Gaussian integer has a potential to enhance the safety of elliptic curve cryptography (ECC) on system under the condition fixing bit length of integral and floating point types, in viewpoint of the order of a finite field. However, there seems to have been no algorithm which makes Gaussian integer ECC safer under the condition. We present the algorithm to enhance the safety of ECC under the condition. Then, we confirm our Gaussian integer ECC is safer in viewpoint of the order of finite field than rational integer ECC or Gaussian integer ECC of naive methods under the condition.

In this letter, a construction of asymmetric Gaussian integer zero correlation zone (ZCZ) sequence sets is presented based on interleaving and filtering. The proposed approach can provide optimal or almost optimal single Gaussian integer ZCZ sequence sets. In addition, arbitrary two sequences from different sets have inter-set zero cross-correlation zone (ZCCZ). The resultant sequence sets can be used in the multi-cell QS-CDMA system to reduce the inter-cell interference and increase the transmission data.

The concept of Gaussian integer sequence pair is generalized from a single Gaussian integer sequence. In this letter, by adopting cyclic difference set pairs, a new construction method for perfect Gaussian integer sequence pairs is presented. Furthermore, the necessary and sufficient conditions for constructing perfect Gaussian integer sequence pairs are given. Through the research in this paper, a large number of perfect Gaussian integer sequence pairs can be obtained, which can greatly extend the existence of perfect sequence pairs.

In this letter, three constructions of perfect Gaussian integer sequences are constructed based on cyclic difference sets. Sufficient conditions for constructing perfect Gaussian integer sequences are given. Compared with the constructions given by Chen et al. [12], the proposed constructions relax the restrictions on the parameters of the cyclic difference sets, and new perfect Gaussian integer sequences will be obtained.

This letter presents two constructions of Gaussian integer Z-periodic complementary sequences (ZPCSs), which can be used in multi-carriers code division multiple access (MC-CDMA) systems to remove interference and increase transmission rate. Construction I employs periodic complementary sequences (PCSs) as the original sequences to construct ZPCSs, the parameters of which can achieve the theoretical bound if the original PCS set is optimal. Construction II proposes a construction for yielding Gaussian integer orthogonal matrices, then the methods of zero padding and modulation are implemented on the Gaussian integer orthogonal matrix. The result Gaussian integer ZPCS sets are optimal and with flexible choices of parameters.

In this letter, based on cyclic difference sets with parameters $(N,rac{N-1}{2},rac{N-3}{4})$ and complex transformations, a method for constructing degree-4 perfect Gaussian integer sequences (PGISs) with good balance property of length $N'equiv2( ext{mod}4)$ are presented. Furthermore, the elements distribution of the proposed Gaussian integer sequences (GISs) is derived.

Based on a perfect Gaussian integer sequence, shift and combination operations are performed to construct Gaussian integer sequences with zero correlation zone (ZCZ). The resultant sequence sets are optimal or almost optimal with respect to the Tang-Fan-Matsufuji bound. Furthermore, the ZCZ Gaussian integer sequence sets can be provided for quasi-synchronous code-division multiple-access systems to increase transmission data rate and reduce interference.

In this letter, by using Whiteman's generalized cyclotomy of order 2 over Zpq, where p, q are twin primes, we construct new perfect Gaussian integer sequences of period pq.

In 2003, Kobayashi et al. proposed a new class of knapsack public-key cryptosystems over Gaussian integer ring. This scheme using two-sequences as the public key. In 2005, Sakamoto and Hayashi proposed an improved version of Kobayashi's scheme. In this paper, we propose the knapsack PKC using l-sequences as the public key and present the low-density attack on it. We have described Schemes R and G for l=2, in which the public keys are constructed over rational integer ring and over Gaussian integer ring, respectively. We discusses on the difference of the security against the low-density attack. We show that the security levels of Schemes R and G differ only slightly.