Simultaneous Scalar Multiplication for Elliptic Curve Cryptography
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Solution Overview
Problem
Scalar multiplication in elliptic curve cryptographic schemes, such as ECDSA verification, is inefficient due to the need for repeated doubling and addition operations when using Montgomery's method for separate point multiplications, which dominates execution time.
Innovation Solution
A method for simultaneous point multiplication that reduces the number of doubling and addition operations by performing Montgomery's method on bit pairs of different scalars, padding shorter scalars with zeros, and discarding most significant bits, allowing for efficient computation of kP + sQ by using pairs like (mP + nQ, (m+1)P + (n+1)Q, which reduces operations by half when bits are the same and three-quarters when different.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If Montgomery's method is used for separate point multiplications to compute kP and sQ, then the scalar multiplication can be performed, but the number of doubling and addition operations increases, leading to longer execution time
Solution Approach 1:
The patent combines two separate scalar multiplications (kP and sQ) into a single simultaneous computation process. By merging the two Montgomery ladder sequences and performing operations on combined points (R0 = M0P + N0Q, R1 = M1P + N1Q), the method reduces the total number of elliptic curve point additions and doublings required, thereby decreasing execution time while maintaining computational accuracy
2Ease of operation
If separate Montgomery's method is applied for each scalar multiplication, then individual computations of kP and sQ are achieved, but the overall operation count doubles, reducing computational efficiency
Solution Approach 1:
The patent creates a universal simultaneous scalar multiplication algorithm that can compute multiple results (kP and sQ) through a single unified process. The method uses a combined Montgomery ladder that processes both scalars k and s simultaneously, with each iteration updating both result pairs (M0, M1) and (N0, N1) together. This multi-functional approach reduces operational complexity while maintaining the ability to compute multiple scalar multiplications
Data Source
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AI summary
In computing point multiples in elliptic curve schemes (e.g. kP and sQ) separately using, for example, Montgomery's method for the purpose of combining kP + sQ, several operations are repeated in computing kP and sQ individually, that could be executed at the same time. A simultaneous scalar multiplication method is provided that reduces the overall number of doubling and addition operations thereby providing an efficient method for multiple scalar multiplication. The elements in the pairs for P and Q method are combined into a single pair, and the bits in k and s are evaluated at each step as bit pairs. When the bits in k and s are equal, only one doubling operation and one addition operation are needed to compute the current pair, and when the bits in k and s are not equal, only one doubling operation is needed and two addition operations.