Elliptic Curve Digital Signature Verification Using Reduced Bit Length Integers
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Solution Overview
Problem
The efficiency of verifying ECDSA signatures is hindered by the need for twice as much computation time compared to signature generation, primarily due to the requirement for two scalar multiplications during the verification process, which is a bottleneck in data communication systems.
Innovation Solution
A method is introduced to verify the equality of a sum of scalar multiples on an elliptic curve by obtaining pair of integers with reduced bit lengths, substituting them into the relationship to obtain an equivalent form, and computing this equivalent form efficiently, potentially using precomputed tables and algorithms like the extended Euclidean algorithm or continued fractions to reduce computational overhead.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If conventional ECDSA verification methods are used, then verification security is maintained, but verification time is doubled compared to signature generation
Solution Approach 1:
The patent changes the parameters of the verification equation by introducing equivalent integers with reduced bit lengths. Instead of directly computing with the original large scalars, the patent transforms the verification into an equivalent form using smaller integers, reducing computational complexity and verification time.
Solution Approach 2:
The patent introduces intermediate values (the equivalent integers derived from the extended Euclidean algorithm) that serve as mediators between the original verification equation and the simplified computation. These intermediaries allow the transformation of the verification process into a more efficient form.
2Reliability
If two scalar multiplications are performed during verification, then verification accuracy is ensured, but computational overhead increases
Solution Approach 1:
The patent transforms the verification parameters from large scalars requiring two full scalar multiplications to equivalent integers with reduced bit lengths. This parameter transformation maintains verification accuracy while reducing computational complexity from O(log n) to O(log sqrt(n)).
Solution Approach 2:
The patent segments the verification process into two parts: first computing the equivalent integers using the extended Euclidean algorithm, then using these segmented smaller values to simplify the scalar multiplication operations. This segmentation breaks down the complex verification into more manageable computational steps.
Data Source
AI summary
Accelerated computation of combinations of group operations in a finite field is provided by arranging for at least one of the operands to have a relatively small bit length. In a elliptic curve group, verification that a value representative of a point R corresponds the sum of two other points uG and vG is obtained by deriving integers w,z of reduced bit length and so that v=w/z. The verification equality R=uG+vQ may then be computed as −zR+(uz mod n) G+wQ=O with z and w of reduced bit length. This is beneficial in digital signature verification where increased verification can be attained.