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February  2020, 14(1): 171-175. doi: 10.3934/amc.2020014

Giophantus distinguishing attack is a low dimensional learning with errors problem

Jintai Ding , , Joshua Deaton and  Kurt Schmidt

University of Cincinnati, Cincinnati, OH 45219, USA

*Corresponding author: Jintai Ding

Received  February 2019 Revised  May 2019 Published  August 2019

In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

Keywords: Lattice, post quantum cryptography, Learning with Errors, Giophantus, Cryptanalysis, NIST..
Mathematics Subject Classification: Primary: 94A60, 68P25; Secondary: 81P94.
Citation: Jintai Ding, Joshua Deaton, Kurt Schmidt. Giophantus distinguishing attack is a low dimensional learning with errors problem. Advances in Mathematics of Communications, 2020, 14 (1) : 171-175. doi: 10.3934/amc.2020014
References:
[1]

K. Akiyama, Y. Goto, S. Okumura, T. Takagi, K. Nuida, G. Hanaoka, H. Shimizu and Y. Ikematsu, A public-key encryption scheme based on non-linear indeterminate equations (Giophantus), Cryptology ePrint Archive, Report 2017/1241, (2017), Available from: https://eprint.iacr.org/2017/1241. Google Scholar

[2]

K. Akiyama, Indeterminate equation public-key cryptosystem "$Giophantus$", First PQC Standardization Conference, Fort Lauderdale FL. USA, (2018), Available from: https://csrc.nist.gov/CSRC/media/Presentations/Giophantus/images-media/Giophantus-April2018.pdf. Google Scholar

[3]

M. R. Albrecht, R. Fitzpatrick and F. Göpfert, On the efficacy of solving LWE by reduction to unique-SVP, International Conference on Information Security and Cryptology, Cryptology ePrint Archive, Report 2013/602, 8565 (2013), 293–310, Available from: https://eprint.iacr.org/2013/602. doi: 10.1007/978-3-319-12160-4_18.  Google Scholar

[4]

W. Beullens, W. Castryck and F. Vercauteren, IND-CPA attack on Giophantus, (2018), Available from: https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/official-comments/Giophantus-official-comment.pdf. Google Scholar

[5]

J. T. Ding, S. Alsayigh, R. V. Saraswathy, S. Fluhrer and X. D. Lin, Leakage of Signal function with reused keys in RLWE key exchange, 2017 IEEE International Conference on Communications (ICC), (2017), Available from: https://eprint.iacr.org/2016/1176. doi: 10.1109/ICC.2017.7996806.  Google Scholar

[6]

S. Fluhrer, Cryptanalysis of ring-LWE based key exchange with key share reuse, Cryptology ePrint Archive: Report 2016/085, (2016), Available from: https://eprint.iacr.org/2016/085. Google Scholar

[7]

P. Nguyen, Giophantus and *LWR-based submissions, (2018), Available from: https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/official-comments/Giophantus-official-comment.pdf. Google Scholar

[8]

O. Regev, On lattices, learning with errors, random linear codes, and cryptography, Proceedings of the Annual ACM Symposium on Theory of Computing, (2005), 84–93. doi: 10.1145/1060590.1060603.  Google Scholar

show all references

References:
[1]

K. Akiyama, Y. Goto, S. Okumura, T. Takagi, K. Nuida, G. Hanaoka, H. Shimizu and Y. Ikematsu, A public-key encryption scheme based on non-linear indeterminate equations (Giophantus), Cryptology ePrint Archive, Report 2017/1241, (2017), Available from: https://eprint.iacr.org/2017/1241. Google Scholar

[2]

K. Akiyama, Indeterminate equation public-key cryptosystem "$Giophantus$", First PQC Standardization Conference, Fort Lauderdale FL. USA, (2018), Available from: https://csrc.nist.gov/CSRC/media/Presentations/Giophantus/images-media/Giophantus-April2018.pdf. Google Scholar

[3]

M. R. Albrecht, R. Fitzpatrick and F. Göpfert, On the efficacy of solving LWE by reduction to unique-SVP, International Conference on Information Security and Cryptology, Cryptology ePrint Archive, Report 2013/602, 8565 (2013), 293–310, Available from: https://eprint.iacr.org/2013/602. doi: 10.1007/978-3-319-12160-4_18.  Google Scholar

[4]

W. Beullens, W. Castryck and F. Vercauteren, IND-CPA attack on Giophantus, (2018), Available from: https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/official-comments/Giophantus-official-comment.pdf. Google Scholar

[5]

J. T. Ding, S. Alsayigh, R. V. Saraswathy, S. Fluhrer and X. D. Lin, Leakage of Signal function with reused keys in RLWE key exchange, 2017 IEEE International Conference on Communications (ICC), (2017), Available from: https://eprint.iacr.org/2016/1176. doi: 10.1109/ICC.2017.7996806.  Google Scholar

[6]

S. Fluhrer, Cryptanalysis of ring-LWE based key exchange with key share reuse, Cryptology ePrint Archive: Report 2016/085, (2016), Available from: https://eprint.iacr.org/2016/085. Google Scholar

[7]

P. Nguyen, Giophantus and *LWR-based submissions, (2018), Available from: https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/official-comments/Giophantus-official-comment.pdf. Google Scholar

[8]

O. Regev, On lattices, learning with errors, random linear codes, and cryptography, Proceedings of the Annual ACM Symposium on Theory of Computing, (2005), 84–93. doi: 10.1145/1060590.1060603.  Google Scholar

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