Pseudo-random sequences of non-maximum length on shift registers with reducible and primitive polynomials

Inhomogeneous pseudo-random sequences of non-maximal length formed by shift registers with linear feedbacks based on a characteristic polynomial of degree n of the form ϕ(x)=ϕ1(x)ϕ2(x), where ϕ1(x) = x m1 ⊕ 1, and ϕ2(x) of degree m 2 is primitive (m 1 = 2 k , k is a positive integer, n = m 1 + m 2) are considered. Three schemes that are equivalent in terms of periodic sequence structures were considered. Of the greatest interest are the shift registers connected in an arbitrary way using a modulo-two adder, the feedbacks in which correspond to the multipliers ϕ1(x) and ϕ2(x) the polynomials ϕ(x). In this case, there is a complex process of forming output sequences, which involves both direct and inverse M-sequences. The statement about the singularity of the generated sequences at m 1 = 4 is proved, which is confirmed by their decimation with an index equal to the period of the primitive polynomial.

Implementation of Reed Solomon Code Over GF(9) and 9-Ary Symmetric Channel

Reed-Solomon and related codes have recently become very important for erasure correction in large disk arrays used in data centers. In this paper, we will implement a 3-error correcting Reed-Solomon encoder and decoder over the field GF(9) generated by the primitive polynomial D^2 + D + 2 over GF(3) and the decoding is carried out by the Berlekamp. We simulate the encoder and decoder using Monte-Carlo simulations over the 9-ary symmetric channel that outputs the correct symbol with probability (1-p), and outputs one of the other 8 possible incorrect symbols with probability p/8. Then, we compare the simulated probability of symbol error P(E) of out code with the union upper bound.

Construction of primitive polynomials over finite fields

In this paper, using the polynomial composition methods some computationally simple and explicit ways for constructing higher degrees primitive polynomials from a given primitive polynomial over [Formula: see text] are given.

Studying of the scrambling coding sequence performance based on the ninth order primitive polynomial in telecommunication networks of information systems

Scrambling coding sequences find wide application in telecommunication networks to improve noise immunity and information transmission concealment. There are many different scrambling coding sequences with different autocorrelation properties from which one can be chosen. In the previous paper the authors researched a possibility of using scrambling coding sequences built on the base of the primitive eighth degree polynomial in telecommunication networks of information systems to enhance noise immunity and concealment of information transmission. This sequence, consisting of two hundred and fifty five chips, showed some good performance in terms of information transmission quality indicators. Nevertheless, some of its limitations were also revealed, which are mostly linked to the number of used chips. This paper aims to overcome the problems by proposing to use scrambling coding sequences based on the ninth degree primitive polynomial. The required polynomial was selected and the needed scrambling sequence generated. This scrambling coding sequence includes five hundred and eleven chips. Computer simulation helped to establish that scrambling coding sequence synthesized using this polynomial permits to substantially improve information transmission quality indicators as compared to the case of the scrambling coding sequence based on the eighth degree polynomial. For example, the interfering signal and internal noise to desired signal ratio, for which reception of transmitted bit sequence is still possible, increased by three decibels. It is clearly linked to the doubling of the number of the used chips. To preserve the transmission rate, the occupied frequency spectrum needs to be doubled. An inference is made as to a possibility of using the synthesized coding sequence in practical realizations of telecommunication channels.

Reversible and Fragile Watermarking for Medical Images

A novel reversible digital watermarking technique for medical images to achieve high level of secrecy, tamper detection, and blind recovery of the original image is proposed. The technique selects some of the pixels from the host image using chaotic key for embedding a chaotically generated watermark. The rest of the pixels are converted to residues by using the Residue Number System (RNS). The chaotically selected pixels are represented by the polynomial. A primitive polynomial of degree four is chosen that divides the message polynomial and consequently the remainder is obtained. The obtained remainder is XORed with the watermark and appended along with the message. The decoder receives the appended message and divides it by the same primitive polynomial and calculates the remainder. The authenticity of watermark is done based on the remainder that is valid, if it is zero and invalid otherwise. On the other hand, residue is divided with a primitive polynomial of degree 3 and the obtained remainder is appended with residue. The secrecy of proposed system is considerably high. It will be almost impossible for the intruder to find out which pixels are watermarked and which are just residue. Moreover, the proposed system also ensures high security due to four keys used in chaotic map. Effectiveness of the scheme is validated through MATLAB simulations and comparison with a similar technique.