In this paper, we describe an image encryption scheme based on the multiple-parameter fractional number-theoretic transform (MFrNTT). In order to define the MFrNTT used in our proposal, we introduce a systematic procedure to construct a number-theoretic transform eigenbasis from chosen parameters. Our results indicate that the proposed scheme can resist to cryptographical attacks usually considered in the related literature; at the same time, our method is very competitive in terms of encryption/decryption speed when compared with other existing approaches. We also demonstrate that our scheme can be easily adjusted to deal with different types of images and that it involves a larger number of free parameters than other NTT-based image encryption techniques.

A generating matrix is a matrix such that, when multiplied by an eigenvector of a discrete transform, a new eigenvector is obtained. In this paper, we introduce a family of generating matrices of DFT eigenvectors. We demonstrate that, if a specific initial set of eigenvectors is chosen, using the referred family of matrices, a Hermite-Gaussian-like DFT eigenbasis is obtained. Such an eigenbasis is then employed to define a discrete fractional Fourier transform which numerically approximates the corresponding continuous transform.

The cosine number transform (CNT) is a cosine-like number-theoretic transform, which has been employed as the basis for multimedia security schemes. In this paper, we propose hardware architectures for computing an 8-point CNT. The architectures include a pipelined approach and are based on a recently introduced fast algorithm, which has been demonstrated to be more efficient than the direct computation of the CNT. We quantify such an efficiency by considering several aspects inherent to modular arithmetic and comparing metrics obtained from field-programmable gate array (FPGA) implementations of the proposed architectures.

In this paper, we introduce fractional number-theoretic transforms (FrNTT) based on matrix functions. The approach, which is a kind of finite field extension of the method presented in [1], does not require the construction of any number-theoretic transform eigenvector set. In this sense, the definition presented in this work is simpler than that of another recently introduced FrNTT. An image encryption scheme based on the proposed FrNTT is suggested.

In this paper, a multiuser communication technique based on the discrete fractional Fourier transform (DFrFT) is discussed. Eigenvectors of the DFrFT transform matrix are used as user sequences, which are transmitted over a real adder channel. Compared to other transforms used in the same context, the advantage of using the DFrFT is that the sequences can be generated from a systematic procedure and the number of generated subspaces, which determines the maximum number of simultaneous users of a scheme, is arbitrary. After describing the basic idea of our approach, we discuss some aspects related to its practical implementation and present preliminary simulation results.

In this paper, a finite field version for the fractional Fourier transform is introduced. We show that, in some aspects, the finite field fractional Fourier transform (4FT) is in perfect analogy with the discrete fractional Fourier transform. On the other hand, we consider some definitions and properties from the finite field context which allow us to discuss particularities of the 4FT.

In this paper, we present a new method for multiplying polynomials in Chebyshev form. Our approach has two steps. First, the well-known Karatsuba's algorithm is applied to polynomials constructed by using Chebyshev coefficients. Then, from the obtained result, extra arithmetic operations are used to write the final result in Chebyshev form. The proposed algorithm has a quadratic computational complexity. We also compare our method to other approaches.

In this paper, a new definition of Chebyshev polynomials over prime finite fields is introduced. Our approach uses a finite field trigonometry and reveals some aspects concerning the security of a recently proposed public-key encryption algorithm based on those polynomials. Particularly, we show that recovering the corresponding plaintext from a given ciphertext involves the discrete logarithm problem.