Perfect Gaussian Integer Sequences Research Articles

A Review Study of Prime Period Perfect Gaussian Integer Sequences

Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of the prime period perfect Gaussian integer sequence (PGIS). When cyclic group {1,2,…,N−1} can be partitioned into k cosets, where N=kf+1 is an odd prime number, the construction of a degree-(k + 1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N=kf+1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGISs of lower degrees to generate new PGISs with a degree of 4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution-based scheme, both degree and pattern of a PGIS vary and can be indeterminate, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications.

Novel Hybrid Public/Private Key Cryptography Based on Perfect Gaussian Integer Sequences

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 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.

Arbitrary Length Perfect Integer Sequences Using All-Pass Polynomial

A novel method is proposed to construct arbitrary length perfect integer sequences based on all-pass polynomial. It uses a fact that associative sequence of an all-pass polynomial is guaranteed to be perfect. In addition, geometric series method is our special case. Moreover, perfect Gaussian integer sequences can also be constructed. Furthermore, this approach can be applied to symmetric perfect sequences design. Finally, perfect sequence with low degree of freedom can be achieved by the difference set. Concrete examples are illustrated.

Perfect Gaussian Integer Sequences Embedding Pre-Given Gaussian Integers

A pre-given Gaussian integer (GI) is a GI that is determined before a sequence is designed, and a sequence embedding a pre-given GI is a sequence that contains the GI as part of its components. In this letter, for an arbitrary pre-given GI, we present two constructions that produce perfect GI sequences (PGISs) embedding the pre-given GI with different embedment frequencies. Typically, for an arbitrary even integer N (N ≥ 4) and arbitrary pre-given GI c, one of our constructions can yield a PGIS of period N and degree 3 that embeds the pre-given GI c N - 2 times. Our constructions provide a high degree of freedom and flexibility for PGIS designs to satisfy the requirements of sequence designs and applications.

Nearly perfect Gaussian integer sequences with arbitrary degree

Based on p-ary pseudorandom sequences, this study proposes a construction of degree-k Gaussian integer sequences of period N = k ( p m − 1 ) / ( p − 1 ) by utilising kth power residue symbol satisfying k | ( p − 1 ) , where p is an odd prime and positive integers m , k ≥ 2 . The periodic autocorrelation values are 0 at shifts τ ≢ 0 ( mod ( N / k ) ) of the resultant sequences. Specially, there is exactly one non-zero out-of-phase periodic autocorrelation value of the resultant sequences for k = 2 . The non-zero elements of the sequences are balanced and can be predefined flexibly. Moreover, the maximum energy efficiency of the proposed sequences is close to ( p − 1 ) / p for sufficiently large m.

Perfect Gaussian Integer Sequence Pairs from Cyclic Difference Set Pairs

Perfect Gaussian Integer Sequence Pairs from Cyclic Difference Set Pairs

Construction of Period <inline-formula> <tex-math notation="LaTeX">$qp$ </tex-math> </inline-formula> PGISs With Degrees Equal to or Larger Than Four

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 $\mathbb {Z}_{N}$ 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 $\mathbb {Z}_{N}$ 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 $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.

Perfect Gaussian Integer Sequences of Period $p^{k}$ With Degrees Equal to or Less Than $k+1$

This paper presents the construction of perfect Gaussian integer sequence (PGIS) of period $N=p^{k}$ with degrees equal to or less than $k+1$ , where $p$ is a prime number and $k\geq 2$ . The study begins with the partitioning of $\mathbb {Z}_{N}$ into $k+1$ subsets, from which $k+1$ base sequences are defined to construct degree- $(k+1)$ PGISs. The $k$ constraint equations derived from matching the ideal periodic autocorrelation function criterion in the time domain are nonlinear. We propose a decomposition method that can transform these $k$ nonlinear equations into $2k-2$ linear equations with $2k-2$ variables. The number of nonlinear constraint equations derived from the frequency domain is $k+1$ , where the $k+1$ sequence coefficients can be derived sequentially by an iteration algorithm. These two proposed schemes simplify the construction of the degree- $(k+1)$ PGIS. PGISs having less than $k+1$ degrees can be constructed in two steps. First, some sequence coefficients are assigned a zero value or one or more pairs of two consecutive base sequences are combined into one. The second step involves the adjustment of new sequence coefficients for the associated sequence to be perfect. We show that there exist at least $2k+(({k(k-1)(k-2)})/{6})$ distinct PGIS patterns of period $N=p^{k}$ with degrees equal to or less than $k+1$ .

Generation of Long Perfect Gaussian Integer Sequences

Recently, the perfect Gaussian integer sequences have been widely used in modern wireless communication systems, such as code division multiple access and orthogonal frequency-division multiplexing systems. This letter presents two different methods to generate the long perfect Gaussian integer sequences with ideal periodic auto-correlation functions. The key idea of the proposed methods is to use a short perfect Gaussian integer sequence together with the polynomial or trace computation over an extension field to construct a family of the long perfect Gaussian integer sequences. The period of the resulting long sequences is not a multiple of that of the short sequence, which has not been investigated so far. Compared with the already existing methods, the proposed methods have three significant advantages that a single short perfect Gaussian integer sequence is employed, the long sequences consist of two distinct Gaussian integers, and their energy efficiency is monotone increasing.

A CDMA scheme based on perfect Gaussian integer sequences

Both m-sequences and Gold sequences have been applied as the pseudonoise (PN) code in the code division multiple access (CDMA) scheme to enable distinct users to simultaneously share the available capacity of a common channel, which is called direct sequence CDMA (DS-CDMA). However, this scheme does not provide the required ideal periodic cross-correlation function property among various members of sequences within the same set. This paper proposes a new perfect Gaussian integer sequence (PGIS)-based CDMA scheme called PGIS-CDMA; in this scheme, the PN code of DS-CDMA is replaced by a set of PGISs. The intrinsic orthogonal property of PGIS can be applied to a CDMA system to achieve perfect cochannel interference separability under the ideal synchronism; it can also be adapted at the receiver to obtain diversity gain and improve the carry-to-interference ratio. These two concepts are further developed to implement two PGIS-CDMA configurations, and they can outperform DS-CDMA on the basis of Gold sequences that possess a favorable cross-correlation property for the CDMA scheme.

New Perfect Gaussian Integer Sequences from Cyclic Difference Sets

New Perfect Gaussian Integer Sequences from Cyclic Difference Sets

Families of Gaussian integer sequences with high energy efficiency

This study extends the authors’ earlier work to show that the Gaussian integer sequences of period p m − 1 with p − 2 non-zero out-of-phase autocorrelation values can be constructed from the known families of two-tuple-balanced p-ary sequences over the finite field 𝔽 p m , where p is an odd prime and m ≥ 2. The proposed Gaussian integer sequences have high energy efficiency and are superior to the perfect Gaussian integer sequences (introduced by Hu et al. in 2012) for the peak-to-average power ratio reduction in orthogonal frequency-division multiplexing systems.

Further results on degree‐2 perfect Gaussian integer sequences

A complex number whose real and imaginary parts are both integers is called a Gaussian integer. A Gaussian integer sequence is said to be perfect if it has an ideal periodic autocorrelation function (PACF) where all out-of-phase values are zero. Further, the degree of a Gaussian integer sequence is defined as the number of distinct non-zero Gaussian integers within one period of the sequence. Recently, the perfect Gaussian integer sequences have been found important practical applications as signal processing tools for orthogonal frequency-division multiplexing systems. The present article generalises the authors’ earlier paper by Lee et al. (2015) related to the Gaussian integer sequences with ideal PACFs. By the applications of two-tuple-balanced binary sequences and cyclic difference sets, a number of new degree-2 perfect Gaussian integer sequences with different periods are obtained.

A Systematic Method for Constructing Sparse Gaussian Integer Sequences With Ideal Periodic Autocorrelation Functions

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set $\{\pm 1, \pm j\}$ . The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$ , where $n$ is any positive integer and most of the sequence elements are nonzero.

Perfect Gaussian Integer Sequences of Arbitrary Length

In this work, we describe some methods to generate zero autocorrelation Gauss integer sequences of arbitrary length. These methods can be combined with each other so that more choices can be made.

Perfect Gaussian Integer Sequences of Arbitrary Composite Length

A composite number can be factored into either N=mp or N=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where p is an odd prime and m, n ≥ 2 are integers. This paper proposes a method for constructing degree-3 and degree-4 perfect Gaussian integer sequences (PGISs) of an arbitrary composite length utilizing an upsampling technique and the base sequence concept proposed by Hu, Wang, and Li. In constructing the PGISs, the degree of the sequence is defined as the number of distinct nonzero elements within one period of the sequence. This paper commences by constructing degree-3 PGISs of odd prime length, followed by degree-2 PGISs of odd prime length. The proposed method is then extended to the construction of degree-3 and degree-4 PGISs of composite length N=mp. Finally, degree-3 and degree-4 PGISs of length N=4 are built to facilitate the construction of degree-3 and degree-4 PGISs of length N=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where n ≥ 3.

Perfect Gaussian Integer Sequences of Odd Period &lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;${2^m} - 1$&lt;/tex&gt; &lt;/formula&gt;

In this letter, some perfect Gaussian integer sequences of period ${2^m} - 1$ are proposed based on the trace representations of Legendre sequences, Hall’s sextic residue sequences, $m$ -sequences, and Gordon-Mills-Welch (GMW) sequences over the finite field ${{\BBF}_{{2^m}}}$ . Moreover, the energy efficiency of these sequences is approximately 1 for sufficiently large $m$ .

A Novel Low-Complexity Precoded OFDM System With Reduced PAPR

The computational complexity and peak-to-average power ratio (PAPR) of conventional precoded orthogonal frequency division multiplexing (OFDM) systems can be reduced using a T-OFDM precoded system based on the Walsh-Hadamard matrix. The present paper proposes a novel precoding scheme for further reducing the computational complexity and PAPR of T-OFDM. In the proposed scheme, the precoding matrix is combined with an inverse discrete Fourier transform to construct a new transform matrix at the transmitter. Notably, the transform matrix is both unitary and circulant, with each column being a perfect Gaussian integer sequence containing just four non-zero elements of {±1,±j}. A low-complexity receiver is additionally constructed for the proposed precoding scheme. A closed-form expression is derived for the bit error rate (BER) in T-OFDM and the proposed precoding scheme under frequency-selective fading channels. The simulation results for the BER are shown to be in good agreement with the mathematical derivations. In addition, it is demonstrated that T-OFDM and the proposed scheme have an equivalent BER performance when their precoding matrices are designed in such a way as to obtain full frequency diversity. However, the proposed scheme has a better PAPR performance and a lower computational complexity than T-OFDM.

New Constructions of Perfect Gaussian Integer Sequences of Even Length

Perfect sequences have wide application fields, including spread spectrum communication, synchronization techniques, channel estimation, and cell search. In this letter, two kinds of (almost) perfect Gaussian integer sequences of length N=2 (mod4) and N=0 (mod4) are proposed and constructed by complex transformations and interleaving techniques, using the perfect Gaussian integer sequences of odd length as the base sequences.

New Perfect Gaussian Integer Sequences of Period &lt;i&gt;pq&lt;/i&gt;

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