Cryptographic Key Generation Simplifying Prime Constraints
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Solution Overview
Problem
The generalized Goldwasser-Micali cryptosystem requires complex prime generation algorithms and relies on both quadratic residuosity and squared Jacobi symbol assumptions for semantic security, making key generation and security proofs cumbersome.
Innovation Solution
A modified key generation procedure that assumes only the quadratic residuosity assumption, allowing for simplified prime generation and proving semantic security when q ≡ 3 (mod 4), which simplifies the security proof and eliminates the need for the squared Jacobi symbol assumption.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the generalized GM cryptosystem uses both quadratic residuosity and squared Jacobi symbol assumptions for semantic security, then security is improved, but device complexity and key generation complexity increase
Solution Approach 1:
The patent extracts and removes the squared Jacobi symbol assumption from the security framework, retaining only the quadratic residuosity assumption. This simplification maintains semantic security while eliminating the complexity associated with the additional assumption, directly resolving the contradiction between security and complexity.
Solution Approach 2:
The patent changes the security parameter configuration by modifying the prime generation requirements. Specifically, it requires that q ≡ 3 (mod 4) while maintaining p ≡ 1 (mod 2^k), which simplifies the security proof and eliminates the need for the squared Jacobi symbol assumption, thereby reducing key generation complexity while preserving security.
2Reliability
If the generalized GM cryptosystem uses special prime generation algorithms, then semantic security is achieved, but manufacturing precision and ease of manufacture deteriorate
Solution Approach 1:
The patent modifies the prime generation parameters by imposing the constraint q ≡ 3 (mod 4) alongside p ≡ 1 (mod 2^k). This parameter change simplifies the prime generation process and security proof while maintaining semantic security, making the system easier to manufacture and implement.
Solution Approach 2:
The patent removes the requirement for complex special prime generation algorithms by extracting only the essential constraint (q ≡ 3 (mod 4)) needed for security. This eliminates unnecessary complexity in the prime generation process while preserving the semantic security property.
3Reliability
If the generalized GM cryptosystem relies on squared Jacobi symbol assumption, then security proof is strengthened, but loss of time in security proofs increases
Solution Approach 1:
The patent extracts and removes the squared Jacobi symbol assumption from the security framework, retaining only the quadratic residuosity assumption. This simplification maintains semantic security while eliminating the time-consuming aspects of proving security under the additional assumption, directly resolving the contradiction between security proof strength and proof time.
Data Source
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AI summary
A method and a device (110) for generation of a cryptographic key pair for use in a (generalized) Goldwasser-Micali cryptosystem. The device (110) generates a first prime ρ ≡ 1 (mod 2 k ), (S10; S20) where k ≥ 1 is an integer, and a second prime q ≡ 3 (mod 4) (S11) or q ≡ 1 (mod 4) (S21); computes (S12; S22) a modulus N = ρq; picks (S13; S23) an integer y formula (I), where formula (II) is a set of integers whose Jacobi symbol is 1 and formula (III) is a set of quadratic residues; and outputs (S14; S24) a public key ρk = {N, y, k} and a private key sk = {ρ, k}.