Modified Mueller Square Root Algorithm for Elliptic Curve Cryptography
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Solution Overview
Problem
Efficient computation of square roots in finite fields for elliptic curve cryptography is challenging, particularly for elliptic curves where the prime parameter p is 1 mod 4 or 1 mod 8, as existing algorithms like Tonelli's and Legendre's become inefficient due to the need for randomization and quadratic extension operations.
Innovation Solution
A modified version of Mueller's square root algorithm utilizing short Lucas sub-sequences is employed to optimize square root computations, where the Lucas parameter Q is set to 1 and the Lucas function is computed using a binary expansion method, reducing the number of squaring and multiplication operations by leveraging the factors of (p-1)/4.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Tonelli's algorithm is used for computing square roots when p = 1 mod 4 or p = 1 mod 8, then square root computation is possible, but the algorithm becomes inefficient due to requiring randomization and handling many factors of 2 in p-1
Solution Approach 1:
The patent changes the approach from using Tonelli's algorithm with randomization to Mueller's algorithm which deterministically computes square roots by finding a specific integer t such that qt² ≡ -1 (mod p), then using Lucas sequences with parameters P = 2qt and Q = 1. This parameter transformation eliminates the inefficiency caused by randomization and the structure of p-1 factors.
2Productivity
If Bernstein's pre-computation approach with larger tables is used, then the number of multiplication operations is reduced, but the memory requirement increases
Solution Approach 1:
The patent replaces Bernstein's table-based pre-computation approach with a mathematical method using Lucas sequences. Instead of storing pre-computed values in memory tables, the algorithm computes square roots through a deterministic sequence of operations based on Lucas sequence properties, substituting memory-intensive operations with computation-intensive but memory-efficient operations.
3Reliability
If standard square root algorithms are used for elliptic curves, then correctness is maintained, but computational time is excessive
Solution Approach 1:
The patent performs preliminary action by finding an integer t such that qt² ≡ -1 (mod p) before computing the square root. This preliminary step enables the use of Lucas sequences with specific parameters (P = 2qt, Q = 1) that guarantee convergence to the square root in a deterministic number of steps, rather than requiring iterative or randomized approaches that may take longer.
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AI summary
A method is presented to compute square roots of finite field elements from the prime finite field of characteristic p over which points lie on a defined elliptic curve. Specifically, while performing point decompression of points that lie on a standardized elliptic curve over a prime finite field of characteristic 2 224 -2 96 +1, the present method utilizes short Lucas subsequences to optimize the implementation of a modified version of Mueller's square root algorithm, to find the square root modulo of a prime number. The resulting method is at least twice as fast as standard methods employed for square root computations performed on elliptic curves.