Binary Decomposition Interval Proof Method
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Solution Overview
Problem
Existing techniques for proving membership in an interval, such as those used in authentication and identification protocols, often result in convincing a verifier that a secret belongs to a larger interval than intended, leading to inefficiencies and increased complexity, particularly for small secret values in applications like electronic voting and electronic money systems.
Innovation Solution
A method that simplifies the proof of knowledge by using binary decomposition of the secret and interval bounds, allowing the verifier and prover to eliminate unnecessary calculations and communications, thereby reducing complexity and improving efficiency for small secret sizes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If classical techniques for proving membership in an interval are used, then the proof can be constructed, but the secret is proven to belong to a larger interval than intended, reducing precision
Solution Approach 1:
The patent applies segmentation by decomposing the secret x and interval bounds a, b into binary representations (x = Σxi·2^i, a = Σai·2^i, b = Σbi·2^i). This binary decomposition allows the proof to be constructed bit-by-bit from most significant to least significant, enabling precise interval membership verification while maintaining manageable complexity through systematic breakdown of the problem into discrete binary components.
2Measurement precision
If Boudot's method with expansion rate δ=1 is used, then exact interval membership can be proven, but the computational complexity remains high even for small secret values
Solution Approach 1:
The patent applies preliminary action by pre-computing and storing lookup tables containing binary decomposition information for the interval bounds a and b before the actual proof process. These pre-computed tables enable the verifier to efficiently check interval membership during the interactive proof without performing complex real-time calculations, significantly improving computational efficiency while maintaining exact precision for small secret values.
Solution Approach 2:
The patent employs dynamics by making the proof protocol adaptive to the size of the secret. The complexity of the proof construction and verification adjusts based on the number of bits in the secret x and the interval bounds, allowing optimal performance for small secrets (e.g., 16-bit secrets in electronic voting) while maintaining security. The protocol dynamically selects the appropriate number of modular exponentiations and proof steps based on the input parameters.
3Productivity
If binary decomposition of the secret is used, then computational complexity can be reduced for small secrets, but the proof construction becomes more complex
Solution Approach 1:
The patent applies self-service by enabling the prover to autonomously generate the proof using the binary decomposition of the secret and interval bounds without requiring complex external verification infrastructure. The prover computes the necessary modular exponentiations and proof components locally based on their own secret and the publicly known interval bounds, then sends the compact proof to the verifier who can efficiently check it using pre-computed lookup tables.
Data Source
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AI summary
The method involves calculating a bond based on a secret of a proover entity, and calculating a knowledge proof of the secret using a binary decomposition of the secret and upper and lower bounds. The proof is recalculated by the verifier entity based on the bond so as to verify the proof received from the proover entity, where the bounds are introduced in the entities, and a preliminary treatment calculating the proof is implemented to simplify an expression of the proof such that the expression of the proof is free of components issued from the decomposition of the bounds. Independent claims are also included for the following: (1) a computer program to be stored in memory of a verifier entity for calculation of proof of knowledge of a secret in a given interval (2) a proover entity to prove knowledge of a secret at a verifier entity in a given interval (3) a verifier entity to verify proof of the secret knowledge provided by the proover entity (4) a system for implementing a informatic entities communication method.