Related papers: Revisiting Fermat's Factorization for the RSA Modu…
Computer algebra systems are really good at factoring polynomials, i.e. writing f as a product of irreducible factors. It is relatively easy to verify that we have a factorisation, but verifying that these factors are irreducible is a much…
We discuss a procedure, which should be called Lenstra's fix, for producing secure RSA moduli even when the random number generation is very poor.
We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of…
Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le…
The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation g^a mod n. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the…
This paper provides a double encryption algorithm that uses the lack of invertibility of the fractional Fourier transform (FRFT) on $L^{1}$. One encryption key is a function, which maps a ``good" $L^{2}$-signal to a ``bad" $L^{1}$-signal.…
In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct…
We present a method for encoding second-quantized fermionic systems in qubits when the number of fermions is conserved, as in the electronic structure problem. When the number $F$ of fermions is much smaller than the number $M$ of modes,…
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.…
We revisit recent works by Don, Fehr, Majenz and Schaffner and by Liu and Zhandry on the security of the Fiat-Shamir transformation of $\Sigma$-protocols in the quantum random oracle model (QROM). Two natural questions that arise in this…
The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby…
The BMR16 circuit garbling scheme introduces gadgets that allow for ciphertext-free modular addition, while the multiplication of private inputs modulo a prime p can be done with 2(p - 1) ciphertexts as described in Malkin, Pastro, and…
We report a quantum-classical hybrid scheme for factorization of bi-prime numbers (which are odd and square-free) using IBM's quantum processors. The hybrid scheme proposed here involves both classical optimization techniques and adiabatic…
This paper proposes an alternative approach to formally establishing the correctness of the RSA public key cryptosystem. The methodology presented herein deviates slightly from conventional proofs found in existing literature. Specifically,…
We introduce the combinatorial notion of a $q$-fatorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with…
It is NP-complete to find non-negative factors $W$ and $H$ with fixed rank $r$ from a non-negative matrix $X$ by minimizing $\|X-WH^\top\|_F^2$. Although the separability assumption (all data points are in the conical hull of the extreme…
Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A…
For two natural numbers $1<p_1<p_2$, with $\alpha = \frac{\log(p_1)}{\log(p_2)}$ irrational, we describe, in main Theorem $\Omega$ and in Note $1.5$, the factorization of two adjacent numbers in the multiplicatively closed subset $S =…
We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model…
In this work, we introduce a new, efficient and practical scheme based on the Rabin cryptosystem without using the Jacobi symbol, message redundancy technique or the needs of extra bits in order to specify the correct plaintext. Our system…