AbstractA digital signature scheme allows a signer to sign electronic messages using his or her secret key, and any verifier can validate the correctness according to a given verification procedure. Although a variety of lattice‐based signature schemes have been proposed in the past few years, there does not exist a scheme that has short signatures and constant‐size public keys simultaneously. In this paper, we propose a new method for constructing short lattice signatures with constant‐size public keys in the standard model. In our scheme, each signature contains a low‐dimensional lattice vector and the public key only contains three matrices plus a vector. Compared with previous constructions, our scheme is very simple and does not require any complex homomorphic computation. In order for security proof to work, we introduce a new hard lattice problem, called variant small integer solution (Variant‐SIS), and give the security reduction from small integer solution to Variant‐SIS. Then we define a family of hash functions based on the hardness of Variant‐SIS and prove its security properties, including one‐wayness and collision resistance. As a matter of independent interest, the proposed hard problem may be useful in many other lattice‐based cryptographic constructions. Last, the comparison with similar works demonstrates the superiority of our scheme. Copyright © 2016 John Wiley & Sons, Ltd.