Digital Signatures Using Multiplicative Semigroups
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Solution Overview
Problem
Quantum computers pose a threat to existing digital signature schemes based on integer factorization and discrete logarithms, such as RSA and ECDSA, as they can potentially break these algorithms quickly, necessitating the development of post-quantum resistant signature schemes.
Innovation Solution
The use of semigroups, specifically multiplicative signature schemes and plactic monoids, which provide a basis for secure digital signatures by employing associative binary operations and structured secure semigroups to create resistant signature schemes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If RSA or ECDSA algorithms are used for digital signatures, then signature verification can be performed efficiently on classical computers, but the security is compromised by quantum computers which can solve integer factorization and discrete logarithm problems quickly
Solution Approach 1:
The patent changes the mathematical foundation from groups (with inverses) to semigroups (without requiring inverses), and specifically uses plactic monoids with their unique factorization properties. This parameter change in the algebraic structure makes the signature scheme resistant to quantum attacks while maintaining classical efficiency
Solution Approach 2:
The patent converts the previously harmful property of quantum computers (ability to solve factorization and discrete log problems) into a benefit by using a different mathematical structure where these problems do not apply. The plactic monoid structure ensures unique factorization that is not vulnerable to Shor's algorithm, turning the quantum threat into an opportunity for post-quantum security
2Reliability
If new post-quantum signature schemes are developed, then quantum resistance is achieved, but the complexity of the cryptographic system increases
Solution Approach 1:
The patent extracts only the essential multiplicative structure needed for signatures from the full group theory, working with semigroups that have fewer requirements (no inverses needed). This extraction simplifies the mathematical foundation while maintaining security, reducing the complexity compared to full group-based post-quantum schemes
Solution Approach 2:
The plactic monoid structure serves multiple functions: it provides unique factorization for security, supports efficient multiplication for signature generation, and enables verification through the endpoint-checker relationship. This multi-functionality reduces the need for separate cryptographic primitives, lowering overall system complexity
Data Source
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Figure 3A~3B
AI summary
A method for verification at a computing device of a signed message received from a first party over a public communications channel, the method including extracting a message digest "a" belonging to a semigroup from the signed message; obtaining a public key [c,e] for the first party, including a fixed value checker "c" and an endpoint "e", checker "c" and endpoint "e" belonging to the semigroup and the endpoint comprising a multiplication of a private key "b" for the first party and the checker "c", multiplying the message digest "a" and the endpoint "e" to create an endmatter "ae"; extracting a signature "d" from the signed message, the signature "d" belonging to the semigroup and being a multiplication of message digest "a" and private key "b"; multiplying the signature "d" and the checker "c" to create a signcheck "dc"; and verifying that the endmatter "ae" matches the signcheck "dc".