# GCKSign: Simple and efficient signatures from generalized compact knapsack problems

## TL;DR

This paper introduces GCKSign, a more efficient lattice-based digital signature scheme that reduces signature and public key sizes while maintaining security.

## Contribution

The paper introduces a new lattice-based assumption called the Module GCK-TMO problem and a more efficient signature scheme that eliminates the need for witness indistinguishability.

## Key findings

- GCKSign achieves 3.4 times shorter signature size compared to prior schemes.
- The public key size is reduced by 2.4 times at the same security level.
- Security is based on the Module GCK-TMO problem in the random oracle model.

## Abstract

In 2009, Lyubashevsky proposed a lattice-based signature scheme using the Schnorr-like identification and the Fiat-Shamir heuristic and proved its security under the collision resistance of a generalized compact knapsack function. However, their security analysis requires the witness indistinguishability property, leading to significant inefficiency and an increase of sizes of public key and signature. To overcome the efficiency issue associated with the WI property, we introduce a new lattice-based assumption, called the target-modified one-wayness problem of the GCK function and show its reduction to well-known lattice-based problems. Additionally, we present a simple and efficient GCK-based signature scheme, GCKSign, whose security is based on the Module GCK-TMO problem in the random oracle model. GCKSign is a natural extension of Lyubashevsky’s scheme in a module setting, but achieves considerable efficiency gains due to eliminating the witness indistinguishability property. As a result, GCKSign achieves approximately 3.4 times shorter signature size and 2.4 times shorter public key size at the same security level.