Linear Feedback Shift Registers

Topics: Randomness, Chaos theory, Pseudorandom number generator Pages: 7 (2310 words) Published: March 7, 2013
Abstract: Linear Feedback Shift Registers (LFSRs) are considered powerful methods for generating pseudo-random bits in cryptography algorithm applications. In this paper it is shown that the linear dependencies in the generated random bit sequences can be controlled by adding a chaotic logistic map to the LFSR’s systems. The structure of the LFSR’s output sequence in combination with a chaotic map is analyzed and proved to have at least as much uniformity than the corresponding set for the linear components individually. In order to understand that using the proposed PRBG is reliable in secure algorithms, the NIST suite test have been taken on the proposed method, finally to compare the proposed PRNG output sequence features with the two types of LFSRs (Fibonacci and Galois). Keywords: Linear Feedback Shift Register, Random Number, Chaotic Map, NIST.

1. Introduction
In the modern world of computers, network security is the main concern which relies on the use of cryptography algorithms. high quality random number generation is a basic subject of cryptography algorithms and the importance of a secure random number generator design cannot be underestimated. Most common generation techniques about RNGs involve truly random and pseudorandom number generators. For a brief introduction in various types of RNGs: Truly Random Number Generators (RNGs) is a computer algorithm, which generates a sequence of statistically independent random numbers. Actually these generators require a naturally occurring source of randomness phenomena (i.e. as a non-deterministic system). Most practical implementations design a hardware device or a software program based on RNGs to produce a bit sequence which is statistically independent. Pseudo-Random Bit Generators (PRBGs) are implemented by an algorithm that is actually a finite state machine; reliable RNGs which are implemented by these methods should pass several statistical tests to prove their usefulness [2-4]. With the mention of these points, the security of the entire cryptographic system such as RSA and DES and the other secure algorithms relies on the randomness quality of the generator [5, 6]. PRNGs are based on the algorithmic function, so the outputs of these methods are not truly random. In the last two decades several works in this area have been implemented based on chaotic systems [7]. Chaotic system is a natural phenomenon that behaves chaotic in the specific system’s parameters [8]. Chaotic maps are sensitive to initial conditions; this makes them sensitive to minimal change of information from the input thus heavily varying the output when input sequence changes by the minute. Chaotic maps compute quickly in the regular machine and are able to create sequences with extremely long cycle lengths [9]. Linear feedback shift register (LFSR) is a shift register which is able to generate random bits (with the mention of amount of registers [1-3]). In the LFSR input bit is a linear function (i.e. it’s an exclusive-or function) of its previous state. It’s a shift register which input bit is driven by the exclusive-or (XOR) of some bits of the overall shift register value. The initial value of the LFSR is called the seed, LFSR’s operation is deterministic, and so the stream of values produced by the register is completely determined by its algorithm and current (or previous) state. The theory of the Linear Feedback Shift Registers (LFSRs) is based on the polynomial form, so in the blow equation p and q are the binary digits: (1)

In this paper we design a new random number generator by using a LFSR generator with a combination of logistic chaotic maps. [10]. The proposed random bit generator is based on a combination of logistic chaotic maps as a chaotic system in the LFSR algorithm, which of course increases the complexity in output sequence of the LFSR and becomes difficult for an intruder to extract information about the cryptography system. In the next section, we...
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