Linear Equivalence of Block Ciphers with Partial Non-Linear Layers: Application to LowMC

$\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ is a block cipher family designed in 2015 by Albrecht et al. It is optimized for practical instantiations of multi-party computation, fully homomorphic encryption, and zero-knowledge proofs. $\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ is used in the $\mathrm{P}\mathrm{I}\mathrm{C}\mathrm{N}\mathrm{I}\mathrm{C}$ signature scheme, submitted to NIST’s post-quantum standardization project and is a substantial building block in other novel post-quantum cryptosystems. Many $\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ instances use a relatively recent design strategy (initiated by Gérard et al. at CHES 2013) of applying the non-linear layer to only a part of the state in each round, where the shortage of non-linear operations is partially compensated by heavy linear algebra. Since the high linear algebra complexity has been a bottleneck in several applications, one of the open questions raised by the designers was to reduce it, without introducing additional non-linear operations (or compromising security).

In this paper, we consider $\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ instances with block size n, partial non-linear layers of size $s\le n$ and r encryption rounds. We redesign LowMC’s linear components in a way that preserves its specification, yet improves LowMC’s performance in essentially every aspect. Most of our optimizations are applicable to all SP-networks with partial non-linear layers and shed new light on this relatively new design methodology.

Our main result shows that when $s<n$, each $\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ instance belongs to a large class of equivalent instances that differ in their linear layers. We then select a representative instance from this class for which encryption (and decryption) can be implemented much more efficiently than for an arbitrary instance. This yields a new encryption algorithm that is equivalent to the standard one, but reduces the evaluation time and storage of the linear layers from $r·n^2$ bits to about $r·n^2-(r-1){(n-s)}^2$. Additionally, we reduce the size of LowMC’s round keys and constants and optimize its key schedule and instance generation algorithms. All of these optimizations give substantial improvements for small s and a reasonable choice of r. Finally, we formalize the notion of linear equivalence of block ciphers and prove the optimality of some of our results.

Comprehensive benchmarking of our optimizations in various $\mathrm{L}\mathrm{O}\mathrm{W}\mathrm{MC}$ applications (such as $\mathrm{P}\mathrm{I}\mathrm{C}\mathrm{N}\mathrm{I}\mathrm{C}$) reveals improvements by factors that typically range between 2x and 40x in runtime and memory consumption.