Discrete Logarithm Algorithm for Efficient Cryptographic Decoding
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Solution Overview
Problem
Computing discrete logarithms in cryptographic systems is inefficient when p−1 has large prime factors, leading to untraceable searches and high computational time.
Innovation Solution
An algorithm that restructures the exponential congruence modulo p=2·p′+1, where p′−1 contains small prime factors, by transforming it into a problem involving new relationships modulo p and concurrent independent congruences, allowing for the selection of primitives of sub-groups to reduce the complexity of exponent searches to a sum of exponents rather than their product.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If the discrete logarithm is computed using conventional methods when p-1 has large prime factors, then the search requires time of the order p'·log p, but this leads to untraceable searches and high computational time
Solution Approach 1:
The patent segments the discrete logarithm problem into multiple independent congruences modulo different factors of p-1. By dividing p-1 into small prime factors q1, q2, ..., qh and solving independent congruences for each factor, the overall computational complexity is reduced from O(p'·log p) to O(log log p·log² p), making the computation feasible even when p-1 has large prime factors
2Ease of operation
If the search for discrete logarithm is performed directly, then the time required is of the order p'·log p, but this makes the search untraceable and computationally infeasible
Solution Approach 1:
The patent divides the search space into smaller segments corresponding to the prime factor decomposition of p-1. Instead of performing a single large-scale search requiring O(p'·log p) time, the method performs multiple smaller independent searches modulo each prime factor, making the search process traceable and computationally feasible
Solution Approach 2:
The patent transforms the one-dimensional search problem (direct discrete logarithm computation) into a multi-dimensional problem by introducing independent congruences for each prime factor of p-1. This dimensional transformation allows the search to be performed in a structured space with complexity O(log log p·log² p) rather than the original O(p'·log p)
Data Source
AI summary
A decoding apparatus having a non-transient memory in which is stored an electromagnetic signal representative of data which were encrypted relying on the difficulty of computing discrete logarithms. The decoding apparatus has a computer in communication with the memory that decodes the encrypted data in the memory by computing the data's discrete logarithm. The decoding apparatus has a display on which the decoded encrypted data are displayed by the computer. A method for decoding.