Asymptotic Analysis of Plausible Tree Hash Modes for SHA-3 Kevin Atighehchi1 and Alexis Bonnecaze2 1 Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000 Caen, France
[email protected] 2 Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
[email protected] Abstract. Discussions about the choice of a tree hash mode of operation for a standardization have recently been undertaken. It appears that a single tree mode cannot address adequately all possible uses and specifications of a system. In this paper, we review the tree modes which have been proposed, we discuss their problems and propose solutions. We make the reasonable assumption that communicating systems have different specifications and that software applications are of different types (securing stored content or live-streamed content). Finally, we propose new modes of operation that address the resource usage problem for three representative categories of devices and we analyse their asymptotic behavior. Keywords: SHA-3 · Hash functions · Sakura · Keccak · SHAKE · Parallel algorithms · Merkle trees · Live streaming 1 Introduction 1.1 Context In this article, we are interested in the parallelism of cryptographic hash functions. De- pending on the nature of the application, we seek to achieve either of the two alternative objectives: • Priority to memory usage. We first reduce the (asymptotic) parallel running time, and then the number of involved processors, while minimizing the required memory of an implementation using as few as one processor. • Priority to parallel time. We retain an asymptotically optimal parallel time while containing the other computational resources. Historically, a cryptographic hash function makes use of an underlying function, de- noted f, having a fixed input size, like a compression function, a block cipher or more recently, a permutation [BDPA13, BDPA11].