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Using Mathematics to Make Computing on Encrypted Data Secure and Practical

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Final rept. Dec 2012-Jun 2015

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In order to make computing on encrypted data more practical to use and more secure from attack, it is necessary to discover, develop, and understand the mathematics on which it is based and the mathematics that can be used to attack it. The security of homomorphic encryption schemes is based on the presumed difficulty of mathematics problems about lattices. Discovering and fully exploring algorithms to solve these mathematical problems allow computing on encrypted data to be performed with confidence, knowing that its cryptographic security is based on sound mathematical foundations. Hendrik Lenstra and Alice Silverberg discovered and developed algorithms to solve some lattices problems under suitable conditions, and investigated the mathematical foundations of these algorithms. A primary method of attack on homomorphic encryption schemes consists of lattice algorithms performed on ideal lattices, which are lattices with a certain type of algebraic structure. Any structure or symmetry is potentially susceptible to exploitation and attack. The work performed here gives algorithms for lattice problems for lattices that have symmetry. Recommendations are that the mathematical foundations of lattices with symmetry be further developed, in order to quantify the security of lattice-based cryptography, including especially the security of homomorphic encryption schemes.

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  • Computer Systems Management and Standards
  • Cybernetics

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