The discrete logarithm problem (DLP) is the basis for several cryptographic primitives. Since Shor’s work, it has been known that the DLP can be solved by combining a polynomial-size quantum circuit and a polynomial-time classical post-processing algorithm. The theoretical result corresponds the situation where a quantum device working with a medium number of qubits of very small errors can solve DLP. However, all the quantum devices that we can use have a limited number of noisy qubits, as of the noisy intermediate-scale quantum (NISQ) era. Thus, evaluating the instance size that the latest quantum device can solve, and give a future prediction of the size along the progress of quantum devices are emerging research topics. This paper contains two proposal to discuss the performance of quantum devices against DLP in the NISQ era. (1) A quantitative measure based on the success probability of the post-processing algorithm to determine whether an experiment on a quantum device (or a classical simulator) succeeded. (2) A procedure to modify bit strings observed from a Shor’s circuit to increase the success probability of a lattice-based post-processing algorithm. We conducted our experiments with theibm_kawasakidevice and discovered that the simplest circuit (7 qubits) from a 2-bit DLP instance achieves a sufficiently high success probability to proclaim the experiment successful. Experiments on another circuit from a slightly harder 2-bit DLP instance, on the other hand, did not succeed, and we determined that reducing the noise level by half is required to achieve a successful experiment. Finally, we give a near-term prediction based on required noise levels to solve some selected small DLP and integer factoring instances.
The discrete logarithm problem is a significant challenge in number theory,and due to the difficulty of solving it,classical computers lack efficient algorithms for this task.As a result,the discrete logarithm problem is widely used in public key cryptosystems,and if it were cracked,it would pose a direct threat to the security of these systems.However,with the advent of quantum computing,researchers have begun exploring quantum computers as a potential solution for the discrete logarithm problem.Currently,most quantum algorithms for solving the discrete logarithm problem are based on Shor’s algorithm.However,due to Shor’s inherent limitations,these algorithms often face issues such as large quantum circuit depth,high qubit usage and complex post-processing steps.This makes it difficult for Shor’s algorithm to be implemented on NISQ computers.To address these issues,this paper proposes a novel approach by introducing a variational quantum algorithm for solving the discrete logarithm pro-blem.This algorithm leverages the parallelism of quantum computing to compute the modular exponentiation of parameterized quantum states.It also designs a marked solution circuit that maps valid solutions of the discrete logarithm problem onto auxiliary qubits.Then,a classical optimizer is used to iteratively adjust the parameters within the parameterized quantum circuit,continuously reducing the value of a designed loss function.Finally,the optimized parameters from the classical optimizer are fed into the measurement circuit,where the solution to the discrete logarithm problem can be obtained with high probability.Compared to Shor’s algorithm,the proposed method significantly reduces the required number of qubits and nearly halves the quantum circuit depth.Furthermore,this paper provides a detailed design of the quantum circuit and verifies the correctness of the proposed algorithm using the Qiskit package in Python.
Solving the discrete logarithm problem (DLP) with quantum computers is a fundamental task with important implications. Beyond Shor's algorithm, many researchers have proposed alternative solutions in recent years. However, due to current hardware limitations, the scale of DLP instances that can be addressed by quantum computers remains insufficient. To overcome this limitation, we propose a distributed quantum discrete logarithm algorithm that reduces the required quantum register size for solving DLPs. Specifically, we design a distributed quantum algorithm to determine whether the solution is contained in a given set. Based on this procedure, our method solves DLPs by identifying the intersection of sets containing the solution. Compared with Shor's original algorithm, our approach reduces the register size and can improve the success probability, while requiring no quantum communication.
Shor's algorithm is one of the most significant quantum algorithms. Shor's algorithm can factor large integers with a certain success probability in polynomial time. However, Shor's algorithm requires an unbearable amount of qubits in the NISQ (Noisy Intermediate-scale Quantum) era. To reduce the resources required for Shor's algorithm, in this paper we first propose a new distributed phase estimation algorithm. Our distributed phase estimation algorithm does not require quantum communication and it reduces the number of qubits of a single node compared to the traditional phase estimation algorithm (non-iterative version). Then we apply our distributed phase estimation algorithm to form a distributed order-finding algorithm for Shor's algorithm. Compared with the traditional Shor's algorithm (non-iterative version), the maximum number of qubits required by a single node of our dristributed order-finding algorithm is reduced by
(2−k2)L−log2k−O(1) when factoring an
L-bit integer (
k is the number of compute nodes). The communication complexity of our distributed order-finding algorithm is
O(kL).
The discrete logarithm problem (DLP) over finite fields, commonly used in classical cryptography, has no known polynomial-time algorithm on classical computers. However, Shor has provided its polynomial-time algorithm on quantum computers. Nevertheless, there are only few examples simulating quantum circuits that operate on general pairs of modulo
pand order
q. In this paper, we constructed such quantum circuits and solved DLPs for all 1,860 possible pairs of
pand
qup to 32 qubits using a quantum simulator with PRIMEHPC FX700. From this, we obtained and verified values of the success probabilities, which had previously been heuristically analyzed by [Ekerå, arXiv:1905.09084 (2019)]. As a result, the detailed waveform shape of the success probability of Shor's algorithm for solving the DLP, known as a periodic function of order
q, was clarified. Additionally, we generated 1,015 quantum circuits for larger pairs of
pand
q, extrapolated the circuit sizes obtained, and compared them for
p=2048bits between safeprime groups and Schnorr groups. While in classical cryptography, the cipher strength of safe-prime groups and Schnorr groups is the same if
pis equal, we quantitatively demonstrated how much the strength of the latter decreases to the bit length of
pin the former when using Shor's quantum algorithm. In particular, it was experimentally and theoretically shown that when a basic adder is used in the addition circuit, the cryptographic strength of a Schnorr group with
p=2048bits under Shor's algorithm is almost equivalent to that of a safe-prime group with
p=1024bits.
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.