Design Of Post-quantum Secure Public Key Cryptosystems From Lattice Assumptions

It seems that we would come into the quantum era. Hence the study of the post-quantum cryptography has been more and more important to the theoretics research and the practical applications. Maybe for that reason the research of the lattice-based public key cryptography has been a hot topic and many schemes have been proposed in recent years. While the research of the lattice-based cryptography still not mature and there are many problems need to be studied hardly. For example, the space size of lattice-based cryptography is so large that needs new tools to be proposed to improve the efficiency of the lattice-based cryptosystems. Moreover, the cryptographic meaning of the lattice-based cryptography also needs to be extent. It is needed for us to study the applications of the lattice-based cryptography fatherly which is important to the establish and the development of the crypto-systems over lattice. So that, we studies the design of the lattice-based cryptographic schemes. The aims of our proposed schemes are that to achieve new cryptographic functions by the proposed schemes, or to reduce the space size efficiently compared with the known schemes.The main results are specified as follows:(1). The construction of the lattice-based identity-based signature (IBS) schemes:To improve the space efficiency of the lattice-based IBS schemes, using a pre-image sampleable function and a lattice-based signature scheme, we propose a lattice-based IBS scheme which is unforgeable in the random oracle model. Compared with the known lattice-based (H)IBS scheme, the signature length is reduced efficiently. Hence, the space efficiency of the proposed scheme would be high. To design an efficient and secure IBS scheme in the standard model over lattice, we first use a new parameter choice rule to improve the bonsai trees signature scheme to get a new signature scheme whose public key size and signature size are reduced efficiently than those of the bonsai trees signature schemes. Next, we transform the proposed improvement scheme of the bonsai trees signature scheme into a lattice-based IBS scheme which is provable secure in the standard model. Since the efficiency of the improvement scheme is higher than that of bonsai tree signature scheme, our proposed IBS scheme is efficient.(2). The design of the "added" functional lattice-based signature schemes, and the main work is following:(a). We transform the bonsai tree signature into a lattice-based ring signature scheme. The unforgeability of the proposed ring signature scheme is proven in the standard model. Moreover, our scheme also satisfied the unconditional anonymity for the signer’s identity.(b). Based on the pre-image sampleable function and the bonsai trees algorithm, we design a strong designate verifier signature(SDVS) over lattice, and then, we transform the proposed SDVS scheme into a lattice-based strong designate verifier ring signature scheme. We also finish the security proof of the proposed SDVS scheme and SDVRS scheme in the random oracle model.(c). Using PSF and its an improvement algorithm, we propose a verifiably en-crypted signature scheme. The unforgeability and the opacity of the proposed VES scheme are proven in the random oracle model.(d). Combine the PSF and a lattice-based hash function, we propose a linearly ho-momorphic signature scheme over binary field. The proposed scheme satisfies the unforgeability and the weakly context hiding properties. Moreover, com-pared with another lattice-based linearly homomorphic signature, the public key length and the signature length are reduced efficiently, so that the space efficiency of the proposed scheme is improved.(e). We propose a lattice-based blind signature scheme whose blind and unforge-ability properties are satisfied. Compared with a known lattice-based blind signature scheme, the sign fail question is avoided in our scheme, furthermore, the signature length is also reduced.(3). We propose two CCA (choice ciphtertext attack) secure crypto-systems which are shown as follows:(a). We propose an efficient CCA secure public key encryption based on the bon-sai trees algorithm. There are some advantages for the public key size and the cipher-text-message expanse fact compared with a known CCA encryption algorithm over lattice.(b). We adopted the definition of the hybrid signcryption, which is designed by tag-KEM, to lattice-based cryptography. We firstly proposed a lattice-based hybrid signcryption scheme over lattice, we also proven that the proposed scheme is indistinguishable under the adaptive choice ciphtertext attack and unforgeable under the choice message attack.