Tropical TEBD Algorithm for Integer Factorization Accuracy
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Solution Overview
Problem
Current classical integer factorization algorithms, such as the Schnorr's algorithm, face challenges in efficiently solving the Closest Vector Problem (CVP) due to their inability to achieve high accuracy, which affects the performance of lattice-based cryptographic protocols, especially when dealing with large integers.
Innovation Solution
Adaptation of the Time-Evolving Block Decimation (TEBD) algorithm to classical optimization problems by employing tropical algebra and implementing a tropical TEBD algorithm, which enhances the accuracy and scalability of solving the CVP step in Schnorr's algorithm.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If classical integer factorization algorithms (e.g., Schnorr's algorithm) are used, then the security of RSA-based cryptosystems is maintained, but the accuracy and efficiency of solving the Closest Vector Problem (CVP) is insufficient
Solution Approach 1:
The patent replaces classical mechanical optimization methods with a quantum-inspired algorithm (TEBD) that uses tensor networks and tropical algebra to solve the CVP. This substitution enables higher accuracy in finding the closest lattice vector while maintaining polynomial time complexity, thus resolving the contradiction between precision and productivity.
Solution Approach 2:
The patent changes the mathematical framework from classical linear algebra to tropical algebra and tensor network formalism. By transforming the CVP into a problem solvable via tensor network contractions and tropical operations, the algorithm achieves both high accuracy and efficient computation, eliminating the trade-off between precision and productivity.
2Productivity
If quantum computers are used to factorize large integers via Shor's algorithm, then the factorization efficiency is dramatically improved, but the available computational resources are insufficient
Solution Approach 1:
The patent creates a classical simulation of quantum algorithms by adapting the TEBD algorithm to work on classical computers. This copying approach captures the essential quantum computational power needed for efficient factorization while running on available classical hardware, thus achieving high productivity without requiring scarce quantum resources.
Solution Approach 2:
The patent introduces tensor networks as an intermediary between quantum algorithms and classical computers. This intermediary framework allows the simulation of quantum computations using classical resources, enabling efficient integer factorization without direct access to quantum hardware, thus resolving the contradiction between productivity and resource availability.
3Loss of time
If the time complexity of factorization algorithms is reduced, then the applicability to larger integers is improved, but the algorithmic complexity increases
Solution Approach 1:
The patent segments the factorization process into distinct phases: lattice basis reduction, CVP formulation, and solution optimization via TEBD. By dividing the complex algorithm into manageable segments, each with clear computational steps, the overall time complexity is reduced while maintaining manageable algorithmic structure through modular design.
Solution Approach 2:
The patent transforms the CVP from a high-dimensional geometric problem into a tensor network contraction problem. By changing the dimensional perspective and using tensor network formalism, the algorithm achieves polynomial time complexity for factorization while the internal structure remains organized through the hierarchical tensor network representation.
Data Source
AI summary
A computer implemented method for solving a classical optimization problem of integer factorization implemented on a digital computer system is described. The method is implemented on a classical processor adapted to execute a time evolving block decimation algorithm. The method comprises in a first step an inputting a lattice basis and a target lattice vector to an input device of the classical processor followed by an implementing a lattice basis reduction algorithm on the lattice basis in an implementation module, thereby obtaining a reduced orthogonal lattice basis. The method further comprises a projecting the target lattice vector on the reduced orthogonal lattice basis followed by a building a closest vector to the target lattice vector and optimizing the closest vector using a tropical time-evolving block decimation algorithm by the classical processor and finally outputting an integer vector.