The Present and Future of Discrete Logarithm Problems on Noisy Quantum Computers

Yoshinori Aono, Sitong Liu, Tomoki Tanaka, Shumpei Uno, Rodney Van Meter, Naoyuki Shinohara, Ryo Nojima

Research output: Contribution to journalArticlepeer-review


The discrete logarithm problem (DLP) is the basis for several cryptographic primitives. Since Shor&#x2019;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&#x2019;s circuit to increase the success probability of a lattice-based post-processing algorithm. We conducted our experiments with the <monospace>ibm_kawasaki</monospace> device 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.

Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalIEEE Transactions on Quantum Engineering
Publication statusAccepted/In press - 2022


  • Computers
  • Cryptography
  • Discrete Logarithm Problem
  • IBM-Quantum
  • Lattice
  • Logic gates
  • Noise level
  • Post-Processing Method
  • Quantum circuit
  • Quantum computing
  • Qubit
  • Shor’s Algorithm

ASJC Scopus subject areas

  • Software
  • Computer Science (miscellaneous)
  • Condensed Matter Physics
  • Engineering (miscellaneous)
  • Mechanical Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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