A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions

Pedro Branco

doi:10.3934/amc.2020046

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February 2021, 15(1): 113-130. doi: 10.3934/amc.2020046

A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions

Pedro Branco

Instituto Superior Técnico - University of Lisbon, SQIG - IT, Lisbon, Portugal

Received May 2019 Revised September 2019 Published February 2021 Early access November 2019

Fund Project: The author is supported by DP-PMI and FCT (Portugal) through the grant PD/BD/135181/2017

In this work, we propose a post-quantum UC-commitment scheme in the Global Random Oracle Model, where only one non-programmable random oracle is available. The security of our proposal is based on two well-established post-quantum hardness assumptions from coding theory: The Syndrome Decoding and the Goppa Distinguisher. We prove that our proposal is perfectly hiding and computationally binding.

Keywords: Commmitment, Universal Composability, global random oracle, code-based cryptography, post-quantum cryptography.

Mathematics Subject Classification: Primary: 94A60; Secondary: 94A15.

Citation: Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046

References:

[1]	P. S. L. M. Barreto, B. David, R. Dowsley, K. Morozov and A. C. A. Nascimento, A Framework for Efficient Adaptively Secure Composable Oblivious Transfer in the ROM, Cryptology ePrint Archive, Report 2017/993, 2017, https://eprint.iacr.org/2017/993.
[2]	A. Becker, A. Joux, A. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How $1+1 = 0$ improves information set decoding, Advances in Cryptology—EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520-536. doi: 10.1007/978-3-642-29011-4_31.
[3]	M. Bellare and P. Rogaway, Random oracles are practical: A paradigm for designing efficient protocols, CCS '93 Proceedings of the 1st ACM Conference on Computer and Communications Security, (1993), 62–73. doi: 10.1145/168588.168596.
[4]	E. R. Berlekamp, R. J. McEliece and H. C. A. van Tilborg, On the inherent intractability of certain coding problems, IEEE Transactions on Information Theory, 24 (1978), 384-386. doi: 10.1109/tit.1978.1055873.
[5]	P. Branco, J. T. Ding, M. Goulão and P. Mateus, Universally Composable Oblivious Transfer Protocol Based on the RLWE Assumption, Cryptology ePrint Archive, Report 2018/1155, 2018, https://eprint.iacr.org/2018/1155.
[6]	Megha Byali, Arpita Patra, Divya Ravi and Pratik Sarkar., Fast and Universally-Composable Oblivious Transfer and Commitment Scheme with Adaptive Security, Cryptology ePrint Archive, Report 2017/1165, 2017, https://eprint.iacr.org/2017/1165.
[7]	J. Camenisch, M. Drijvers, T. Gagliardoni, A. Lehmann and G. Neven, The wonderful world of global random oracles, Advances in cryptology—EUROCRYPT 2018. Part I, Lecture Notes in Comput. Sci., Springer, Cham, 10820 (2018), 280-312.
[8]	R. Canetti, Universally composable security: A new paradigm for cryptographic protocols, 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, (2001), 136–145.
[9]	R. Canetti and M. Fischlin, Universally composable commitments, Advances in Cryptology—CRYPTO 2001, Berlin, Heidelberg, Springer Berlin Heidelberg, (2001), 19–40. doi: 10.1007/3-540-44647-8_2.
[10]	R. Canetti, A. Jain and A. Scafuro, Practical UC security with a global random oracle, Proceedings of the 2014 ACM SIGSAC Conference on Computer and Communications Security, CCS '14, New York, NY, USA, ACM, (2014), 597–608. doi: 10.1145/2660267.2660374.
[11]	R. Canetti, Y. Lindell, R. Ostrovsky and A. Sahai, Universally composable two-party and multi-party secure computation, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, (2002), 494–503. doi: 10.1145/509907.509980.
[12]	I. Cascudo, I. Damgård, B. David, I. Giacomelli, J. B. Nielse and R. Trifiletti, Additively homomorphic uc commitments with optimal amortized overhead, Public-Key Cryptography—PKC 2015, Berlin, Heidelberg, Springer Berlin Heidelberg, 9020 (2015), 495-515. doi: 10.1007/978-3-662-46447-2_22.
[13]	I. Cascudo, I. Damgård, B. David, N. Döttling and J. B.Nielsen, Rate-1, linear time and additively homomorphic uc commitments, Advances in Cryptology—CRYPTO 2016, Berlin, Heidelberg, Springer Berlin Heidelberg, (2016), 179–207.
[14]	N. T. Courtois, M. Finiasz and N. Sendrier, How to achieve a McEliece-based digital signature scheme, Advances in Cryptology—ASIACRYPT 2001, Berlin, Heidelberg, Springer Berlin Heidelberg, 2248 (2001), 157–174. doi: 10.1007/3-540-45682-1_10.
[15]	I. Damgård and J. B. Nielsen, Perfect hiding and perfect binding universally composable commitment schemes with constant expansion factor, Advances in Cryptology—CRYPTO 2002, Berlin, Heidelberg, Springer Berlin Heidelberg, 2442 (2002), 581-596. doi: 10.1007/3-540-45708-9_37.
[16]	T. Debris-Alazard, N. Sendrier and J.-P. Tillich, Wave: A New Code-Based Signature Scheme, Cryptology ePrint Archive, Report 2018/996, 2018, https://eprint.iacr.org/2018/996.
[17]	A. Esser, R. Kübler and A. May, LPN decoded, Advances in cryptology—CRYPTO 2017. Part II, Lecture Notes in Comput. Sci., Springer, Cham, 10402 (2017), 486-514.
[18]	J.-C. Faugère, V. Gauthier-Umaña, A. Otmani, L. Perret and J. Tillich, A distinguisher for high rate McEliece cryptosystems, IEEE Trans. Inform. Theory, 59 (2013), 6830-6844. doi: 10.1109/TIT.2013.2272036.
[19]	M. Fischlin, B. Libert and M. Manulis, Non-interactive and re-usable universally composable string commitments with adaptive security, Advances in Cryptology—ASIACRYPT 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7073 (2011), 468-485. doi: 10.1007/978-3-642-25385-0_25.
[20]	E. Fujisaki, Improving practical UC-secure commitments based on the DDH assumption, Security and Cryptography for Networks, Lecture Notes in Comput. Sci., Springer, [Cham], 9841 (2016), 257-272. doi: 10.1007/978-3-319-44618-9_14.
[21]	J. A. Garay, Y. Ishai, R. Kumaresan and H. Wee, On the complexity of UC commitments, Advances in Cryptology—EUROCRYPT 2014, Lecture Notes in Comput. Sci., Springer, Heidelberg, 8441 (2014), 677-694. doi: 10.1007/978-3-642-55220-5_37.
[22]	D. Hofheinz and J. Müller-Quade, Universally composable commitments using random oracles, Theory of cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 2951 (2004), 58-76. doi: 10.1007/978-3-540-24638-1_4.
[23]	A. Jain, S. Krenn, K. Pietrzak and A. Tentes, Commitments and efficient zero-knowledge proofs from learning parity with noise, Advances in Cryptology—ASIACRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7658 (2012), 663-680. doi: 10.1007/978-3-642-34961-4_40.
[24]	J. Kilian, Founding cryptography on oblivious transfer, Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC '88, New York, NY, USA, ACM, (1988), 20–31.
[25]	E. Kirshanova, Improved quantum information set decoding, Post-quantum cryptography, Lecture Notes in Comput. Sci., Springer, Cham, 10786 (2018), 507-527. doi: 10.1002/fld.4317.
[26]	Y. Lindell, Highly-efficient universally-composable commitments based on the DDH assumption, Advances in cryptology—EUROCRYPT 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6632 (2011), 446–466. doi: 10.1007/978-3-642-20465-4_25.
[27]	P. Loidreau and N. Sendrier, Weak keys in the McEliece public-key cryptosystem, IEEE Transactions on Information Theory, 47 (2001), 1207-1211. doi: 10.1109/18.915687.
[28]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[29]	R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, JPL DSN Progress Report, 44 (1978).
[30]	C. Peikert, V. Vaikuntanathan and B. Waters, A framework for efficient and composable oblivious transfer, Advances in cryptology—CRYPTO 2008, Lecture Notes in Comput. Sci., Springer, Berlin, 5157 (2008), 554-571. doi: 10.1007/978-3-540-85174-5_31.
[31]	P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, 26 (1997), 1484-1509. doi: 10.1137/S0097539795293172.
[32]	X. Xie, R. Xue and M. Q. Wang, Zero knowledge proofs from ring-LWE, Cryptology and network security, Lecture Notes in Comput. Sci., Springer, Cham, 8257 (2013), 57-73. doi: 10.1007/978-3-319-02937-5_4.

show all references

References:

[1]	P. S. L. M. Barreto, B. David, R. Dowsley, K. Morozov and A. C. A. Nascimento, A Framework for Efficient Adaptively Secure Composable Oblivious Transfer in the ROM, Cryptology ePrint Archive, Report 2017/993, 2017, https://eprint.iacr.org/2017/993.
[2]	A. Becker, A. Joux, A. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How $1+1 = 0$ improves information set decoding, Advances in Cryptology—EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520-536. doi: 10.1007/978-3-642-29011-4_31.
[3]	M. Bellare and P. Rogaway, Random oracles are practical: A paradigm for designing efficient protocols, CCS '93 Proceedings of the 1st ACM Conference on Computer and Communications Security, (1993), 62–73. doi: 10.1145/168588.168596.
[4]	E. R. Berlekamp, R. J. McEliece and H. C. A. van Tilborg, On the inherent intractability of certain coding problems, IEEE Transactions on Information Theory, 24 (1978), 384-386. doi: 10.1109/tit.1978.1055873.
[5]	P. Branco, J. T. Ding, M. Goulão and P. Mateus, Universally Composable Oblivious Transfer Protocol Based on the RLWE Assumption, Cryptology ePrint Archive, Report 2018/1155, 2018, https://eprint.iacr.org/2018/1155.
[6]	Megha Byali, Arpita Patra, Divya Ravi and Pratik Sarkar., Fast and Universally-Composable Oblivious Transfer and Commitment Scheme with Adaptive Security, Cryptology ePrint Archive, Report 2017/1165, 2017, https://eprint.iacr.org/2017/1165.
[7]	J. Camenisch, M. Drijvers, T. Gagliardoni, A. Lehmann and G. Neven, The wonderful world of global random oracles, Advances in cryptology—EUROCRYPT 2018. Part I, Lecture Notes in Comput. Sci., Springer, Cham, 10820 (2018), 280-312.
[8]	R. Canetti, Universally composable security: A new paradigm for cryptographic protocols, 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, (2001), 136–145.
[9]	R. Canetti and M. Fischlin, Universally composable commitments, Advances in Cryptology—CRYPTO 2001, Berlin, Heidelberg, Springer Berlin Heidelberg, (2001), 19–40. doi: 10.1007/3-540-44647-8_2.
[10]	R. Canetti, A. Jain and A. Scafuro, Practical UC security with a global random oracle, Proceedings of the 2014 ACM SIGSAC Conference on Computer and Communications Security, CCS '14, New York, NY, USA, ACM, (2014), 597–608. doi: 10.1145/2660267.2660374.
[11]	R. Canetti, Y. Lindell, R. Ostrovsky and A. Sahai, Universally composable two-party and multi-party secure computation, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, (2002), 494–503. doi: 10.1145/509907.509980.
[12]	I. Cascudo, I. Damgård, B. David, I. Giacomelli, J. B. Nielse and R. Trifiletti, Additively homomorphic uc commitments with optimal amortized overhead, Public-Key Cryptography—PKC 2015, Berlin, Heidelberg, Springer Berlin Heidelberg, 9020 (2015), 495-515. doi: 10.1007/978-3-662-46447-2_22.
[13]	I. Cascudo, I. Damgård, B. David, N. Döttling and J. B.Nielsen, Rate-1, linear time and additively homomorphic uc commitments, Advances in Cryptology—CRYPTO 2016, Berlin, Heidelberg, Springer Berlin Heidelberg, (2016), 179–207.
[14]	N. T. Courtois, M. Finiasz and N. Sendrier, How to achieve a McEliece-based digital signature scheme, Advances in Cryptology—ASIACRYPT 2001, Berlin, Heidelberg, Springer Berlin Heidelberg, 2248 (2001), 157–174. doi: 10.1007/3-540-45682-1_10.
[15]	I. Damgård and J. B. Nielsen, Perfect hiding and perfect binding universally composable commitment schemes with constant expansion factor, Advances in Cryptology—CRYPTO 2002, Berlin, Heidelberg, Springer Berlin Heidelberg, 2442 (2002), 581-596. doi: 10.1007/3-540-45708-9_37.
[16]	T. Debris-Alazard, N. Sendrier and J.-P. Tillich, Wave: A New Code-Based Signature Scheme, Cryptology ePrint Archive, Report 2018/996, 2018, https://eprint.iacr.org/2018/996.
[17]	A. Esser, R. Kübler and A. May, LPN decoded, Advances in cryptology—CRYPTO 2017. Part II, Lecture Notes in Comput. Sci., Springer, Cham, 10402 (2017), 486-514.
[18]	J.-C. Faugère, V. Gauthier-Umaña, A. Otmani, L. Perret and J. Tillich, A distinguisher for high rate McEliece cryptosystems, IEEE Trans. Inform. Theory, 59 (2013), 6830-6844. doi: 10.1109/TIT.2013.2272036.
[19]	M. Fischlin, B. Libert and M. Manulis, Non-interactive and re-usable universally composable string commitments with adaptive security, Advances in Cryptology—ASIACRYPT 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7073 (2011), 468-485. doi: 10.1007/978-3-642-25385-0_25.
[20]	E. Fujisaki, Improving practical UC-secure commitments based on the DDH assumption, Security and Cryptography for Networks, Lecture Notes in Comput. Sci., Springer, [Cham], 9841 (2016), 257-272. doi: 10.1007/978-3-319-44618-9_14.
[21]	J. A. Garay, Y. Ishai, R. Kumaresan and H. Wee, On the complexity of UC commitments, Advances in Cryptology—EUROCRYPT 2014, Lecture Notes in Comput. Sci., Springer, Heidelberg, 8441 (2014), 677-694. doi: 10.1007/978-3-642-55220-5_37.
[22]	D. Hofheinz and J. Müller-Quade, Universally composable commitments using random oracles, Theory of cryptography, Lecture Notes in Comput. Sci., Springer, Berlin, 2951 (2004), 58-76. doi: 10.1007/978-3-540-24638-1_4.
[23]	A. Jain, S. Krenn, K. Pietrzak and A. Tentes, Commitments and efficient zero-knowledge proofs from learning parity with noise, Advances in Cryptology—ASIACRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7658 (2012), 663-680. doi: 10.1007/978-3-642-34961-4_40.
[24]	J. Kilian, Founding cryptography on oblivious transfer, Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC '88, New York, NY, USA, ACM, (1988), 20–31.
[25]	E. Kirshanova, Improved quantum information set decoding, Post-quantum cryptography, Lecture Notes in Comput. Sci., Springer, Cham, 10786 (2018), 507-527. doi: 10.1002/fld.4317.
[26]	Y. Lindell, Highly-efficient universally-composable commitments based on the DDH assumption, Advances in cryptology—EUROCRYPT 2011, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6632 (2011), 446–466. doi: 10.1007/978-3-642-20465-4_25.
[27]	P. Loidreau and N. Sendrier, Weak keys in the McEliece public-key cryptosystem, IEEE Transactions on Information Theory, 47 (2001), 1207-1211. doi: 10.1109/18.915687.
[28]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[29]	R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, JPL DSN Progress Report, 44 (1978).
[30]	C. Peikert, V. Vaikuntanathan and B. Waters, A framework for efficient and composable oblivious transfer, Advances in cryptology—CRYPTO 2008, Lecture Notes in Comput. Sci., Springer, Berlin, 5157 (2008), 554-571. doi: 10.1007/978-3-540-85174-5_31.
[31]	P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, 26 (1997), 1484-1509. doi: 10.1137/S0097539795293172.
[32]	X. Xie, R. Xue and M. Q. Wang, Zero knowledge proofs from ring-LWE, Cryptology and network security, Lecture Notes in Comput. Sci., Springer, Cham, 8257 (2013), 57-73. doi: 10.1007/978-3-319-02937-5_4.

Figure 1. Sigma protocol structure. The value $ x $ is public information and $ w $ is usually called the witness. Let $ \sim $ be a relation and $ \mathcal{R} = \{(x, w):x\sim w\} $. A transcript $ T = ( {com}, {ch}, {resp}) $ is the tuple of messages exchange and we say that it is valid when $ \mathsf{V}_2(x, T) = 1 $

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Figure 2. $ \left( \mathsf{P}, \mathsf{V}\right)_{ \mathsf{xLPN}} $ scheme. Let $ \mathbf{A}\in\{0,1\}^{k\times n} $, $ \mathbf{s}\in \{0,1\}^k $, $ \mathbf{e}\in \{0,1\}^n $ such that $ \mathbf{e}\in \mathfrak{B}_{ = \omega}^n $ and let $ \mathbf{y} \in \{0,1\}^{n} $ such that $ \mathbf{s} \mathbf{A}+ \mathbf{e} = \mathbf{y} $. By $ \mathcal{C}_ \mathbf{A} $ we denote the code defined by $ \mathbf{A} $. Let $ S $ be the set of permutations of size $ n $ and let $ ( \mathsf{Com}, \mathsf{Ver}) $ be a commitment scheme where $ \mathsf{Com} $ is the commitment algorithm and $ \mathsf{Ver} $ is the opening algorithm

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Figure 3. Commitment phase. $ \mathsf{GenTd} $ creates a matrix $ \mathbf{A} $ and a trapdoor, according to Lemma 1 and $ \mathsf{gRO} $ is the global random oracle. The random strings $ t_1 $, $ t_2 $ and $ u_1 $ are omitted

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Figure 4. Opening phase. $ ( \mathsf{P}^\varepsilon, \mathsf{V}^\varepsilon) $ is the sigma-protocol $ ( \mathsf{P}, \mathsf{V})_ \mathsf{xLPN} $ repeated $ \mathcal{O}(1/\varepsilon) $, where $ \mathsf{P}^\varepsilon = ( \mathsf{P}^\varepsilon_1, \mathsf{P}^\varepsilon_2) $ and $ \mathsf{V}^\varepsilon = ( \mathsf{V}^\varepsilon_1, \mathsf{V}^\varepsilon_2) $. The random strings $ t_1 $, $ t_2 $, $ t_3 $, $ u_1 $ and $ u_2 $ are omitted

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