Fully Homomorphic Encryption without Modulus Switching

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This scheme proposed by Brakerski [1] has a number of advantages over previous candidates such as BGV. In particular, it uses the same modulus throughout the evaluation process, so there's no need for modulus switching. Security of these scheme is baed on the hardness of the GapSVP problem.

Preliminaries

For an integer , write . This is not the same with the ring . For any , write for the unique value in that is congruent to modulo .

If are two -dimensional vectors, then the tensor product is the dimensional vector containing all elements of the form . Note that

Building Blocks of a homomorphic encryption scheme

We start by presenting Regev's [2] basic public-key encryption scheme.

References

  1. Z. Brakerski, Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP. In: Safavi-Naini R., Canetti R. (eds) Advances in Cryptology – CRYPTO 2012. CRYPTO 2012. Lecture Notes in Computer Science, vol 7417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32009-5_50
  2. O. Regev. On lattices, learning with errors, random linear codes, and cryptography. In Harold N. Gabow and Ronald Fagin, editors, STOC, pages 84–93. ACM, 2005