Packing Messages and Optimizing Bootstrapping in GSW-FHE

We construct the first fully homomorphic encryption (FHE) scheme that encrypts matrices and supports homomorphic matrix addition and multiplication. This is a natural extension of packed FHE and thus supports more complicated homomorphic operations. We optimize the bootstrapping procedure of Alperin-Sheriff and Peikert (CRYPTO 2014) by applying our scheme. Our optimization decreases the lattice approximation factor from \(\tilde{O}(n^{3})\) to \(\tilde{O}(n^{2.5})\). By taking a lattice dimension as a larger polynomial in a security parameter, we can also obtain the same approximation factor as the best known one of standard lattice-based public-key encryption without successive dimension-modulus reduction, which was essential for achieving the best factor in prior works on bootstrapping of standard lattice-based FHE.

[1]  Ron Rothblum,et al.  Homomorphic Encryption: from Private-Key to Public-Key , 2011, Electron. Colloquium Comput. Complex..

[2]  Chris Peikert,et al.  Faster Bootstrapping with Polynomial Error , 2014, CRYPTO.

[3]  Craig Gentry,et al.  A fully homomorphic encryption scheme , 2009 .

[4]  Vinod Vaikuntanathan,et al.  Fully Homomorphic Encryption from Ring-LWE and Security for Key Dependent Messages , 2011, CRYPTO.

[5]  Oded Regev,et al.  On lattices, learning with errors, random linear codes, and cryptography , 2009, JACM.

[6]  Daniele Micciancio,et al.  Pseudorandom Knapsacks and the Sample Complexity of LWE Search-to-Decision Reductions , 2011, CRYPTO.

[7]  Craig Gentry,et al.  Fully Homomorphic Encryption over the Integers , 2010, EUROCRYPT.

[8]  Craig Gentry,et al.  Better Bootstrapping in Fully Homomorphic Encryption , 2012, Public Key Cryptography.

[9]  Brent Waters,et al.  A Framework for Efficient and Composable Oblivious Transfer , 2008, CRYPTO.

[10]  Jean-Sébastien Coron,et al.  Practical Multilinear Maps over the Integers , 2013, CRYPTO.

[11]  Craig Gentry,et al.  Packed Ciphertexts in LWE-Based Homomorphic Encryption , 2013, Public Key Cryptography.

[12]  D. A. Barrington.  BOUNDED WIDTH BRANCHING PROGRAMS , 1986 .

[13]  Brent Waters,et al.  Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based , 2013, CRYPTO.

[14]  Vinod Vaikuntanathan,et al.  Efficient Fully Homomorphic Encryption from (Standard) LWE , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[15]  Vinod Vaikuntanathan,et al.  Lattice-based FHE as secure as PKE , 2014, IACR Cryptol. ePrint Arch..

[16]  Craig Gentry,et al.  Fully homomorphic encryption using ideal lattices , 2009, STOC '09.

[17]  Craig Gentry,et al.  Graph-Induced Multilinear Maps from Lattices , 2015, TCC.

[18]  Craig Gentry,et al.  (Leveled) fully homomorphic encryption without bootstrapping , 2012, ITCS '12.

[19]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[20]  Frederik Vercauteren,et al.  Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes , 2010, Public Key Cryptography.

[21]  Chris Peikert,et al.  Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller , 2012, IACR Cryptol. ePrint Arch..

[22]  Craig Gentry,et al.  Candidate Multilinear Maps from Ideal Lattices , 2013, EUROCRYPT.

[23]  Vinod Vaikuntanathan,et al.  On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption , 2012, STOC '12.

[24]  Zvika Brakerski,et al.  Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP , 2012, CRYPTO.

[25]  Chris Peikert,et al.  Public-key cryptosystems from the worst-case shortest vector problem: extended abstract , 2009, STOC '09.