US Patent 11005654 Outsourcing exponentiation in a private group

A method for outsourcing exponentiation in a private group includes executing a query instruction to retrieve a query element stored on an untrusted server by selecting a prime factorization of two or more prime numbers of a modulus associated with the query element stored on the server, obtaining a group element configured to generate a respective one of the prime numbers, generating a series of base values using the prime factorization and the group element, and transmitting the series of base values from the client device to the server. The server is configured to determine an exponentiation of the group element with an exponent stored on the server using the series of base values. The method also includes receiving a result from the server based on the exponentiation of the group element with the exponent.