Attacks on the Firekite Cipher

Abstract

Firekite is a synchronous stream cipher using a pseudo-random number generator (PRNG) whose security is conjectured to rely on the hardness of the Learning Parity with Noise (LPN) problem. It is one of a few LPN-based symmetric encryption schemes, and it can be very efficiently implemented on a low-end SoC FPGA. The designers, Bogos, Korolija, Locher and Vaudenay, demonstrated appealing properties of Firekite, such as requiring only one source of cryptographically strong bits, small key size, high attainable throughput, and an estimate for the bit level security depending on the selected practical parameters.
We propose distinguishing and key-recovery attacks on Firekite by exploiting the structural properties of its PRNG. We adopt several birthday-paradox techniques to show that a particular sum of Firekite’s output has a low Hamming weight with higher probability than the random case. We achieve the best distinguishing attacks with complexities 2^66.75 and 2^106.75 for Firekite’s parameters corresponding to 80-bit and 128-bit security, respectively. By applying the distinguishing attacks and an additional algorithm we describe, one can also recover the secret matrix used in the Firekite PRNG, which is built from the secret key bits. This key recovery attack works on most large instances of Firekite parameters and has slightly larger complexity, for instance, 2^69.87 on the 80-bit security parameters n = 16,384, m = 216, k = 216.