Exploiting Collisions in Addition Chain-Based Exponentiation Algorithms Using a Single Trace

Public key cryptographic algorithms are typically based on group exponentiation algorithms where the exponent is unknown to an adversary. A collision attack applied to an instance of an exponentiation is typically where an adversary seeks to determine whether two operations in the exponentiation have the same input. In this paper, we extend this to an adversary who seeks to determine whether the output of one operation is used as the input to another. We describe implementations of these attacks applied to a 192-bit scalar multiplication over an elliptic curve that only require a single power consumption trace to succeed with a high probability. Moreover, our attacks do not require any knowledge of the input to the exponentiation algorithm. These attacks would, therefore, be applicable to algorithms, such as EC-DSA, where an exponent is ephemeral, or to implementations where an exponent is blinded. We then demonstrate that a side-channel resistant implementation of a group exponentiation algorithm will require countermeasures that introduce enough noise such that an attack is not practical, as algorithmic countermeasures are not possible. (The work described in this paper was conducted when the last two authors were part of the Cryptography Group at the University of Bristol, United Kingdom.)