Sparse w-NAF Key Generation for Encryption Speed
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Solution Overview
Problem
Existing encryption methods, such as Diffie-Hellman and Elliptic Curve Cryptography, face increased computational complexity due to long key lengths, which slows down operations, especially in mobile devices with slower processors.
Innovation Solution
A method for generating a key using a string of coefficients where at most one coefficient corresponds to a positive odd integer or a non-divisible integer, reducing computational complexity by using Unsigned w-NAF or τ-adic w-NAF keys for exponential and scalar multiplication operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the key length is increased to ensure stable information security, then security reliability is improved, but computational complexity increases and operation speed decreases
Solution Approach 1:
The patent segments the key representation into groups of w coefficients, where each group contains at most one non-zero coefficient. This segmentation transforms a dense key representation into a sparse one, reducing the number of coefficients that need to be processed during exponential operations while maintaining the same security level through sufficient key length.
Solution Approach 2:
The patent changes the parameter representation from standard binary form to w-NAF (width-w Non-Adjacent Form) with sparse coefficients. By representing the key exponent in this specialized form where coefficients are mostly zero and non-zero coefficients are spaced at least w positions apart, the number of required multiplicative operations is reduced from approximately n operations to approximately n/w operations, directly improving operation speed.
2Reliability
If the key length is increased to ensure stable information security, then security reliability is improved, but device complexity increases
Solution Approach 1:
The patent divides the key processing into segments of w coefficients, with each segment containing at most one non-zero coefficient. This segmentation reduces computational complexity by allowing the system to skip over groups of zero coefficients and only perform operations on the sparse non-zero coefficients, thereby reducing the overall computational burden while maintaining security through adequate key length.
Solution Approach 2:
The patent applies partial action by performing cryptographic operations only on the non-zero coefficients of the w-NAF representation rather than processing all coefficients. Since at most one out of every w coefficients is non-zero, this partial processing approach reduces computational complexity by approximately a factor of w compared to processing the full key length, while still achieving the required security level.
Data Source
AI summary
A method for generating, operating, and using a sparse w-NAF key for encryption is disclosed. The method for generating a key comprises generating a string of a number of coefficients, in which at most one coefficient, excluding 0, among a consecutive w number of coefficients, corresponds to a positive odd integer equal to or less than 2w (w being a natural number equal to or more than 2); and outputting the generated string as a key. Accordingly, an encryption is executed through an exponential operation or scalar multiplication using a sparse w-NAF key having the scarce coefficients, excluding 0, such that an encryption pace is improved.