MPC Arithmetic Comparison Using Quadratic Residues
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Solution Overview
Problem
Existing multi-party computation (MPC) protocols for securely comparing shared secrets rely on binary representations, which are inefficient for arithmetic operations, and there is a lack of protocols that do not depend on binary input representation, especially for large prime-order fields.
Innovation Solution
The development of novel arithmetic protocols for securely computing equal-to-zero and greater-than-zero using a quadratic residue test and probabilistic rounding of Farey rationals, which utilize a constant number of core operations like multiplication, independent of input size, and operate over a domain of Farey rationals.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If binary representation is used for MPC protocols, then security is maintained, but efficiency for arithmetic operations deteriorates
Solution Approach 1:
The patent changes the representation parameter from binary to a domain based on Farey rationals and quadratic residues. This parameter change enables efficient arithmetic operations while maintaining security through the mathematical properties of quadratic residues, which provide a different basis for secure computation that is naturally suited for arithmetic operations.
Solution Approach 2:
The patent replaces the mechanical binary representation system with a number-theoretic system based on quadratic residues and Farey rationals. This substitution eliminates the need for binary-to-arithmetic conversion overhead while maintaining the security guarantees through the computational hardness of quadratic residue problems.
2Adaptability or versatility
If binary input representation is used, then existing protocols work, but the number of core operations increases with input size
Solution Approach 1:
The patent segments the comparison operation into independent arithmetic operations (addition, multiplication, quadratic residue testing) that operate directly on the encoded rational numbers. This segmentation eliminates the need for binary decomposition and recombination steps, reducing the total number of core operations to a constant independent of input size.
Solution Approach 2:
The patent transitions from one-dimensional binary representation to a two-dimensional domain involving Farey rationals and quadratic residues. This dimensional change allows arithmetic operations to be performed directly in the mathematical domain without conversion overhead, reducing operational complexity while maintaining adaptability through the universal applicability of the encoding.
3Reliability
If protocols are designed for large prime-order fields, then security is improved, but efficiency deteriorates due to lack of binary-independent protocols
Solution Approach 1:
The patent creates a universal protocol framework that works efficiently for large prime-order fields without requiring binary representation. The Farey rational encoding and quadratic residue testing provide a multi-functional approach that handles both security requirements and arithmetic efficiency in a single unified system, eliminating the trade-off between field size and performance.
Data Source
AI summary
Disclosed are methods and system for providing secure arithmetic equality and comparison using probabilistic rounding in the domain of Farey rationals with quadratic residues for a Multi-Party Computation (MPC) system. Embodiments securely provide for evaluation of encoded rational numbers for arithmetic equality equal to zero or not, probabilistic modulo of two rational numbers, probabilistic modulo of a rational number by a public value of two, a test for whether a rational number is greater-than-zero for quadratic residues and a general greater-than-zero test for a rational number which are quadratic non-residues.