Modular Inverse Computation for ECC Digital Signatures
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Solution Overview
Problem
The computational complexity of verifying digital signatures using elliptic curve cryptography (ECC) is high due to the intensive mathematical operations required, which can be a constraint for devices with limited resources such as mobile devices, IoT devices, and others.
Innovation Solution
A method that reduces computational complexity by using a modified Euclidean algorithm, referred to as the efficient extended Euclid algorithm or mutual Euclid method, which performs iterative processes using only the most significant bits of large integers, thereby reducing the number of iterations and computational demands.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional ECC verification is used, then security and authentication are ensured, but computational complexity and processing time increase significantly
Solution Approach 1:
The patent segments the modular inverse computation into two phases: a pre-computation phase that creates a lookup table of inverses for smaller integers, and a verification phase that uses this table to quickly resolve inverses during signature verification. This segmentation reduces the computational burden during the critical verification process while maintaining security.
Solution Approach 2:
The patent performs preliminary action by pre-computing and storing modular inverses in a lookup table before the actual signature verification occurs. This pre-computation allows the verification process to simply retrieve pre-calculated values rather than performing complex modular inverse operations in real-time, significantly reducing verification computational complexity.
2Measurement precision
If traditional modular inverse computation is used, then accurate results are obtained, but memory usage and power consumption increase
Solution Approach 1:
The patent changes the parameters of the computation by working with smaller integers in the pre-computation phase to create the lookup table, rather than directly computing with the full-size cryptographic integers during verification. This parameter change reduces the computational complexity and power consumption of the verification process while maintaining the mathematical correctness of the results through the properties of modular arithmetic.
Data Source
AI summary
A method includes receiving a message and a digital signature associated with a signing party and the message, verifying authenticity of the digital signature using elliptic curve cryptography (ECC), and authenticating use of the message based, at least in part, on the confirmed authenticity of the digital signature. The verifying includes one or more computations involving computing modular inverses. Computing modular inverses includes identifying first and second integer of a modular inverse operation, performing a first iterative process that, at each iteration: (i) initializes a third integer with a pre-defined number of most significant bits of the first integer and a fourth integer with the pre-defined number of most significant bits of the second integer and (ii) computes a quotient and a remainder, determining a resultant inverse value using the quotient; and confirming the authenticity of the digital signature based, at least in part, on the resultant inverse value.