This paper proposes a novel hybrid public/private key cryptography scheme based on perfect Gaussian integer sequences (PGISs) of period N = pq. First, a review study of construction degree-4 PGIS is addressed. We show that circular convolution over PGISs is a trapdoor one-way permutation function that enables simultaneous cipher encryption and digital signatures. To implement the proposed cipher encryption scheme, a private PGIS is assigned as the encryption key sequence for circular convolution with the plaintext to generate the ciphertext. The reverse decryption key sequence involves the time reflection and complex conjugation of the encryption sequence, which can be regenerated using a pair of public and private keys. The security level of the proposed scheme is the same as that of the Rivest-Shamir-Adleman (RSA) system; however, the capacity of a cryptosystem based on PGISs may outperform that of based on RSA, because the abundant PGISs are available. Simulation results show that the approximation error when finite digits are used to represent the irrational coefficients of a normalized PGIS can be relatively small compared with the noise. This contributes to the simplicity of this scheme's implementation. With the fast development of IoT (internet of things), the adaptation and applicability of the proposed scheme to IoT platforms are also addressed, where lightweight cryptographic functions are preferable due to the limited resources of IoT devices.

Multiuser communication systems or multiple access scheme systems favor sequences possessing the ideal periodic cross-correlation function (PCCF) property. In comparison, channel estimation, equalization, and synchronization applications favor sequences possessing the ideal periodic auto-correlation function (PACF) property. However, there is no set of sequences possessing both the ideal PCCF and PACF properties simultaneously, where auto-correlation and cross-correlation balance each other. In this work, sequences possessing the ideal PACF property are used as the base sequences. Then, a modulation technique is applied upon these base sequences to construct a set of zero circular convolution (ZCC) sequences within which an arbitrary pair of two sequences possesses the ideal PCCF property. Compared with least squares (LS) and minimum mean squared error (MMSE) algorithm, the simulation results show that the channel estimation performance of ZCC is better than MMSE and LS algorithms, and the computational complexity of the algorithm is the same as LS algorithm, but far lower than MMSE algorithm. This is the first study on ZCC sequences reported in the literature, in which their fundamental theorems, properties, construction, and applications are investigated. The advantage of possessing the desired PACF and the ideal PCCF properties allows the ZCC sequences to be used in a broader range of applications than other sets of sequences.

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