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Chinese Journal of Computers

Quantum Algorithm for Discrete Logarithm Over Z_N

Xiang Fu

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

By applying multiple Fourier transform and variable transform,aquantum algorithm for discrete logarithm problem over ZNis presented.The relationship between order r and success probability is quantitatively characterized,where r is the order of the selective element.Particularly, when r is a prime number,the success probability is close to 1and the scale of elementary quantum gates is O(L3),where L=[log N]+1.The new algorithm doesn't need to perform inverse quantum Fourier transform on|f(x1,x2)〉,which is superior to the existing algorithm.

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By applying multiple Fourier transform and variable transform,aquantum algorithm for discrete logarithm problem over ZNis presented.The relationship between order r and success probability is quantitatively characterized,where r is the order of the selective element.Particularly, when r is a prime number,the success probability is close to 1and the scale of elementary quantum gates is O(L3),where L=[log N]+1.The new algorithm doesn't need to perform inverse quantum Fourier transform on|f(x1,x2)〉,which is superior to the existing algorithm.
Assume the flagged polygonal-transform and exact-factoring interfaces specified in the paper. We construct a polynomial-time uniform quantum circuit family over one fixed finite gate set that computes discrete logarithms in prime fields with probability one and polynomial worst-case resources. For prime-order inputs, each verified Fourier-bin label has probability at least 1/(81q). Given its verified logarithm, that probability is computable exactly by a bounded number of lattice-point counts in dimension at most thirteen. Dyadic completion produces success probability exactly one quarter, followed by one exact amplification step. All sampled probabilities are dyadic; no input-dependent rotation or postselection is used.
We heuristically show that Shor's algorithm for computing general discrete logarithms achieves an expected success probability of approximately 60% to 82% in a single run when modified to enable efficient implementation with the semi-classical Fourier transform. By slightly increasing the number of group operations that are evaluated quantumly and performing a single limited search in the classical post-processing, or by performing two limited searches in the post-processing, we show how the algorithm can be further modified to achieve a success probability that heuristically exceeds 99% in a single run. We provide concrete heuristic estimates of the success probability of the modified algorithm, as a function of the group order rrr, the size of the search space in the classical post-processing, and the additional number of group operations evaluated quantumly. In the limit as r→∞r \rightarrow \inftyr→∞, we heuristically show that the success probability tends to one. In analogy with our earlier works, we show how the modified quantum algorithm may be heuristically simulated classically when the logarithm ddd and rrr are both known. Furthermore, we heuristically show how slightly better tradeoffs may be achieved, compared to our earlier works, if rrr is known when computing ddd. We generalize our heuristic to cover some of our earlier works, and compare it to the non-heuristic analyses in those works.
Abstract We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.
We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.
In this paper,based on the quantum Fourier transform,we give a new quantum algorithm for prime factorization.The algorithm turns r to be phase factor under repeated application of Fourier transform and variate transform,where r is the order of a selective element in the ring of integers modulo N.Furthermore the amplitude of the non-target states except zero state is modified to be 0.Our algorithm′s success probability,which is more than 3/4 and higher than Shor′s algorithm,doesn′t depend on the size of r other than Shor′s algorithm.Meanwhile,we present a comparison of the required resource between the new algorithm and Shor′s algorithm.
The quantum algorithm with polynomial time for discrete logarithm problem proposed by Shor is one of the most significant quantum algorithms, but a large number of qubits may be required in the Noisy Intermediate-scale Quantum (NISQ) era. The main contributions of this paper are as follows: (1) A distributed quantum algorithm for discrete logarithm problem is designed, and its space complexity and success probability exhibit certain advantages to some extent; (2) The classical error correction technique proposed by Xiao and Qiu et al. is generalized by extending three bits to more than three bits.