Abstract. We propose the signature scheme Hawk, a concrete instantiation of proposals to use the Lattice Isomorphism Problem (LIP) as a

foundation for cryptography that focuses on simplicity. This simplicity

stems from LIP, which allows the use of lattices such as Z


, leading to

signature algorithms with no floats, no rejection sampling, and compact

precomputed distributions. Such design features are desirable for constrained devices, and when computing signatures inside FHE or MPC.

The most significant change from recent LIP proposals is the use of module lattices, reusing algorithms and ideas from NTRUSign and Falcon.

Its simplicity makes Hawk competitive. We provide cryptanalysis with

experimental evidence for the design of Hawk and implement two parameter sets, Hawk-512 and Hawk-1024. Signing using Hawk-512 and

Hawk-1024 is four times faster than Falcon on x86 architectures, produces signatures that are about 15% more compact, and is slightly more

secure against forgeries by lattice reduction attacks. When floating-points

are unavailable, Hawk signs 15 times faster than Falcon.

We provide a worst case to average case reduction for module LIP. For

certain parametrisations of Hawk this applies to secret key recovery and

we reduce signature forgery in the random oracle model to a new problem

called the one more short vector problem.

Keywords: Post-Quantum Cryptography, Signatures, Module Lattice

Isomorphism Problem, Concrete Design, Quadratic Forms.