Let $\\G$ be a group of prime order $q$, and let $g_1,\\ldots,g_n$ berandom 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.