Modular Exponentiation Randomization via Key Product Auxiliary Number
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Solution Overview
Problem
Existing cryptographic systems face challenges in protecting secret data from side-channel attacks, particularly in modular exponentiation processes, where randomization of exponents is necessary but often requires calculation of Carmichael or Euler's phi functions, which can be cumbersome and resource-intensive.
Innovation Solution
A method for determining a result of modular exponentiation using a randomization auxiliary number calculated as the product of public and private keys minus 1, which is a multiple of the Carmichael function, allowing for efficient randomization without the need for additional function calculations, and can be implemented using the Chinese Remainder Theorem for enhanced security.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If randomization of the exponent is performed using Carmichael function or Euler's phi function, then security against side-channel attacks is improved, but computational complexity and resource consumption increase
Solution Approach 1:
The patent extracts only the necessary component for randomization (the auxiliary number derived from key product minus one) without requiring full calculation of Carmichael function or Euler's phi function. This selective extraction maintains security requirements while reducing computational overhead by eliminating unnecessary function calculations.
Solution Approach 2:
The auxiliary number for randomization is pre-calculated as the product of public and private keys minus one, which is a multiple of the Carmichael function. This preliminary calculation allows subsequent randomization operations to proceed efficiently without repeatedly computing complex number-theoretic functions during actual cryptographic operations.
2Productivity
If randomization auxiliary number is calculated as product of keys minus one, then computational overhead is reduced, but requirement for pre-known key values increases
Solution Approach 1:
The system uses its own key materials (public key e and private key d) to generate the randomization auxiliary number through the calculation (e×d−1). This self-service approach eliminates dependency on external pre-computed values like Carmichael function or Euler's phi function, allowing the system to autonomously perform efficient randomization using only its existing key pairs.
Data Source
AI summary
For the determination of a result of a modular exponentiation, a randomization auxiliary number is employed for the randomization of the exponent on the basis of the product of the public key and the private key less “1”. This randomization auxiliary number may be derived from the private RSA dataset without special functionalities. Thus, low-overhead exponent randomization may be performed for each security protocol universally, to perform a digital signature secure against side-channel attacks.