Simulation study of the functioning of LFSR for grade 4 irreducible polynomials MIRELLA AMELIA MIOC Computer Science Department “Politehnica” University of Timisoara Bd. V. Parvan 2, RO-300223 ROMANIA mmioc@cs.utt.ro http://www.cs.upt.ro/ro/Staff/~mmioc Abstract: - It all began in Colossus, with a code-breaking machine that was one of the first known examples of a shift register. During the ages, the shift registers demonstrated their capacity to be the heart of any digital system. Now the applications are well-known as well in cryptography (Rijndael Algorithm) as in error correcting and in wireless communication systems. This paper contains an analysis of functioning for Linear Feedback Shift Register and Multiple Input-output Shift Register using grade 4 irreducible polynomials. Three kinds of scheme for implementation of a LFSR were analyzed and a formula that linked the results was verified. Key-Words: - Shift registers, Calculate, Irreducible polynomials, Cryptosystem, Simulate, Rijndael, Pseudo- random sequence, Error correcting. 1 Introduction Even in the 40’s, in Colossus, a code-breaking machine appeared as one of the first forms of a shift register. It was a five-stage device built of thyratrons and vacuum tubes. Along the years many different implementation forms were developed. An LFSR is composed of memory cells connected together as a shift register with linear feedback. In digital circuits a shift register is formed by flip- flops and EXOR gates chained together with a synchronous clock. Shift registers, like counters, are a form of sequential logic like counters. Always the shift registers produce a discrete delay of a digital signal or waveform. Considering that a shift register has n stages, the waveform is delayed by n discrete clock times. The main utility of using a LFSR is to create test patterns. Behind the construction of a LFSR there are some mathematical grounds based on communication theory. Also, the table containing taps for LFSR constructions is based on communication theory protocol. Usually the naming of the shift register follows a type of convention shown normally in digital logic, with the least significant bit on the left. According to the communication protocol, the signals will be addressed, not the registers. There are n+1 signals for each n-bit register. Always the next state of an LFSR is uniquely determined from the previous one by the feedback network. Any LFSR will generate a sequence of different states starting with the initial one, called seed. An LFSR can be represented as a polynomial of variable x referred to as the characteristic polynomial or the generator polynomial. Almost all applications of using shift registers representing generator polynomials need to be developed in a finite field. Evariste Galois demonstrated that a field is an algebra with both addition and multiplication forming a group. Some ground information from Algebra demonstrated the importance of working with irreducible polynomials and primitive polynomials. Also the importance of using shift registers in cryptosystems based on irreducible polynomials is demonstrated in increasing the security obtained[7], [17], [18]. The most important applications of using the LFSR are: · Pattern Generators[14]; · Counters; · Testing[15], [21]; · Encryption[9], [10], [11], [13]; · Built-in Self-Test(BIST)[8]; · Compression; · Checksums; · Pseudo-Random Bit Sequences (PRBS)[1], Proceedings of the 8th WSEAS Int. Conference on SOFTWARE ENGINEERING, PARALLEL and DISTRIBUTED SYSTEMS ISSN: 1790-5117 27 ISBN: 978-960-474-052-9