Abstract: The existing type 1 heterogeneous signcryption schemes are all based on the traditional number theoretic assumptions, so that they cannot resist a quantum computer attacks. To solve this problem, based on the quantum-resistant lattice hard problems, learning with errors problem and inhomogeneous small integer solution problem, we use the techniques of lattice-based signcryption and lattice basis delegation in fixed dimension, build the first lattice-based heterogeneous signcryption scheme. We also provide its correctness and security analysis. The scheme actualizes the property of quantum resistance, and gives theoretical support for anti-quantum communication from public key infrastructure (PKI) systems to identity-based systems.