# 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.

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