Abstract ˗ Polar codes are provably capacity achieving linear block codes. The generator matrix of these codes is specified by knowing the parameters of transmission channel, length and dimension of the used code. On the other hand, for the cryptosystems based on general decoding problem (i.e. code based cryptosystems), the generator matrix of the applied code should be properly hidden from the attacker. Moreover, in the computational security, it is assumed that an attacker with restricted processing power has unlimited access to transmission media. Thus, an attacker can construct the generator matrix of polar codes, especially for Binary Erasure Channel on which this matrix can be efficiently specified. In this paper, we introduce a novel method to hide the generator matrix of polar codes in such a way that an attacker cannot construct it in polynomial time even by knowledge of the channel parameters, dimension and length of the used code. By the help of this method, a secret key cryptosystem based on nonsystematic polar codes over Binary Erasure Channel is proposed which provides both data security and reliability in one process simultaneously. In fact, the main goal of this research is to achieve the acceptable level of security and reliability by taking advantage of the interesting properties of polar codes. The proposed scheme resists against the typical attacks on the cryptosystems based on error correcting codes. Also, by employing some efficient methods, the key length of our scheme is decreased compared to Rao-Nam secret key cryptosystem. Moreover, our scheme benefits from high code rate, proper error performance, faster processing and efficient implementation.

Abstract—Polar codes are a new class of error correcting linear block codes, whose generator matrix is specified by the knowledge of transmission channel parameters, code length and code dimension. Moreover, regarding computational security, it is assumed that an attacker with a restricted processing power has unlimited access to the transmission media. Therefore, the attacker can construct the generator matrix of polar codes, especially in the case of Binary Erasure Channels, on which this matrix can be easily constructed. In this paper, we introduce a novel method to keep the generator matrix of polar codes in secret in a way that the attacker cannot access the required information to decode the intended polar code. With the help of this method, a secret key cryptosystem is proposed based on non-systematic polar codes. In fact, the main objective of this study is to achieve an acceptable level of security and reliability through taking advantage of the special properties of polar codes. The analyses revealed that our scheme resists the typical attacks on the secret key cryptosystems based on linear block codes. In addition, by employing some efficient methods, the key length of the proposed scheme is decreased compared to that of the previous cryptosystems. Moreover, this scheme enjoys other advantages including high code rate, and proper error performance as well.