Abstract
We study non-malleable secret sharing against joint leakage and joint tampering attacks. Our main result is the first threshold secret sharing scheme in the plain model achieving resilience to noisy-leakage and continuous tampering. The above holds under (necessary) minimal computational assumptions (i.e., the existence of one-to-one one-way functions), and in a model where the adversary commits to a fixed partition of all the shares into non-overlapping subsets of at most \(t-1\) shares (where t is the reconstruction threshold), and subsequently jointly leaks from and tampers with the shares within each partition.
We also study the capacity (i.e., the maximum achievable asymptotic information rate) of continuously non-malleable secret sharing against joint continuous tampering attacks. In particular, we prove that whenever the attacker can tamper jointly with \(k > t/2\) shares, the capacity is at most \(t - k\). The rate of our construction matches this upper bound.
An important corollary of our results is the first non-malleable secret sharing scheme against independent tampering attacks breaking the rate-one barrier (under the same computational assumptions as above).
The first and last author acknowledge support by Sapienza University under the grant SPECTRA.
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Notes
- 1.
The only (necessary) restriction is that the experiment self-destructs after the first tampering query yielding an invalid secret sharing.
- 2.
In this setting, once a subset of shares has been tampered with jointly, that subset is always either tampered jointly or not modified by future tampering queries.
- 3.
i.e., one-joint leakage and tampering attacks.
- 4.
In the hybrid experiment, one also needs to adjust the reconstruction algorithm so that an attacker cannot distinguish between the hybrid and the original experiment by mauling \(\sigma _1,\sigma _2\) without changing the underlying shared value. In the description, we omit these details to simplify the exposition.
- 5.
Actually, Goyal et al. prove security in the more general setting of \(t\)-cover-free tampering, in which, intuitively, the subsets of the partition of the shares may overlap.
- 6.
Note that in case the tampered commitments within one of the subsets \(\mathcal {B} _i\) are not equal, the reduction can immediately self-destruct.
- 7.
Note that in case the tampered ciphertexts within one of the subsets \(\mathcal {B} _i\) are not equal, the reduction can immediately self-destruct.
- 8.
They require that the leakage on the i-th share does not drop the conditional average min-entropy of the share i conditioned on all other shares \(j\ne i\) by too much. This additional requirement makes their rate compiler incompatible with the non-malleable secret sharing scheme by Brian, Faonio, Obremski, Simkin and Venturi [6] which does not satisfy this property.
- 9.
Goyal and Kumar [18] originally gave a simulation-based definition of non-malleability (for the case of one-time tampering). It is folklore that this flavor of non-malleability can be shown to be equivalent to the indistinguishability-based notion we define (even in the setting of continuous tampering), so long as the message length is super-logarithmic in the security parameter.
- 10.
It is always possible to modify the reconstruction algorithm \(\mathsf {Rec}\) so that, upon input more than \(t\) shares, say \(\sigma _{i_1}, \ldots , \sigma _{i_t}, \ldots \), with \(i_1< i_2< \ldots< i_t< \ldots \), it only considers the first \(t\) shares \(\sigma _{i_1}, \ldots , \sigma _{i_t}\) in order to reconstruct the message.
- 11.
For the information dispersal, it suffices to define \(\mathsf {IDisp}(\mu ) := (\mu , \ldots , \mu )\) (i.e., the same message repeated \(n\) times) and \(\mathsf {IRec}(\mu ) := \mu \).
- 12.
For the sake of simplicity, assume \(t\) even. When \(t\) is odd, we obtain \(t^* = (t- 1)/2\).
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Acknowledgments
We thank Mark Simkin and Maciej Obremski for their valuable comments.
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Brian, G., Faonio, A., Venturi, D. (2021). Continuously Non-malleable Secret Sharing: Joint Tampering, Plain Model and Capacity. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13043. Springer, Cham. https://doi.org/10.1007/978-3-030-90453-1_12
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