Cube Attack on Stream Ciphers using a Modified Linearity Test

Abstract

There have been various attempts to attack reduced variants of Trivium stream cipher using cube attack. During the preprocessing phase of cube attack, we need to test the linearity of a superpoly. The linearity testing problem is to check whether a function is close to linear by asking oracle queries to the function. This is the BLR linearity test for Boolean functions, which has a time complexity of O(22k + c) cipher operations, where k is the length of the key and c is the size of the cube. In this paper we present a method which is supposed to be a sufficient condition for testing a superpoly for linearity in F2 with a time complexity O(2c + 1 (k2 + k)). Our analysis on Trivium cipher reduced to 576 rounds using cube attack gives 69 extremely sparse linearly independent linear equations for smaller cubes, which recovers 69 bits of the key and reduces the attack complexity in the online phase to 211.