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.

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of N = qp, where q and p are distinct primes. The study begins with the partitioning of a ring ℤ _N into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of ℤ _N should be carried out; here, two cases, the even period N = 2p and the odd period N = qp, are treated separately. We can adopt the Legendre sequences of the prime period p to construct PGISs of period 2p with degrees larger than four. For the case of period qp, we introduce the Jacobi symbols to partition ℤ _N into seven subsets and construct PGISs with more diverse degrees.

A sequence is said to be perfect if it has an ideal periodic autocorrelation function (PACF). In addition, the degree of a sequence is defined as the number of distinct nonzero elements within each period of the sequence. This study presents a systematic method for constructing perfect sequences (PSs) of odd prime periods, where the general constraint equations for the sequence coefficients are derived, providing a solid theoretical foundation to the construction of PSs. The proposed scheme commences by partitioning a cyclic group Z_P={1,2,\ldots,P-1} into K cosets of cardinality M, where P=KM+1 is an odd prime. Based on this partition, the degree-(K+1) PSs are then constructed. Finally, case studies are presented to illustrate the proposed construction.

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.

A novel data encryption scheme using a set of exponential numbers is proposed in this paper which aims at achieving secure communications over public networks. The study begins with the definition of pattern of exponential number, where the decimal pattern of an exponential number g^e is the result of taking e − 1 times multiplication the base element g by itself. The multiplication operation of two large integers involves the complicated carry conditions in each step, which is digit-dependence and can only be analyzed case by case. This implies that the pattern of a large exponential number is unpredictable. Next, we prove that nature number can be spanned by a set of exponential numbers, and there exists numerous spanning sets for one nature number, demonstrating that the expression of nature number using a set of exponential numbers is not unique. The unpredictable nature of exponential patterns and the abundant available sets of exponential numbers can contribute the concealment of information when data streams are encrypted by the exponential numbers. In addition, we prove that the base elements of a set of exponential numbers applied for spanning a nature number can be arbitrarily assigned. With this property, we can distribute a secret key sequence in advance as the base elements for data encryption, or this secret key can be shared using the Diffie-Hellman Key-Exchange system over public network. Finally, the encrypted message consists of a sequence of exponents and the index data, which indicates the relationship between the base elements and the secret key for decryption. The ciphertext is transmitted via the public channel to the receiver end; however, it can only be decrypted by those who own the secret key. Some examples are shown for demonstration the feasibility of the proposed encryption scheme.