Paper 2014/565

Kangaroos in Side-Channel Attacks

Tanja Lange, Christine van Vredendaal, and Marnix Wakker

Abstract

Side-channel attacks are a powerful tool to discover the cryptographic secrets of a chip or other device but only too often do they require too many traces or leave too many possible keys to explore. In this paper we show that for side channel attacks on discrete-logarithm-based systems significantly more unknown bits can be handled by using Pollard's kangaroo method: if $b$ bits are unknown then the attack runs in $2^{b/2}$ instead of $2^b$. If an attacker has many targets in the same group and thus has reasons to invest in precomputation, the costs can even be brought down to $2^{b/3}$. Usually the separation between known and unknown keybits is not this clear cut -- they are known with probabilities ranging between 100\% and 0\%. Enumeration and rank estimation of cryptographic keys based on partial information derived from cryptanalysis have become important tools for security evaluations. They make the line between a broken and secure device more clear and thus help security evaluators determine how high the security of a device is. For symmetric-key cryptography there has been some recent work on key enumeration and rank estimation, but for discrete-logarithm-based systems these algorithms fail because the subkeys are not independent and the algorithms cannot take advantage of the above-mentioned faster attacks. We present $\epsilon$-enumeration as a new method to compute the rank of a key by using the probabilities together with (variations of) Pollard's kangaroo algorithm and give experimental evidence.