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.

LFSR Next Bit Prediction through Deep Learning

Pseudorandom bit sequences are generated using deterministic algorithms to simulate truly random sequences. Many cryptographic algorithms use pseudorandom sequences, and the randomness of these sequences greatly impacts the robustness of these algorithms. Important crypto primitive Linear Feedback Shift Register (LFSR) and its combinations have long been used in stream ciphers for the generation of pseudorandom bit sequences. The sequences generated by LFSR can be predicted using the traditional Berlekamp Massey Algorithm, which solves LFSR in 2×n number of bits, where n is the degree of LFSR. Many different techniques based on ML classifiers have been successful at predicting the next bit of the sequences generated by LFSR. However, the main limitation in the existing approaches is that they require a large number (as compared to the degree of LFSR) of bits to solve the LFSR. In this paper, we have proposed a novel Pattern Duplication technique that exponentially reduces the input bits requirement for training the ML Model. This Pattern Duplication technique generates new samples from the available data using two properties of the XOR function used in LFSRs. We have used the Deep Neural Networks (DNN) as the next bit predictor of the sequences generated by LFSR along with the Pattern Duplication technique. Due to the Pattern Duplication technique, we need a very small number of input patterns for DNN. Moreover, in some cases, the DNN model managed to predict LFSRs in less than 2n bits as compared to the Berlekamp Massey Algorithm. However, this technique was not successful in cases where LFSRs have primitive polynomials with a higher number of tap points.

A Gist of Analogues μ_pto Mobius function μ

If f (x) [x] p Z is an irreducible polynomial, the number of polynomials g(x) with deg(g(x))  deg( f (x))  (g(x), f (x)) =1 is the order of the multiplicative group of [x]/( f (x)) p Z. In this paper we introducing analogues p  to Mobius function  defined on [x], p Z the set of all primitive polynomials in [x] p Z .

Performance Study of Spread Spectrum Systems with Hard Limiters

Use of spread spectrum systems in telecommunications is studied. It is shown that spread spectrum techniques can substantially enhance noise and interference immunity in the currently deployed information transmission networks. Primitive polynomials are proposed to obtain respective spreading codes. A spreading code consisting of 1023 chips is synthesized and its characteristics are studied. It is deduced that powerful interferences can exceed dynamic range of the receiving part of the system and, as a result, deteriorate information transmission. To overcome this problem, utilization of limiters is proposed, in which limitation level equals that of the internal noise of the receiving part. Computer simulation is employed to test the performance of the proposed solution. Performance of the spread spectrum system for information transfer is studied both without the limiter and with the limiter. Research results show that, for binary modulation, performance of the system with limitation and without limitation is nearly identical while limiters can substantially reduce requirements for the dynamic range. Compared to the existing approaches, it is proposed to use the synthesized spreading coding sequence with the limitation technique in practical implementations of those telecommunication networks, in which noise immunity and transmission concealment are required, such as in unmanned aerial vehicles. This can replace currently used approaches, such as frequency hopping, transmission power adjustment and antenna pattern changes.

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.