RSA cryptanalysis — Fermat factorization exact bound and the role of integer sequences in factorization problem

Integer factorization as subset-sum problem

We explore certain sequences of integers which appear in the number theory.We start by exploring properties of Beatty sequences.We concentrate on looking at the sum of primes from a Beatty sequence and properties of certain multiplicative functions on a Beatty sequence.We move on to the Robin and Nicolas inequalities and consider sequences with certain properties which must satisfy these.Next is we explore certain sequences of composite integers which are similar to those of the primes, mainly Carmichael, Guiga, and Lucas numbers.Finally we discuss Descartes numbers, and determine all such numbers with certain other properties.

Notice of Violation of IEEE Publication Principles<br><br>"Modified Integer Factorization Algorithm using V-Factor Method"<br> by Prashant Sharma, Amit Kumar Gupta and Ashish Vijay<br> in the Proceedings of the Second International Conference on Advanced Computing and Communication Technology, January 2012, pp. 423-425<br><br> After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles.<br><br> This paper is a duplication of the original text from the paper cited below. The original text was copied without attribution (including appropriate references to the original author(s) and/or paper title) and without permission.<br><br> Due to the nature of this violation, reasonable effort should be made to remove all past references to this paper, and future references should be made to the following article:<br><br> "VFactor: A Simple Algorithm for Integer Factorization"<br> by Vikas Pareek and Manisha Sharma<br> in the Proceedings of the International Conference on Computer Engineering and Technology (ICCET), November 2010<br><br> <br/> RSA is the asymmetric cryptography system. The security of RSA public key cryptosystem is based on the assumption that factoring of a large number (modulus). Integer Factorization is an important problem mainly due to its connection with RSA algorihm of Public key cryptography. We present a new special purpose algorithm (VFactor) for factoring. We compare this algorithm with Fermat's Factorization algorithm (FFM) and trial division algorithm (TDM) and we show that VFactor's runtime depends on the difference of factors and is independent of size of the modulus. So it's effective whenever factors are close to each other. In that case VFactor outperforms FFM and TDM. Keywords: Integer factorization, RSA algorithm, Fermat's Method of Factorization, Public key cryptography, TDM.

A distributed enumeration algorithm and applications to all pairs shortest paths, diameter…

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor &gt; 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 &lt; 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.

We consider three popular types of shared memory that support one of the following sets of operations: {CAS, read, write}, {LL, SC, VL, read, write}, or {RLL, RSC, read, write}. We present algorithms that, together with Moir's [Moi97], efficiently implement each shared memory above from any of the other two. Our implementations are wait-free and have constant time and space complexity. Thus, concurrent programs developed for one of the above memories can be ported to any other without incurring any increase in time complexity. Further, since our implementations are wait-free, a wait-free concurrent program remains wait-free even after porting.

Optimization has been widely used to generate smooth trajectories for motion planning. However, existing trajectory optimization methods show weakness when dealing with large-scale long trajectories. Recent advances in parallel computing have accelerated optimization in some fields, but how to efficiently solve trajectory optimization via parallelism remains an open question. In this paper, we propose a novel trajectory optimization framework based on the Consensus Alternating Direction Method of Multipliers (CADMM) algorithm, which decomposes the trajectory into multiple segments and solves the subproblems in parallel. The proposed framework reduces the time complexity to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1)$</tex-math></inline-formula> per iteration with respect to the number of segments, compared to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(N)$</tex-math></inline-formula> of the state-of-the-art (SOTA) approaches. Furthermore, we introduce a closed-form solution that integrates convex linear and quadratic constraints to speed up the optimization, and we also present a numerical solution for general convex inequality constraints. A series of simulations and experiments demonstrate that our approach outperforms the SOTA approach in terms of efficiency and smoothness. Especially for a large-scale trajectory, with one hundred segments, achieving over a tenfold speedup. To fully explore the potential of our algorithm on modern parallel computing architectures, we deploy our framework on a GPU and show high performance with thousands of segments.

Cognitive radio networks (CRNs) allow unlicensed users to opportunistically access the licensed spectrum without causing disruptive interference to the primary users (PUs). One of the main challenges in CRNs is the ability to detect PU transmissions. Recent works have suggested the use of secondary user (SU) cooperation over individual sensing to improve sensing accuracy. In this paper, we consider a CRN consisting of multiple PUs and SUs to study the problem of maximizing the total expected system throughput. First, we study the sensing decision problem for maximizing the system throughput subject to a constraint on the PU throughput, and we design a Bayesian decision rule-based algorithm. The problem is shown to be strongly NP-hard and solved via a greedy algorithm with time complexity O([(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> )/(log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> [1/(1-ε)])]), where N is the total number of SUs. The algorithm achieves a throughput strictly greater than 1/2(1-ε) of the optimal solution and results in a small constraint violation that goes to zero with ε. We then investigate the more general problem with constraints on both PU throughput and the sensing time overhead, which limits the number of SUs that can participate in cooperative sensing. We illustrate the efficacy of the performance of our algorithms and provide sensitivity analysis via a numerical investigation.

This paper presents a new algorithm that reduces multivalued consensus to binary consensus in an asynchronous message-passing system made up of n processes where up to t may commit Byzantine failures. This algorithm has the following noteworthy properties: it assumes t<n/3 and is consequently optimal from a resilience point of view, uses On2 messages, has a constant time complexity, and does not use signatures. The design of this reduction algorithm relies on two new all-to-all communication abstractions. The first one allows the non-faulty processes to reduce the number of proposed values to c, where c is a small constant. The second communication abstraction allows each non-faulty process to compute a set of proposed values such that, if the set of a non-faulty process contains a single value, then this value belongs to the set of any non-faulty process. Both communication abstractions have an On2 message complexity and a constant time complexity. The reduction of multivalued Byzantine consensus to binary Byzantine consensus is then a simple sequential use of these communication abstractions. To the best of our knowledge, this is the first asynchronous message-passing algorithm that reduces multivalued consensus to binary consensus with On2 messages and constant time complexity measured with the longest causal chain of messages in the presence of up to t<n/3 Byzantine processes, and without using cryptography techniques. Moreover, this reduction algorithm uses a single instance of the underlying binary consensus, and tolerates message re-ordering by Byzantine processes.

It is well known that Discrete Fourier Transform (DFT) techniques may be used to multiply large integers. We introduce the concept of Discrete Weighted Transforms (DWTs) which, in certain situations, substantially improve the speed of multiplication by obviating costly zero-padding of digits. In particular, when arithmetic is to be performed modulo Fermat Numbers 2 2 m + 1 {2^{{2^m}}} + 1 , or Mersenne Numbers 2 q − 1 {2^q} - 1 , weighted transforms effectively reduce FFT run lengths. We indicate how these ideas can be applied to enhance known algorithms for general multiplication, division, and factorization of large integers.

Cardiac extracellular matrix (cECM) derived hydrogel has been investigated to treat myocardial infarction through animal studies and clinical trials. The tissue harvesting site commonly selects porcine left ventricle (LV) because heart attack majorly takes place in LV. However, little is known about whether the region of cardiac tissue harvesting is critical for downstream applications. In this work, in vitro studies to compare cECM hydrogels derived from adult porcine whole heart (WH), LV, and right ventricle (RV) were performed. The cECM from WH has similar chemical composition compared with cECM from LV and RV. All three types of cECM hydrogels share many similarities in terms of their microstructure, gelation time, and mechanical properties. WH-derived cECM hydrogels have larger variations in storage modulus (G') and complex modulus (G*) compared with the other two types of cECM hydrogels. Both human cardiomyocytes and mesenchymal stem cells could maintain high cell viability on all hydrogels without significant difference. In terms of above results, the cECM hydrogels from WH, LV and RV exhibited similarity in material properties and cell response in vitro. Thus, future fabrication of cECM hydrogels from WH would increase the yield, which would decrease processing time and production cost.

Similar to the classical meet-in-the-middle algorithm, the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm. Aiming at the target vector of fixed weight, based on the quantum meet-in-the-middle algorithm, the algorithm for searching all n-product vectors with the same weight is presented, whose complexity is better than the exhaustive search algorithm. And the algorithm can reduce the storage complexity of the quantum meet-in-the-middle search algorithm. Then based on the algorithm and the knapsack vector of the Chor-Rivest public-key crypto of fixed weight d, we present a general quantum meet-in-the-middle search algorithm based on the target solution of fixed weight, whose computational complexity is with Σdi=0 Cki memory cost. And the optimal value of k is given. Compared to the quantum meet-in-the-middle search algorithm for knapsack problem and the quantum algorithm for searching a target solution of fixed weight, the computational complexity of the algorithm is lower. And its storage complexity is smaller than the quantum meet-in-the-middle-algorithm.

Global roundings of sequences

The addition chain for a given exponent $n$ is an increasing sequence of positive integers: its first term is $1$, and each subsequent term is obtained by adding the previous two terms which can be the same , so that the last element of the sequence is equal to $n$. Constructing the shortest addition chain for a fixed exponent $n$ is the most efficient method for computing $x^n$ in some group under multiplication. Therefore, the addition chain plays a crucial role in modern cryptography. It can improve the computational efficiency of cryptographic algorithms that require fast exponentiation, such as RSA, ElGamal, Paillier, and ECC, etc. However, the problem of finding the shortest additive chain is textbf{NP}-Complete. Moreover, the existing evolutionary algorithms can not work well for finding short addition chains for large integers. This paper integrates genetic algorithms with the window method to obtain an efficient strategy for the addition chain problem involving large integers. We do experiments on an RSA-1536 modulus to verify the efficiency and practicability of our algorithm.

Certificateless public key cryptography (CL-PKC) can solve the problems of certificate management in a public key infrastructure (PKI) and of key escrows in identity-based public key cryptography (ID-PKC). In CL-PKC, the key generation center (KGC) does not know the private keys of all users, and their public keys need not be certificated by certification authority (CA). At present, however, most certificateless encryption schemes are based on large integer factorization and discrete logarithms that are not secure in a quantum environment and the computation complexity is high. To solve these problems, we propose a new certificate-less encryption scheme based on lattices, more precisely, using the hardness of the learning with errors (LWE) problem. Compared with schemes based on large integer factorization and discrete logarithms, the most operations are matrixvector multiplication and inner products in our scheme, our approach has lower computation complexity. Our scheme can be proven to be indistinguishability chosen ciphertext attacks (IND-CPA) secure in the random oracle model.