Related papers: Revisiting Fermat's Factorization for the RSA Modu…
Shor's factorisation algorithm is a combination of classical pre- and post-processing and a quantum period finding (QPF) subroutine which allows an exponential speed up over classical factoring algorithms. We consider the stability of this…
Rivest-Shamir-Adleman (RSA) cryptosystem uses modular multiplication for encryption and decryption. So, performance of RSA can be drastically improved by optimizing modular multiplication. This paper proposes a new parallel, high-radix…
Prime factorization on quantum processors is typically implemented either via circuit-based approaches such as Shor's algorithm or through Hamiltonian optimization methods based on adiabatic, annealing, or variational techniques. While…
The factorizations of the polynomial $X^n-1$ and the cyclotomic polynomial $\Phi_n$ over a finite field $\mathbb F_q$ have been studied for a very long time. Explicit factorizations have been given for the case that $\mathrm{rad}(n)\mid…
In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the…
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by…
We factorize a 48-bit integer using 10 trapped-ion qubits on a Quantinuum's quantum computer. This result outperforms the recent achievement by B. Yan et al., arXiv:2212.12372 (2022), increasing the success probability by a factor of 6 with…
Recently, a quantum multi-party summation protocol based on the quantum Fourier transform has been proposed [Quantum Inf Process 17: 129, 2018]. The protocol claims to be secure against both outside and participant attacks. However, a…
For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n $\times$ log n $\times$ log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved…
A new scheme of probabilistic subgroup-related encryption is introduced. Some applications of this scheme based on the RSA, Diffie-Hellman and ElGamal encryption algorithms are described. Security assumptions and main advantages of this…
In this paper, we provide a generalization of Proth's theorem for integers of the form $Kp^n+1$. In particular, a primality test that requires only one modular exponentiation similar to that of Fermat's test without the computation of any…
We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let $a,n,m$ be positive integers and suppose $m$ factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters $t_i\in [1,…
The cryptosystem RSA is a very popular cryptosystem in the study of Cryptography. In this article, we explore how the idea of a primitive mth root of unity in a ring can be integrated into the Discrete Fourier Transform, leading to the…
We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The…
We analyze the performance of a quantum computer architecture combining a small processor and a storage unit. By focusing on integer factorization, we show a reduction by several orders of magnitude of the number of processing qubits…
Feynman's prescription for a quantum simulator was to find a hamitonian for a system that could serve as a computer. P\'olya and Hilbert conjecture was to demonstrate Riemann's hypothesis through the spectral decomposition of hermitian…
Secure two-party scalar product (S2SP) is a promising research area within secure multiparty computation (SMC), which can solve a range of SMC problems, such as intrusion detection, data analysis, and geometric computations. However,…
We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the…
A generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in Quantum Mechanics. The method is applied in the study of radial oscillator, Morse…
We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Sch\"{o}nhage-Strassen. Both these algorithms use modular arithmetic.…