Nowadays, Smooth Projective Hash Functions (SPHFs) play an important role in constructing cryptographic tools such as secure Password-based Authenticated Key Exchange (PAKE) protocol in the standard model, oblivious transfer, and zero-knowledge proofs. Specifically, in this article, we focus on constructing PAKE protocol; that is, a kind of key exchange protocol which needs only a low entropy password to produce a cryptographically strong shared session key. In spite of relatively good progress of SPHFs in applications, it seems there has been little effort to build them upon quantum-resistant assumptions such as lattice-based cryptography and code-based cryptography to make them secure against quantum computer attacks. More precisely, there are two proposals based on lattice assumptions that utilize the SPHFs to construct PAKE secured in standard model. Considering quantum-resistant assumptions is less than straightforward and needs some relaxations. In this article, we introduce two new Approximate SPHF (ASPHFs) from error-correcting codes. Upon designing ASPHF, we can construct two efficient PAKE protocols. The security of our protocols could be proved based on the hardness of bounded decoding (BD) problem and learning with parity (LPN) problem in the standard model.

Code‐based digital signatures suffer from two main drawbacks: large public key size and slow signature generation. Large public key size is inherent in almost all the code‐based cryptosystems and other post‐quantum alternatives; however, slow signature generation is due to their specific structure. Most of the current code‐based signature schemes are constructed based on Courtois, Finiasz, and Sendrier (CFS) signature. CFS uses a counter to produce decodable syndromes or the complete decoding technique that imposes some extra computational cost to the signing algorithm for many choices of codes. In this study, the authors propose an efficient digital signature, PolarSig, which can reduce both public key size and signing time simultaneously. PolarSig uses some specific instances of polar codes that enable us to decode every random syndrome. Moreover, they apply puncturing and randomised omitting of frozen bits to protect the authors’ scheme from commonplace attacks targeting former cryptosystems based on polar codes. Besides, they prove that their signature is existentially unforgeable under a chosen message attack secure in the random oracle model.

Polar codes are novel and efficient error‐correcting codes with low encoding and decoding complexities. These codes have a channel‐dependent generator matrix, which is determined by the code dimension, code length and transmission channel parameters. A variant of the McEliece public‐key cryptosystem based on polar codes, called the PKC‐PC, is studied. Since the structure of the polar codes’ generator matrix depends on the parameters of the channel, the authors have used an efficient approach to conceal their generator matrix. The proposed approach is based on a random selection of rows of the matrix by which a random generator matrix is constructed. Using the characteristics of polar codes and introducing an efficient approach, they could reduce the public and secret key sizes, and computational complexity compared to the McEliece cryptosystem. Moreover, they show that PKC‐PC yields an increased security level against conventional attacks as well as possible vulnerabilities to the code‐based public‐key cryptosystems. Furthermore, they prove the security of the authors’ cryptosystem and show that its security is reduced to solve NP‐complete problems, called polar parameterised syndrome decoding and polar parameterised codeword existence.

One of the approaches to modify the McEliece cryptosystem to overcome its large key size is replacing binary Goppa codes with a new structured code. However, this modification makes such cryptosystems encounter some new attacks. There are a few modified McEliece cryptosystem variants which are known to be secure. One of them is the cryptosystem introduced by Baldi et al . which uses quasi‐cyclic low‐density parity check (QC‐LDPC) codes. This cryptosystem is still unbroken as no efficient attack has been reported against it since 2008. In this study, an attack has been applied to this cryptosystem which is feasible when the code length is a multiple of a power of 2. Also an important weakness of this kind of cryptosystem has been pointed out, namely utilising a too low‐weight intentional error vector. The authors have established a new security level for this cryptosystem which is applicable to other McEliece‐like cryptosystems using QC‐LDPC codes. This security level for instance is 2 9.18 times lower than previous ones in the case of n = 4 × 4096 when only one ciphertext is available. The gain of the attack in this study can be increased if more than one ciphertext is available.