Abstract
Let \(\mathbb{G}\) be a group of prime order q, and let g 1,…,g n be random elements of \(\mathbb{G}\). We say that a vector x = \((x_1,\ldots,x_n)\in \mathbb{Z}_q^n\) is a discrete log representation of some some element \(y\in\mathbb{G}\) (with respect to g 1,…,g n ) if \(g_1^{x_1}\cdots g_n^{x_n} = y\). Any element y has many discrete log representations, forming an affine subspace of \(\mathbb{Z}_q^n\). We show that these representations have a nice continuous leakage-resilience property as follows. Assume some attacker \(\mathcal{A}(g_1,\ldots,g_n,y)\) can repeatedly learn L bits of information on arbitrarily many random representations of y. That is, \(\mathcal{A}\) adaptively chooses polynomially many leakage functions \(f_i:\mathbb{Z}_q^n\rightarrow \{0,1\}^L\), and learns the value f i (x i ), where x i is a fresh and random discrete log representation of y. \(\mathcal{A}\) wins the game if it eventually outputs a valid discrete log representation x* of y. We show that if the discrete log assumption holds in \(\mathbb{G}\), then no polynomially bounded \(\mathcal{A}\) can win this game with non-negligible probability, as long as the leakage on each representation is bounded by \(L\approx (n-2)\log q = (1-\frac{2}{n})\cdot\) |x|.
As direct extensions of this property, we design very simple continuous leakage-resilient (CLR) one-way function (OWF) and public-key encryption (PKE) schemes in the so called “invisible key update” model introduced by Alwen et al. at CRYPTO’09. Our CLR-OWF is based on the standard Discrete Log assumption and our CLR-PKE is based on the standard Decisional Diffie-Hellman assumption. Prior to our work, such schemes could only be constructed in groups with a bilinear pairing.
As another surprising application, we show how to design the first leakage-resilient traitor tracing scheme, where no attacker, getting the secret keys of a small subset of decoders (called “traitors”) and bounded leakage on the secret keys of all other decoders, can create a valid decryption key which will not be traced back to at least one of the traitors.
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Agrawal, S., Dodis, Y., Vaikuntanathan, V., Wichs, D. (2013). On Continual Leakage of Discrete Log Representations. In: Sako, K., Sarkar, P. (eds) Advances in Cryptology - ASIACRYPT 2013. ASIACRYPT 2013. Lecture Notes in Computer Science, vol 8270. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-42045-0_21
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