Private Group Exponentiation Outsourcing for Efficient PIR
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Solution Overview
Problem
Conventional private information retrieval (PIR) schemes face challenges in efficiently retrieving data from untrusted servers without revealing access patterns, as they either consume significant bandwidth or require excessive computation, especially when dealing with large exponents in exponentiation operations.
Innovation Solution
A method is introduced to outsource exponentiation in a private group by generating a series of base values using the prime factorization of the modulus and reducing them by the modulus, allowing an untrusted server to perform the exponentiation while keeping the prime factorization secret, thus reducing bandwidth and computation costs.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of information
If conventional PIR schemes are used to retrieve data from untrusted servers, then data privacy is protected, but bandwidth consumption increases significantly
Solution Approach 1:
The exponentiation operation is segmented into multiple smaller exponentiations with precomputed base values. Instead of transmitting a single large exponent, the client generates a series of base values (g1, g2, ..., gm) where each corresponds to a digit position in the exponent's numeral system representation. The server performs m separate exponentiations with these smaller bases and combines the results, reducing the bandwidth required to transmit the exponent while maintaining privacy.
2Loss of information
If conventional PIR schemes are used to retrieve data from untrusted servers, then data privacy is protected, but computation requirements increase significantly
Solution Approach 1:
The client performs preliminary actions by precomputing the base values (g1, g2, ..., gm) based on the group element g and the prime factorization of the modulus M before contacting the server. These base values are prepared in advance according to the numeral system representation of the exponent, allowing the server to perform simpler exponentiations rather than one large exponentiation, thus improving overall computation efficiency.
Solution Approach 2:
Instead of the server performing the original exponentiation g^E directly, the client creates multiple copies of the base element g raised to different powers (g1, g2, ..., gm) that correspond to digits of the exponent. The server then combines these copies through multiplication to achieve the final result, distributing the computational workload and reducing the complexity of any single exponentiation operation.
3Loss of information
If the entire database is downloaded to avoid revealing access patterns, then privacy is maintained, but retrieval time increases excessively
Solution Approach 1:
The invention extracts only the necessary computational work (exponentiation) to the server while keeping the critical secret (prime factorization of M) on the client side. The client generates base values and transmits only these smaller values to the server, extracting the expensive exponentiation operation from the client while maintaining privacy, thus avoiding the need to download the entire database.
4Reliability
If large exponents are used in exponentiation operations, then security is improved, but computation and communication costs increase
Solution Approach 1:
The invention changes the parameters of the exponentiation operation by representing the large exponent E in a numeral system (binary, decimal, or hexadecimal) and breaking it into digits. This allows the computation to be performed as multiple smaller exponentiations with precomputed bases corresponding to each digit position, reducing the computational burden while maintaining the security provided by the large effective exponent through the combination of results.
Data Source
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AI summary
A method (500) for outsourcing exponentiation in a private group includes executing a query instruction (130) to retrieve a query element (152) stored on an untrusted server (111) by selecting a prime factorization of two or more prime numbers (135) of a modulus (134) associated with the query element stored on the server, obtaining a group element (133) configured to generate a respective one of the prime numbers, generating a series of base values (162) using the prime factorization and the group element, and transmitting the series of base values from a client device (10) to the server. The server is configured to determine an exponentiation of the group element with an exponent (154) stored on the server using the series of base values. The method also includes receiving a result (192) from the server based on the exponentiation of the group element with the exponent.