This research paper proposes a novel approach for constructing substitution boxes (S-boxes) over Gaussian integers, which are complex numbers with integer coefficients. The proposed method is based ...on the properties of the Gaussian integers and their arithmetic operations and ensures the S-boxes exhibit strong cryptographic properties. Furthermore, the paper demonstrates how these S-boxes can be utilized for image encryption through a substitution-permutation network (SPN) over Gaussian integers. The SPN involves iteratively applying the S-box and a permutation layer to the input image, which effectively scrambles the image data. Experimental results show that the proposed method achieves high security and robustness against various attacks while providing efficient encryption and decryption performance. This research thus provides a promising avenue for developing secure image encryption schemes based on 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.
Gaussian integers are complex numbers of the form \gamma=x+iy where x and y are integers and i^2=-1. The set of Gaussian integers is usually denoted by \mathbb{Z}i. A Gaussian integer ...\gamma=a+ib\in\mathbb{Z}i is prime if and only if either \gamma=\pm(1\pm i),N(\gamma)= a^2+b^2 is a prime integer congruent to 1(mod4), or \gamma=p+0i or =0+pi where p is a prime integer and |p|\equiv3(mod4). Let D=(V,A) be a digraph with |V|=n. An injective function f:V(D)\rightarrow\left\gamma_n\right is said to be a Gaussian twin neighborhood prime labeling of D, if it is both Gaussian in and out neighborhood prime labeling. A digraph which admits a Gaussian twin neighborhood prime labeling is called a Gaussian twin neighborhood prime digraph. In this paper, we introduce some definitions of fan digraphs. Further, we establish the Gaussian twin neighborhood prime labeling in fan digraphs using Gaussian integers.
A Boolean generator for a large number of standard complementary QAM sequences of length <inline-formula> <tex-math notation="LaTeX">2^{K} </tex-math></inline-formula> is proposed. This Boolean ...generator is derived from the authors' earlier paraunitary generator, which is based on matrix multiplications. Both generators are based on unitary matrices. In contrast to previous Boolean QAM algorithms which represent complementary sequences as a weighted sum, our algorithm has a multiplicative form. Any element of a sequence can be generated efficiently by indexing the entries of unitary matrices with the binary representation of the discrete time index (which is easily implemented as a binary counter). Our 1Qum (based on one QAM unitary matrix) and 2Qum (based on two QAM unitary matrices) algorithms generate generalized <xref ref-type="other" rid="other6">Case I <xref ref-type="other" rid="other7"/>-<xref ref-type="other" rid="other8">III sequences and generalized <xref ref-type="other" rid="other11">Case IV and <xref ref-type="other" rid="other9">V sequences, respectively, as specified by Liu et al. in 2013, in addition to many new 2Qum sequences. The ratio of the numbers of sequences that are generated by our new construction and the previous construction increases with the constellation size. For example, for a 1024-QAM sequence of length 1024, this ratio is 4.4. However, if we compare only 2Qum sequences to <xref ref-type="other" rid="other11">Case IV and <xref ref-type="other" rid="other9">V sequences, this ratio is 267.
This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards ...this goal, we establish an asymptotic estimate for the first momentM1,I2(S)=∑rs,r′s′∈GSconsec12|s|2+12|s′|2, where the sum is taken over pairs of fractions associated with ‘consecutive’ Ford spheres of radius less than or equal to 12S2.
Recently, there is an increased interest in the study of modulo analog to digital converters (ADCs). These new systems can reconstruct a signal whose amplitude is much higher than the conventional ...ADC's dynamic range. Modulo ADCs are characterized by their modulo threshold and in the current literature, all existing works are limited to real-valued moduli. In this paper, we propose multi-channel modulo samplers with complex-valued moduli to sample a band-limited complex signal. Specifically, we discuss the construction of complex divisors from Gaussian integers and propose their efficient implementations. A memory-efficient, closed-form recovery algorithm is also proposed. Simulation results demonstrate that the proposed systems can provide stable reconstruction of a high dynamic range complex-valued signal at low sampling rates.
The classical number-theoretic functions – a number of divisors τ(n), sum of the divisors σ(n) and product of the divisors π(n) of a positive integer n – were generalized to the ring Zi of Gaussian ...integers. For the evaluation of the corresponding functions τ*(α), σ*m(α) and π*(α), obtained were the explicit formulae that use the canonical representation of α. A number of properties of these functions were studied, in particular, estimates from above for the functions τ*(α) and σ*m(α) and the properties connected with divisibility of their values by certain numbers. Researched are also sums of products of powers of the divisors for α∈Zi.
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