and modular multiplicative inversion operations. The image is segmented into blocks of 521 bits. Each of these blocks is encrypted using a 521-bit key, ensuring high entropy and low predictability. The inclusion of cyclic bit shifting and XOR operations in the encryption/decryption process enhances the diffusion properties and resistance against attacks. This approach was experimentally proven to secure the image data while preserving the image structure. The experimental results demonstrate significant improvements in security metrics, including key sensitivity and correlation coefficients, confirming the technique’s effectiveness against cryptographic attacks. Overall, this method offers a scalable and secure solution for encrypting high-resolution digital images without compromising computational efficiency. Full article

It is shown that a serial-parallel processor, comparable in bit capacity to a 16-bit binary processor, can be implemented based on an algorithm built on the residue number system, a distinctive feature of which is the use of the first four quasi-Mersenne numbers, i.e., prime numbers representable as $p_k=2^k+1, k=1,2,3,4$. Such a set of prime numbers satisfies the criterion $2p_1{p}_2{p}_3{p}_4+1=P$, where $P$ is also a prime number. Fulfillment of this criterion ensures the possibility of convenient use of the considered RNS for calculating partial convolutions developed for the convenience of using convolutional neural networks. It is shown that the processor of the proposed type can be based on the use of a set of adders modulo a quasi-Mersenne number, each of which operates independently. A circuit of a modulo $2^k+1$ adder is proposed, which can be called a trigger circuit, since its peculiarity is the existence (at certain values of the summed quantities) of two stable states. The advantage of such a circuit, compared to known analogs, is the simplicity of the design. Possibilities for further development of the proposed approach related to the use of the digital logarithm operation, which allows reducing the operations of multiplication modulo $2^k+1$ to addition operations, are discussed. Full article

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing. Full article

A new generalized definition of Mersenne numbers is proposed of the form $\left(a^n,-,{\left(a,-,1\right)}^n\right)$, called global generalized Mersenne numbers and noted $G{M}_{a,n}$ with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, $\left(G,M_{a,n},-,1\right)$ is even and divisible by n, a and $\left(a,-,1\right)$ for any prime $n>2$, and by $\left(a,\left(a,-,1\right),+,1\right)$ for any prime $n>5$. The remaining factor is a function of triangular numbers of $\left(a,-,1\right)$, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that $G{M}_{a,n}\equiv \left(1,,\mathrm{or},,7\right)\left(\mathrm{mod},,12\right)$ depending on the congruence of $a\left(\mathrm{mod},,4\right)$; second, that $\left(G,M_{a,n},-,1\right)$ are divisible by 10 if $n\equiv 1\left(\mathrm{mod},,4\right)$ and, if $n\equiv 3\left(\mathrm{mod},,4\right)$, $G{M}_{a,n}\equiv \left(1,,\mathrm{or},,7,,\mathrm{or},,9\right)\left(\mathrm{mod},,10\right)$, depending on the congruence of $a\left(\mathrm{mod},,5\right)$; third, that all factors $c_i$ of $G{M}_{a,n}$ are of the form $\left(2,n,f_i,+,1\right)$ such that $c_i$ is either prime or the product of primes of the form $\left(2,n,j,+,1\right)$, with $f_i,j$ natural integers; fourth, that for prime $n>2$, all $G{M}_{a,n}$ are periodically congruent to $\left(\pm ,1,,\mathrm{or},,\pm ,3\right)\left(\mathrm{mod},,8\right)$ depending on the congruence of $a\left(\mathrm{mod},,8\right)$; and fifth, that the factors of a composite $G{M}_{a,n}$ are of the form $\left(2,n,f_i,+,1\right)$ with $f_i\equiv u\left(\mathrm{mod},,4\right)$ with $u=0$, 1, 2 or 3 depending on the congruences of $n\left(\mathrm{mod},,4\right)$ and of $a\left(\mathrm{mod},,8\right)$. The potential use of generalized Mersenne primes in cryptography is shortly addressed. Full article

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if $\mathbb{F}$ is a subfield of a local field of characteristic $\ne 2$, then the special upper triangular group ${\mathrm{ST}}^{+}(n,\mathbb{F})$ is minimal precisely when the special linear group $SL(n,\mathbb{F})$ is. We provide criteria for the minimality (and total minimality) of $SL(n,\mathbb{F})$ and ${\mathrm{ST}}^{+}(n,\mathbb{F}),$ where $\mathbb{F}$ is a subfield of $\mathbb{C}$. Let ${\mathcal{F}}_{\pi }$ and ${\mathcal{F}}_c$ be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for $\mathcal{A}\in \{{\mathcal{F}}_{\pi },{\mathcal{F}}_c\}$: $\mathcal{A}$ is finite; ${\prod }_{{\mathbb{F}}_n\in \mathcal{A}}SL(F_n-1,\mathbb{Q}\left(i\right))$ is minimal, where $\mathbb{Q}\left(i\right)$ is the Gaussian rational field; and ${\prod }_{{\mathbb{F}}_n\in \mathcal{A}}{\mathrm{ST}}^{+}({\mathbb{F}}_n-1,\mathbb{Q}\left(i\right))$ is minimal. Similarly, denote by ${\mathcal{M}}_{\pi }$ and ${\mathcal{M}}_c$ the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let $\mathcal{B}\in \{{\mathcal{M}}_{\pi },{\mathcal{M}}_c\}.$ Then the following conditions are equivalent: $\mathcal{B}$ is finite; ${\prod }_{M_p\in \mathcal{B}}SL(M_p+1,\mathbb{Q}\left(i\right))$ is minimal; and ${\prod }_{M_p\in \mathcal{B}}{\mathrm{ST}}^{+}(M_p+1,\mathbb{Q}\left(i\right))$ is minimal. Full article

Lockable obfuscation, a new primitive that occurs in cryptography, makes it possible to execute arbitrary polynomial-sized functions and recover a secret under specific equality conditions. More concretely, if the function executed over a specific input produces an output that matches an expected target value, here denoted by a, some secret string of bits s is exposed. Written in algebraic terms, if $f:\mathcal{X}\to \mathcal{A}$ has the property that for some $x,f\left(x\right)=a$, s is revealed. This work explores the possibility for safely decrypting ciphertexts, and based on the recovered plaintext’s equality to a stored message, to reveal some secret. Concretely, this work provides a review of existing, well-known public key encryption schemes and argues for the efficiency of a new one relying on the ratio Mersenne hypothesis (RMERS), which is to be used in conjunction with a lockable obfuscator. This work explores the advantage conferred by this scheme, especially in the minimization of the branching program’s number of levels that need to be obfuscated. The drawbacks of such schemes are also pointed out, given that they currently require the $\mathsf{LWE}$ evaluations level-per-level, one output bit at a time. Full article

This paper proposes a code defined on a finite ring ${\mathbb{Z}}_{p_M}$, where $p_M$ = $2^m-1$ is a Mersenne prime, and m is a binary size of ring elements. The code is based on a splitting sequence (splitting set) $\mathcal{S}$, defined for the given multiplier set $\mathcal{E}=\left\{\pm 2^0, \pm 2^1,\dots , \pm 2^{m-1}\right\}$. The elements of $\mathcal{E}$ correspond to the weights of binary error patterns that can be corrected, with the bidirectional single-bit error being the representative that occurs the most. The splitting set splits the code-word into sub-words, which inspired the name splitting code. Each sub-word, provided with auxiliary control symbols that are a byproduct of the coding procedure, corrects a single symbol error. The code can be defined, with some constraints, for general Mersenne numbers as well, while the multiplier set can be adjusted for adjacent binary errors correction. The application proposed for this code is a hybrid three-stage incremental ARQ procedure that transmits the code-word in the first stage, auxiliary control symbols in the second stage, and retransmits the sub-words detected as incorrect in the third stage. At each stage, error correction can be turned on or off, keeping both the retransmission rate and residual error rate at a low level. Full article

The order and disorder of binary representations of the natural numbers < 2⁸ is measured using the BiEntropy function. Significant differences are detected between the primes and the non-primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a Monte Carlo simulation of a sample of natural numbers < 2³² and in trinary for all natural numbers < 3⁹ with similar but cubic results. We found a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical basis of these results and show how they generalise to give a tight bound on the variance of Pi(x)–Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices. Full article

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer $k\ge 2$ . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the $p^e$ -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form ${\displaystyle \frac{p^{em}-1}{p^e-1}}$ with some integer $m\ge 1$ . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ${(x+1)}^{p^e}\equiv {(x^p+1)}^{p^{e-1}}(\mathrm{mod}{p}^e)$ of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. Full article