Related papers: Improving the Success Probability for Shor's Facto…
The integer factorization problem (IFP) underpins the security of RSA, yet becomes efficiently solvable on a quantum computer through Shor's algorithm. Regev's recent high-dimensional variant reduces the circuit size through lattice-based…
Lenstra's integer factorization algorithm is asymptotically one of the fastest known algorithms, and is ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase.…
The purpose of this note was to give a proof that Shor's algorithm for period search is polynomial using only the standard $2^{n}$ quantum Fourier thansform and some simple trigonometry. There is an error that was pointed out to the author…
In this paper we included some basic properties for weighted greatest common divisors, and discuss how to speed up computing the weighted greatest common divisor. By ordering the 'weights' we are able to significantly shorten the operations…
The objective of this paper concerns at first the motivation and the method of Shor's algorithm including an excursion into quantum mechanics and quantum computing introducing an algorithmic description of the method. The corner stone of…
Let $p$ be a prime number, $p=2^nq+1$, where $q$ is odd. D. Shanks described an algorithm to compute square roots $\pmod{p}$ which needs $O(\log q + n^2)$ modular multiplications. In this note we describe two modifications of this…
For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…
The theoretical aspects of four integer factorization algorithms are discussed in details in this note. The focus is on the performances of these algorithms on the subset of hard to factor balanced integers N = pq, p < q < 2p. The running…
We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log (1/epsilon)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2^n…
We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…
Almost all public secure communication relies on the inability to factor large numbers. There is no known analytic or classical numeric method to rapidly factor large numbers. Shor[1] has shown that a quantum computer can factor numbers in…
The security of RSA algorithm depends upon the positive integer N, which is the multiple of two precise large prime numbers. Factorization of such great numbers is a problematic process. There are many algorithms has been implemented in the…
We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…
We construct uniquely satisfiable $k$-CNF formulas that are hard for the algorithm PPSZ. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most $(2 + \epsilon) / k$; the saving of an algorithm on an input formula…
We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from…
We show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further…
Generalized sorting problem, also known as sorting with forbidden comparisons, was first introduced by Huang et al. together with a randomized algorithm which requires $\tilde O(n^{3/2})$ probes. We study this problem with additional…
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…
Let $x\ge y>0$ be integers. A positive integer is $y$-smooth if all its prime divisors are at most $y$. Let $\Psi(x,y)$ count the number of $y$-smooth integers up to $x$. We present several algorithms that will generate an integer $n\le x$…
The success probability of a search of $M$ targets from a database of size $N$, using Grover's search algorithm depends critically on the number of iterations of the composite operation of the oracle followed by Grover's diffusion…