Paper 2012/367

On Continual Leakage of Discrete Log Representations

Shweta Agrawal, Yevgeniy Dodis, Vinod Vaikuntanathan, and Daniel Wichs


Let $\G$ be a group of prime order $q$, and let $g_1,\ldots,g_n$ be random elements of $\G$. We say that a vector $\vecx = (x_1,\ldots,x_n)\in \Z_q^n$ is a {\em discrete log representation} of some some element $y\in\G$ (with respect to $g_1,\ldots,g_n$) if $g_1^{x_1}\cdots g_n^{x_n} = y$. Any element $y$ has many discrete log representations, forming an affine subspace of $\Z_q^n$. We show that these representations have a nice {\em continuous leakage-resilience} property as follows. Assume some attacker $\A(g_1,\ldots,g_n,y)$ can repeatedly learn $L$ bits of information on arbitrarily many random representations of $y$. That is, $\A$ adaptively chooses polynomially many leakage functions $f_i:\Z_q^n\rightarrow \{0,1\}^L$, and learns the value $f_i(\vecx_i)$, where $\vecx_i$ is a {\em fresh and random} discrete log representation of $y$. $\A$ wins the game if it eventually outputs a valid discrete log representation $\vecx^*$ of $y$. We show that if the discrete log assumption holds in $\G$, then no polynomially bounded $\A$ can win this game with non-negligible probability, as long as the leakage on each representation is bounded by $L\approx (n-2)\log q = (1-\frac{2}{n})\cdot |\vecx|$. 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 {\em traitor tracing} scheme, where no attacker, getting the secret keys of a small subset of decoders (called ``traitors'') {\em 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.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Leakage Resilience
Contact author(s)
wichs @ cs nyu edu
2012-06-29: received
Short URL
Creative Commons Attribution


      author = {Shweta Agrawal and Yevgeniy Dodis and Vinod Vaikuntanathan and Daniel Wichs},
      title = {On Continual Leakage of Discrete Log Representations},
      howpublished = {Cryptology ePrint Archive, Paper 2012/367},
      year = {2012},
      note = {\url{}},
      url = {}
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