Multilinear Maps Using Ideal Lattices without Encodings of Zero

Garg, Gentry and Halevi (GGH) described the first candidate multilinear maps using ideal lattices. However, Hu and Jia recently presented an efficient attack for two applications based on the GGH map, multipartite Diffie-Hellman key exchange and an instance of witness encryption using 3-exact cover problem. In this paper, we describe a modification construction of multilinear maps from ideal lattices without encodings of zero by introducing random matrices to avoid the zeroing attack problem. The security of our construction depends upon new hardness assumption, which is seemingly closely related to hardness problems of lattices. Furthermore, we present multipartite Diffie-Hellman key exchange protocol using our construction, and an instance of witness encryption using 3-exact cover problem based on a variant of our construction.

[1]  Ron Rothblum,et al.  On the Circular Security of Bit-Encryption , 2013, TCC.

[2]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

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

[4]  Joseph H. Silverman,et al.  NTRU: A Ring-Based Public Key Cryptosystem , 1998, ANTS.

[5]  Nigel P. Smart,et al.  AN IDENTITY BASED AUTHENTICATED KEY AGREEMENT PROTOCOL BASED ON THE WEIL PAIRING , 2001 .

[6]  Phong Q. Nguyen,et al.  BKZ 2.0: Better Lattice Security Estimates , 2011, ASIACRYPT.

[7]  Roberto Tamassia,et al.  Optimal Authenticated Data Structures with Multilinear Forms , 2010, Pairing.

[8]  Jean-Sébastien Coron,et al.  Cryptanalysis of Two Candidate Fixes of Multilinear Maps over the Integers , 2014, IACR Cryptol. ePrint Arch..

[9]  Jung Hee Cheon,et al.  Cryptanalysis of the Multilinear Map over the Integers , 2014, EUROCRYPT.

[10]  Craig Gentry,et al.  Zeroizing without zeroes: Cryptanalyzing multilinear maps without encodings of zero , 2014, IACR Cryptol. ePrint Arch..

[11]  Brice Minaud,et al.  Cryptanalysis of the New CLT Multilinear Map over the Integers , 2016, EUROCRYPT.

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

[13]  Yupu Hu,et al.  Cryptanalysis of GGH Map , 2016, EUROCRYPT.

[14]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[15]  Guy N. Rothblum,et al.  Virtual Black-Box Obfuscation for All Circuits via Generic Graded Encoding , 2014, TCC.

[16]  Antoine Joux.  A One Round Protocol for Tripartite Diffie-Hellman , 2000, ANTS.

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

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

[19]  Brent Waters,et al.  Candidate Indistinguishability Obfuscation and Functional Encryption for all Circuits , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[20]  Dan Boneh,et al.  Applications of Multilinear Forms to Cryptography , 2002, IACR Cryptol. ePrint Arch..

[21]  Ron Steinfeld,et al.  Making NTRU as Secure as Worst-Case Problems over Ideal Lattices , 2011, EUROCRYPT.

[22]  Dan Boneh,et al.  Immunizing Multilinear Maps Against Zeroizing Attacks , 2014, IACR Cryptol. ePrint Arch..

[23]  Craig Gentry,et al.  Fully Secure Functional Encryption without Obfuscation , 2014, IACR Cryptol. ePrint Arch..

[24]  Tatsuaki Okamoto,et al.  Multilinear Maps on LWE , 2013 .

[25]  Brent Waters,et al.  Attribute-Based Encryption for Circuits from Multilinear Maps , 2012, CRYPTO.

[26]  Yupu Hu,et al.  A Comment on Gu Map-1 , 2015, IACR Cryptol. ePrint Arch..

[27]  Mark Zhandry,et al.  Multiparty Key Exchange, Efficient Traitor Tracing, and More from Indistinguishability Obfuscation , 2014, Algorithmica.

[28]  Craig Gentry,et al.  Fully Key-Homomorphic Encryption, Arithmetic Circuit ABE and Compact Garbled Circuits , 2014, EUROCRYPT.

[29]  Ron Steinfeld,et al.  GGHLite: More Efficient Multilinear Maps from Ideal Lattices , 2014, IACR Cryptol. ePrint Arch..

[30]  Brent Waters,et al.  Witness encryption and its applications , 2013, STOC '13.

[31]  Jean-Sébastien Coron,et al.  New Multilinear Maps Over the Integers , 2015, CRYPTO.

[32]  Markus Rückert,et al.  Aggregate and Verifiably Encrypted Signatures from Multilinear Maps Without Random Oracles , 2009, IACR Cryptol. ePrint Arch..

[33]  Leonid A. Levin,et al.  A Pseudorandom Generator from any One-way Function , 1999, SIAM J. Comput..