Speeding up the Number Theoretic Transform for Faster Ideal Lattice-Based Cryptography

The Number Theoretic Transform (NTT) provides efficient algorithms for cyclic and nega-cyclic convolutions, which have many applications in computer arithmetic, e.g., for multiplying large integers and large degree polynomials. It is commonly used in cryptographic schemes that are based on the hardness of the Ring Learning With Errors (R-LWE) problem to efficiently implement modular polynomial multiplication.

[1]  Guido Bertoni,et al.  Keccak , 2013, EUROCRYPT.

[2]  Léo Ducas,et al.  Lattice Signatures and Bimodal Gaussians , 2013, IACR Cryptol. ePrint Arch..

[3]  Craig Costello,et al.  Post-Quantum Key Exchange for the TLS Protocol from the Ring Learning with Errors Problem , 2015, 2015 IEEE Symposium on Security and Privacy.

[4]  Tim Güneysu,et al.  High-Performance Ideal Lattice-Based Cryptography on 8-Bit ATxmega Microcontrollers , 2015, LATINCRYPT.

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

[6]  Chris Peikert,et al.  Hardness of SIS and LWE with Small Parameters , 2013, CRYPTO.

[7]  Chris Peikert,et al.  SWIFFT: A Modest Proposal for FFT Hashing , 2008, FSE.

[8]  W. M. Gentleman,et al.  Fast Fourier Transforms: for fun and profit , 1966, AFIPS '66 (Fall).

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

[10]  Ron Steinfeld,et al.  Improved Security Proofs in Lattice-Based Cryptography: Using the Rényi Divergence Rather than the Statistical Distance , 2015, Journal of Cryptology.

[11]  David Harvey,et al.  Faster arithmetic for number-theoretic transforms , 2012, J. Symb. Comput..

[12]  Jintai Ding,et al.  Authenticated Key Exchange from Ideal Lattices , 2015, EUROCRYPT.

[13]  Erdem Alkim,et al.  Post-quantum Key Exchange - A New Hope , 2016, USENIX Security Symposium.

[14]  Paul Barrett,et al.  Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor , 1986, CRYPTO.

[15]  H. Nussbaumer,et al.  Fast polynomial transform algorithms for digital convolution , 1980 .

[16]  Zhe Liu,et al.  Efficient Ring-LWE Encryption on 8-Bit AVR Processors , 2015, CHES.

[17]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

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

[19]  Tancrède Lepoint,et al.  NFLlib: NTT-Based Fast Lattice Library , 2016, CT-RSA.

[20]  P. L. Montgomery.  Modular multiplication without trial division , 1985 .

[21]  Tim Güneysu,et al.  Towards Practical Lattice-Based Public-Key Encryption on Reconfigurable Hardware , 2013, Selected Areas in Cryptography.

[22]  Vadim Lyubashevsky,et al.  Lattice Signatures Without Trapdoors , 2012, IACR Cryptol. ePrint Arch..

[23]  J. Wrench.  Table errata: The art of computer programming, Vol. 2: Seminumerical algorithms (Addison-Wesley, Reading, Mass., 1969) by Donald E. Knuth , 1970 .

[24]  Oded Regev,et al.  On lattices, learning with errors, random linear codes, and cryptography , 2005, STOC '05.

[25]  Chris Peikert,et al.  On Ideal Lattices and Learning with Errors over Rings , 2010, JACM.

[26]  Chris Peikert,et al.  Lattice Cryptography for the Internet , 2014, PQCrypto.

[27]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[28]  Frederik Vercauteren,et al.  Compact Ring-LWE Cryptoprocessor , 2014, CHES.

[29]  Peter Schwabe,et al.  Software Speed Records for Lattice-Based Signatures , 2013, PQCrypto.

[30]  Jintai Ding,et al.  A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem , 2012, IACR Cryptol. ePrint Arch..

[31]  Phong Q. Nguyen,et al.  Advances in Cryptology – EUROCRYPT 2013 , 2013, Lecture Notes in Computer Science.