Simulating Regev's quantum factoring algorithm and Ekerå–Gärtner's extensions to discrete logarithm finding, order finding and factoring via order finding

The repository contains Sage scripts that implement a simulator for the quantum part of Regev's multi-dimensional variation [Regev23] of Shor's factoring algorithm [Shor94], and of Ekerå–Gärtner's extensions [EG23p] of Regev's algorithm to discrete logarithm finding, order finding, and factoring via order finding. The scripts furthermore implement the lattice-based post-processing algorithms from the aforementioned works.

The simulator works by constructing a basis for the lattice for the problem instance to be simulated. For the simulator to be able to construct such a basis, it requires the modulus that defines the problem instance to be on special form. More specifically,

• when the modulus is a prime $p$, the simulator requires $p - 1$ to be smooth, and
• when the modulus is a composite $N = {\prod}_{i=1}^t \, p_i$, for the $p_i$ distinct prime factors, the simulator requires each $p_i - 1$ to be smooth and to not share any factor with $p_j - 1$ for $j \neq i$ except for a factor of two.

Imposing the above requirements enables the simulator to efficiently compute discrete logarithms in $\mathbb Z_N^*$ and $\mathbb Z_p^*$, respectively, which in turn enables it to construct a basis for the lattice. Given this basis, a basis for the dual can be trivially computed, and noisy vectors sampled from the dual.

Note that imposing the above requirements makes the problem instance classically tractable: The simulator can hence not be used to break classically hard problem instances. This having been said, we expect its performance (in terms for the success probability of the post-processing for a given problem instance size and set of parameters) to be representative of that for Regev's algorithm, and of Ekerå–Gärtner's extensions thereof, also for classically hard problem instances.

For factoring, the simulator is sufficiently efficient to simulate Regev's algorithm, and our extensions of it to factoring via order finding, for 2048-bit RSA integers. For discrete logarithms, the simulator is sufficiently efficient to simulate computing discrete logarithms in safe-prime groups and Schnorr groups, as used in Diffie–Hellman and DSA, with 2048-bit moduli.

The high-level functionality for factoring, logarithm finding, and order finding (including factoring via order finding), respectively, is implemented in the scripts

• factoring.sage,
• logarithm-finding.sage, and
• order-finding.sage.

These scripts also contain convenience test functions, and functions for finding the minimum constant $C$ that suffices for the post-processing to be successful for a given problem instance and parameterization.

The aforementioned high-level scripts in turn depend on a number of other scripts, such as

• simulator.sage that implements the simulator,
• problem-instances.sage that samples problem instances of special form,
• parameter-search.sage that implements the search for $C$, and
• common.sage that defines default parameters.

To support the factoring.sage and order-finding.sage scripts, this repository also contains a Sage script dependencies/factor.sage, alongside supporting Python scripts, that are copied directly from the factoritall repository. These scripts implement procedures from [E21b]. They are used by to improve the post-processing for Regev's factoring algorithm, so as to use all available factoring relations to attempt to factor completely. Furthermore, they are used to factor completely via order finding via Ekerå–Gärtner's extensions.

The aforementioned scripts were developed for academic research purposes. They grew out of our research project in an organic manner as research questions were posed and answered. They are distributed "as is" without warranty of any kind, either expressed or implied. For further details, see the license.

Prerequisites

To install Sage under Ubuntu 22.04 LTS, simply execute:

$ sudo apt install sagemath

For other Linux and Unix distributions, or operating systems, you may need to download Sage and install it manually. These scripts were developed for Sage 9.5.

Loading the scripts

Launch Sage in this directory and load the main scripts by executing:

$ sage
(..)
sage: load("factoring.sage")
sage: load("logarithm-finding.sage")
sage: load("order-finding.sage")

Examples

For examples that illustrate how to use the simulator, please see

• examples/factoring.md for factoring,
• examples/logarithm-finding.md for finding discrete logarithms, and
• examples/order-finding.md for finding orders, and for factoring completely via order finding.

About and acknowledgments

These scripts were developed by Martin Ekerå and Joel Gärtner, in part at KTH, the Royal Institute of Technology, in Stockholm, Sweden. Martin Ekerå thanks Sam Jaques for useful discussions.

Funding and support was provided by the Swedish NCSA that is a part of the Swedish Armed Forces.