Related papers: Integer Factorisation, Fermat & Machine Learning o…
Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic.…
We propose a semiclassical version of Shor's quantum algorithm to factorize integer numbers, based on spin-1/2 SU(2) generalized coherent states. Surprisingly, we find evidences that the algorithm's success probability is not too severely…
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as…
We revisit Fermat's factorization method for a positive integer $n$ that is a product of two primes $p$ and $q$. Such an integer is used as the modulus for both encryption and decryption operations of an RSA cryptosystem. The security of…
A deterministic algorithm for factoring $n$ using $n^{1/3+o(1)}$ bit operations is presented. The algorithm tests the divisibility of $n$ by all the integers in a short interval at once, rather than integer by integer as in trial division.…
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of…
Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required…
Current asymmetric cryptography is based on the principle that while classical computers can efficiently multiply large integers, the inverse operation, factorization, is significantly more complex. For sufficiently large integers, this…
Probabilistic computing has been introduced to operate functional networks using a probabilistic bit (p-bit), generating 0 or 1 probabilistically from its electrical input. In contrast to quantum computers, probabilistic computing enables…
Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
Polynomial factorization over $ZZ$ is of great historical and practical importance. Currently, the standard technique is to factor the polynomial over finite fields first and then to lift to integers. Factorization over finite fields can be…
Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…
Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n x m binary…
Numerous methods have been considered to create a fast integer factorization algorithm. Despite its apparent simplicity, the difficulty to find such an algorithm plays a crucial role in modern cryptography, notably, in the security of RSA…
This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer…
We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where…
Large integer factorization is a prominent research challenge, particularly in the context of quantum computing. This holds significant importance, especially in information security that relies on public key cryptosystems. The classical…
Mixed-integer optimisation problems can be computationally challenging. Here, we introduce and analyse two efficient algorithms with a specific sequential design that are aimed at dealing with sampled problems within this class. At each…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…