# Bounded Tamper Resilience: How to Go Beyond the Algebraic Barrier

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## Abstract

Related key attacks (RKAs) are powerful cryptanalytic attacks where an adversary can change the secret key and observe the effect of such changes at the output. The state of the art in RKA security protects against an a-priori unbounded number of certain algebraic induced key relations, e.g., affine functions or polynomials of bounded degree. In this work, we show that it is possible to go beyond the algebraic barrier and achieve security against *arbitrary* key relations, by restricting the number of tampering queries the adversary is allowed to ask for. The latter restriction is necessary in case of arbitrary key relations, as otherwise a generic attack of Gennaro et al. (TCC 2004) shows how to recover the key of almost any cryptographic primitive. We describe our contributions in more detail below. (1) We show that standard ID and signature schemes constructed from a large class of \(\Sigma \)-protocols (including the Okamoto scheme, for instance) are secure even if the adversary can *arbitrarily* tamper with the prover’s state a *bounded* number of times and obtain some bounded amount of leakage. Interestingly, for the Okamoto scheme we can allow also independent tampering with the public parameters. (2) We show a *bounded* tamper and leakage resilient CCA-secure public key cryptosystem based on the DDH assumption. We first define a weaker CCA-like security notion that we can instantiate based on DDH, and then we give a general compiler that yields CCA security with tamper and leakage resilience. This requires a public tamper-proof common reference string. (3) Finally, we explain how to boost bounded tampering and leakage resilience [as in (1) and (2) above] to *continuous* tampering and leakage resilience, in the so-called *floppy model* where each user has a personal hardware token (containing leak- and tamper-free information) which can be used to refresh the secret key. We believe that bounded tampering is a meaningful and interesting alternative to avoid known impossibility results and can provide important insights into the security of existing standard cryptographic schemes.

## Keywords

Related key security Bounded tamper resilience Public key encryption Identification schemes## Notes

### Acknowledgments

This work was done while the last author was a postdoc at the Computer Science Department of Aarhus University, supported by the Danish Council for Independent Research (under the DFF Starting Grant 10-081612). Ivan Damgård acknowledges support from the Danish National Research Foundation, the National Science Foundation of China (under the Grant 61061130540), and also from the CFEM research center. Sebastian Faust was partially funded by the above grants. Pratyay Mukherjee’s work at Aarhus University was supported by a European Research Commission Starting Grant (no. 279447) and the above grants. Part of this work was done while this author was at the University of Warsaw and was supported by the WELCOME/2010-4/2 Grant founded within the framework of the EU Innovative Economy Operational Programme.

## References

- 1.D. Aggarwal, Y. Dodis, T. Kazana, M. Obremski, Non-malleable reductions and applications, in
*STOC*(2015)Google Scholar - 2.D. Aggarwal, Y. Dodis, S. Lovett, Non-malleable codes from additive combinatorics, in
*STOC*, (2014), pp. 774–783Google Scholar - 3.D. Aggarwal, S. Dziembowski, T. Kazana, M. Obremski, Leakage-resilient non-malleable codes, in
*TCC*, (2015), pp. 398–426Google Scholar - 4.S. Agrawal, D. Gupta, H.K. Maji, O. Pandey, M. Prabhakaran, Explicit non-malleable codes against bit-wise tampering and permutations, in
*CRYPTO*, (2015), pp 538–557.Google Scholar - 5.S. Agrawal, D. Gupta, H.K. Maji, O. Pandey, M. Prabhakaran, A rate-optimizing compiler for non-malleable codes against bit-wise tampering and permutations, in
*TCC*, (2015), pp. 375–397Google Scholar - 6.S. Agrawal, Y. Dodis, V. Vaikuntanathan, D. Wichs, On continual leakage of discrete log representations, in
*ASIACRYPT*, (2013), pp. 401–420Google Scholar - 7.J. Alwen, Y. Dodis, D. Wichs, Leakage-resilient public-key cryptography in the bounded-retrieval model, in
*CRYPTO*, (2009), pp. 36–54Google Scholar - 8.R. Anderson, M. Kuhn, Tamper resistance: a cautionary note, in
*WOEC’96: Proceedings of the 2nd conference on Proceedings of the Second USENIX Workshop on Electronic Commerce*, (USENIX Association, Berkeley, 1996), p. 1Google Scholar - 9.B. Applebaum, D. Harnik, Y. Ishai, Semantic security under related-key attacks and applications, in
*ICS*, (2011), pp. 45–60Google Scholar - 10.M. Bellare, D. Cash, Pseudorandom functions and permutations provably secure against related-key attacks, in
*CRYPTO*, (2010), pp. 666–684Google Scholar - 11.M. Bellare, D. Cash, R. Miller, Cryptography secure against related-key attacks and tampering. In
*ASIACRYPT*, (2011), pp. 486–503Google Scholar - 12.M. Bellare, T. Kohno, A theoretical treatment of related-key attacks: RKA-PRPs, RKA-PRFs, and applications, in
*EUROCRYPT*, (2003), pp. 491–506Google Scholar - 13.M. Bellare, K.G. Paterson, S. Thomson. RKA security beyond the linear barrier: IBE, encryption and signatures, in
*ASIACRYPT*, (2012), pp. 331–348Google Scholar - 14.R. Bhattacharyya, A. Roy, Secure message authentication against related key attack, in
*FSE*(2013)Google Scholar - 15.D. Boneh, R.A. DeMillo, R.J. Lipton, On the importance of eliminating errors in cryptographic computations.
*J. Cryptol.*,**14**(2):101–119 (2001)Google Scholar - 16.D. Boneh, S. Halevi, M. Hamburg, R. Ostrovsky, Circular-secure encryption from decision diffie-hellman, in
*CRYPTO*, (2008), pp. 108–125Google Scholar - 17.J. Camenisch, N. Chandran, V. Shoup, A public key encryption scheme secure against key dependent chosen plaintext and adaptive chosen ciphertext attacks, in
*EUROCRYPT*, (2009), pp. 351–368Google Scholar - 18.E. Chattopadhyay, D. Zuckerman. Non-malleable codes against constant split-state tampering, in
*FOCS*, (2014), pp. 306–315Google Scholar - 19.M. Cheraghchi, V. Guruswami, Capacity of non-malleable codes, in
*Innovations in Theoretical Computer Science, ITCS*, (2014), pp. 155–168Google Scholar - 20.M. Cheraghchi, V. Guruswami, Non-malleable coding against bit-wise and split-state tampering, in
*TCC*, (2014), pp. 440–464Google Scholar - 21.S.G. Choi, A. Kiayias, T. Malkin, BiTR: Built-in tamper resilience, in
*ASIACRYPT*, (2011), pp. 740–758Google Scholar - 22.S. Coretti, Y. Dodis, B. Tackmann, D. Venturi, Non-malleable encryption: simpler, shorter, stronger.
*IACR Cryptol. ePrint Archive*,**772**(2015)Google Scholar - 23.S. Coretti, U. Maurer, B. Tackmann, D. Venturi, From single-bit to multi-bit public-key encryption via non-malleable codes, in
*TCC*, (2015), pp. 532–560Google Scholar - 24.R. Cramer,
*Modular Design of Secure yet Practical Cryptographic Protocols*. PhD thesis, University of Amsterdam, (1996)Google Scholar - 25.G. Di Crescenzo, R.J. Lipton, S. Walfish, Perfectly secure password protocols in the bounded retrieval model, in
*TCC*, (2006), pp. 225–244Google Scholar - 26.D. Dachman-Soled, Y.T. Kalai, Securing circuits against constant-rate tampering, in
*CRYPTO*, (2012), pp. 533–551Google Scholar - 27.D. Dachman-Soled, Y.T. Kalai, Securing circuits and protocols against 1/poly(k) tampering rate, in
*TCC*, (2014), pp. 540–565Google Scholar - 28.D. Dachman-Soled, F.-H. Liu, E. Shi, H.-S. Zhou, Locally decodable and updatable non-malleable codes and their applications, in
*TCC*, (2015), pp. 427–450Google Scholar - 29.I. Damgård, S. Faust, P. Mukherjee, D. Venturi, Bounded tamper resilience: How to go beyond the algebraic barrier, in
*ASIACRYPT*, (2013), pp. 140–160Google Scholar - 30.I. Damgård, S. Faust, P. Mukherjee, D. Venturi, The chaining lemma and its application, in
*ICITS*, (2015), pp. 181–196Google Scholar - 31.Y. Dodis, K. Haralambiev, A. López-Alt, D. Wichs, Cryptography against continuous memory attacks, in
*FOCS*, (2010), pp. 511–520Google Scholar - 32.Y. Dodis, K. Haralambiev, A. López-Alt, D. Wichs, Efficient public-key cryptography in the presence of key leakage, in
*ASIACRYPT*, (2010), pp. 613–631Google Scholar - 33.Y. Dodis, R. Ostrovsky, L. Reyzin, A. Smith, Fuzzy extractors: How to generate strong keys from biometrics and other noisy data.
*SIAM J. Comput.***38**(1), 97–139 (2008)Google Scholar - 34.S. Dziembowski, Intrusion-resilience via the bounded-storage model, in
*TCC*, (2006), pp. 207–224Google Scholar - 35.S. Dziembowski, T. Kazana, M. Obremski, Non-malleable codes from two-source extractors, in
*CRYPTO*, (2013), pp. 239–257Google Scholar - 36.S. Dziembowski, T. Kazana, D. Wichs, One-time computable self-erasing functions, in
*TCC*, (2011), pp. 125–143Google Scholar - 37.S. Dziembowski, K. Pietrzak, D. Wichs, Non-malleable codes, in
*ICS*, (2010), pp. 434–452Google Scholar - 38.S. Faust, M. Kohlweiss, G.A. Marson, D. Venturi, On the non-malleability of the fiat-shamir transform, in
*INDOCRYPT*, (2012), pp. 60–79Google Scholar - 39.S. Faust, P. Mukherjee, J.B. Nielsen, D. Venturi, Continuous non-malleable codes, in
*TCC*(2014)Google Scholar - 40.S. Faust, P. Mukherjee, J.B. Nielsen, D. Venturi, A tamper and leakage resilient von Neumann architecture, in
*PKC*, (2015), pp. 579–603Google Scholar - 41.S. Faust, P. Mukherjee, D. Venturi, D. Wichs, Efficient non-malleable codes and key-derivation for poly-size tampering circuits, in
*EUROCRYPT*, (2014), pp. 111–128Google Scholar - 42.S. Faust, K. Pietrzak, D. Venturi, Tamper-proof circuits: How to trade leakage for tamper-resilience. In
*ICALP (1)*, (2011), pp. 391–402Google Scholar - 43.A. Fiat, A. Shamir, How to prove yourself: practical solutions to identification and signature problems, in
*CRYPTO*, (1986), pp. 186–194Google Scholar - 44.M. Fischlin, R. Fischlin, The representation problem based on factoring, in
*CT-RSA*, (2002), pp. 96–113Google Scholar - 45.D. Genkin, Y. Ishai, M. Prabhakaran, A. Sahai, E. Tromer, Circuits resilient to additive attacks with applications to secure computation, in
*STOC*, (2014), pp. 495–504Google Scholar - 46.R. Gennaro, A. Lysyanskaya, T. Malkin, S. Micali, T. Rabin, Algorithmic tamper-proof (ATP) security: theoretical foundations for security against hardware tampering, in
*TCC*, (2004), pp. 258–277Google Scholar - 47.V. Goyal, A. O’Neill, V. Rao, Correlated-input secure hash functions, in
*TCC*, (2011), pp. 182–200Google Scholar - 48.J. Groth, Simulation-sound NIZK proofs for a practical language and constant size group signatures, in
*ASIACRYPT*, (2006), pp. 444–459Google Scholar - 49.L.C. Guillou, J.-J. Quisquater, A “paradoxical” identity-based signature scheme resulting from zero-knowledge, in
*CRYPTO*, (1988), pp. 216–231Google Scholar - 50.Y. Ishai, M. Prabhakaran, A. Sahai, D. Wagner, Private circuits II: keeping secrets in tamperable circuits, in
*EUROCRYPT*, (2006), pp. 308–327Google Scholar - 51.Z. Jafargholi, D. Wichs, Tamper detection and continuous non-malleable codes, in
*TCC*, (2015), pp. 451–480,Google Scholar - 52.Y.T. Kalai, B. Kanukurthi, A. Sahai, Cryptography with tamperable and leaky memory, in
*CRYPTO*, (2011), pp. 373–390Google Scholar - 53.J. Katz, V. Vaikuntanathan, Signature schemes with bounded leakage resilience, In
*ASIACRYPT*, (2009), pp. 703–720Google Scholar - 54.A. Kiayias, Y. Tselekounis, Tamper resilient circuits: the adversary at the gates, in
*ASIACRYPT*, (2013), pp. 161–180Google Scholar - 55.F.-H. Liu, A. Lysyanskaya, Tamper and leakage resilience in the split-state model, in
*CRYPTO*, (2012), pp. 517–532Google Scholar - 56.S. Lucks, Ciphers secure against related-key attacks, in
*FSE*, (2004), pp. 359–370Google Scholar - 57.M. Naor, G. Segev, Public-key cryptosystems resilient to key leakage, in
*CRYPTO*, (2009), pp. 18–35Google Scholar - 58.T. Okamoto, Provably secure and practical identification schemes and corresponding signature schemes, in
*CRYPTO*, (1992), pp. 31–53Google Scholar - 59.K. Pietrzak, Subspace LWE, in
*TCC*, (2012), pp. 548–563Google Scholar - 60.S. Pohlig, M. Hellman, An improved algorithm for computing logarithms over and its cryptographic significance.
*IEEE Trans. Inform. Theory*,**24**(1), 106–110 (1978)Google Scholar - 61.B. Qin, S. Liu, T.H. Yuen, R.H. Deng, K. Chen, Continuous non-malleable key derivation and its application to related-key security, in
*PKC*, (2015), pp. 557–578Google Scholar - 62.H. Wee, Public key encryption against related key attacks, in
*PKC*, (2012), pp. 262–279Google Scholar