Linear Feedback Shift Register

Information about Linear Feedback Shift Register

A linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state.

The only linear functions of single bits are xor and inverse-xor; thus it is a shift register whose 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, and because the operation of the register is deterministic, the sequence of values produced by the register is completely determined by its current (or previous) state. Likewise, because the register has a finite number of possible states, it must eventually enter a repeating cycle. However, a LFSR with a well-chosen feedback function can produce a sequence of bits which appears random and which has a very long cycle.

Applications of LFSRs include generating pseudo-random numbers, pseudo-noise sequences, fast digital counters, and whitening sequences. Both hardware and software implementations of LFSRs are common.

Fibonacci LFSRs

The list of the bits positions that affect the next state is called the tap sequence. In the diagram below, the sequence is [16,14,13,11,0]. In a Fibonacci LFSR, as below, the taps are XOR'd sequentially with the output and then feed back into the leftmost bit.

The outputs that influence the input are called taps (blue in the diagram below).
A maximal LFSR produces an n-sequence (i.e. cycles through all possible $\text{[math]}$ states within the shift register except the state where all bits are zero), unless it contains all zeros, in which case it will never change.

The sequence of numbers generated by a LFSR can be considered a binary numeral system just as valid as Gray code or the natural binary code.

The tap sequence of an LFSR can be represented as a polynomial mod 2. This means that the coefficients of the polynomial must be 1's or 0's. This is called the feedback polynomial or characteristic polynomial. For example, if the taps are at the 16th, 14th, 13th and 11th bits (as below), the resulting LFSR polynomial is:

: $\text{[math]}$

The 'one' in the polynomial does not correspond to a tap - it corresponds to the input to the first bit (i.e. x⁰, which is equivalent to 1). The powers of the terms represent the tapped bits, counting from the left. The first and last bits are always connected as an input and tap respectively.

If (and only if) this polynomial is a primitive, then the LFSR is maximal
The LFSR will only be maximal if the number of taps is even
There can be more than one maximal tap sequence for a given LFSR length
Once one maximal tap sequence has been found, another automatically follows. If the tap sequence, in an n-bit LFSR, is [n,A,B,C,0], where the 0 corresponds to the $\text{[math]}$ term, then the corresponding 'mirror' sequence is [n,n-C,n-B,n-A,0]. So the tap sequence [32,3,2,0] has as its counterpart [32,30,29,0]. Both give a maximal sequence.

Output-stream properties

Ones and zeroes occur in 'runs'. The output stream 0110100, for example consists of five runs of lengths 1,2,1,1,2, in order. In one period of a maximal LFSR, $\text{[math]}$ runs occur (for example, a six bit LFSR will have 32 runs). Exactly $\text{[math]}$ of these runs will be one bit long, $\text{[math]}$ will be two bits long, up to a single run of zeroes $\text{[math]}$ bits long, and a single run of ones $\text{[math]}$ bits long. This same property is statistically expected in a truly random sequence.
LFSR outputs streams are deterministic. If you know the present state, you can predict the next state. This is not possible with truly random events such as nuclear decay.
The output stream is reversible; an LFSR with mirrored tap sequence will cycle through the states in reverse order.

A drop in replacement for Gray Code counters

Some applications need to mark individual locations along a certain distance with unique values. For example, most tape measures mark each inch or centimeter with a unique number using the decimal numeral system. When computer index or framing locations need to be machine-readable, they are often marked using a LFSR sequence, because LFSR counters are simpler and faster than any other kind of binary counter. LFSRs are faster than natural binary counters and Gray code counters. Given an output sequence you can construct a LFSR of minimal size by using the Berlekamp-Massey algorithm.

Galois LFSRs

Named after the French mathematician Évariste Galois, a Galois LFSR, or an LFSR in Galois configuration, is an alternate structure that can generate the same output sequences as a conventional LFSR. In the Galois configuration, when the system is clocked, bits that are not taps are shifted as normal to the next flip-flop. The taps, on the other hand, are XOR'd with the new output, which also becomes the new input. These won't be shifted in until the next clock cycle.

To generate the same output sequence, the order of the taps is the counterpart (see above) of the order for the conventional LFSR, otherwise the sequence will be in reverse. Note that the internal state of the LFSR is not necessarily the same. The Galois register above has the same output as the Fibonnacci register in the first section.

Galois LFSRs do not concatenate every tap to produce the new input (the XOR'ing is done within the LFSR and no XOR gates are run in serial, therefore the propagation times are reduced to that of one XOR rather than a whole chain), thus it is possible for each tap to be computed in parallel, increasing the speed of execution.
In a software implementation of an LFSR, the Galois form is more efficient as the XOR operations can be implemented a word at a time: only the output bit must be examined individually.

Below is example of 32-bit maximal period Galois LFSR simulated in C:


unsigned int lfsr = 1;
while(1)
lfsr = (lfsr >> 1) ^ (-(signed int)(lfsr & 1) & 0xd0000001u); /* taps 32 31 29 1 */

Applications

LFSRs can be implemented in hardware, and this makes them useful in applications that require very fast generation of a pseudo-random sequence, such as direct-sequence spread spectrum radio.

The Global Positioning System uses a LFSR to rapidly transmit a sequence that indicates high-precision relative time offsets. The Nintendo Entertainment System video game console also has a LFSR as part of its sound system. ([1])

Uses in cryptography

LFSRs have long been used as a pseudo-random number generator for use in stream ciphers (especially in military cryptography), due to the ease of construction from simple electromechanical or electronic circuits, long periods, and very uniformly distributed outputs. However the outputs of LFSRs are completely linear, leading to fairly easy cryptanalysis.

Three general methods are employed to reduce this problem in LFSR based stream ciphers

Non-linear combination of several bits from the LFSR state;
Non-linear combination of the outputs of two or more LFSRs; or
Irregular clocking of the LFSR, as in the alternating step generator.

Important LFSR-based stream ciphers include A5/1, A5/2, E0, and the shrinking generator.

Uses in digital broadcasting and communications

To prevent short repeating sequences (e.g., runs of 0's or 1's) from forming spectral lines that may complicate symbol tracking at the receiver or interfere with other transmissions, linear feedback registers are often used to "randomize" the transmitted bitstream. This randomization is removed at the receiver after demodulation. When the LFSR runs at the same rate as the transmitted symbol stream, this technique is referred to as scrambling. When the LFSR runs considerably faster than the symbol stream, expanding the bandwidth of the transmitted signal, this is direct-sequence spread spectrum.

Neither scheme should be confused with encryption or encipherment; scrambling and spreading with LFSRs do not protect the information from eavesdropping.

Digital broadcasting systems that use linear feedback registers

ATSC Standards (HDTV transmission system – North America)
DAB (Digital audio broadcasting system -- for radio)
DVB-T (HDTV transmission system – Europe, Australasia)
NICAM (digital audio system for television)

Other digital communications systems using LFSR:

IBS (INTELSAT business service)
IDR (Intermedaite Data Rate service)
SDI (Serial Digital Interface transmission)
Data transfer over PSTN (according to the ITU-T V-series recommendations)

External links

International Telecommunications Union Recommendation O.151 (August 1992)
Maximal Length LFSR table with length from 3 to 168
Maximal Length LFSR table with length from 1 to 786, also 1024 and 2048.
Pseudo-Random Number Generation Routine
http://www.ee.ualberta.ca/~elliott/ee552/studentAppNotes/1999f/Drivers_Ed/lfsr.html
http://www.quadibloc.com/crypto/co040801.htm
Simple explanation of LFSRs for Engineers
Feedback terms
General LFSR Theory
Table of Maximal Tap Sequences
Shift register code generator

In digital circuits a shift register is a group of flip flops set up in a linear fashion which have their inputs and outputs connected together in such a way that the data are shifted down the line when the circuit is activated.
..... Click the link for more information.

In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
..... Click the link for more information.

exclusive disjunction, also called exclusive or, (symbolized XOR or EOR), is a type of logical disjunction on two operands that results in a value of "true" if and only if exactly one of the operands has a value of "true.
..... Click the link for more information.

A pseudorandom process is a process that appears random but is not. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process.
..... Click the link for more information.

In cryptography, pseudorandom noise (PRN) is a signal similar to noise which satisfies one or more of the standard tests for statistical randomness.

Although it seems to lack any definite pattern, pseudorandom noise consists of a deterministic sequence of pulses that will
..... Click the link for more information.

A maximum length sequence (MLS) is a type of pseudorandom binary sequence.

They are polynomial rings generated using maximal linear feedback shift registers and are so called because they are periodic and reproduce every binary sequence that can be reproduced by the
..... Click the link for more information.

binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
..... Click the link for more information.

The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one digit.

The reflected binary code was originally designed to prevent spurious output from electromechanical switches.
..... Click the link for more information.

In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. is a polynomial.
..... Click the link for more information.

Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
..... Click the link for more information.

In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the extension field GF(p^m).
..... Click the link for more information.

In mathematics, the parity of an object refers to whether it is even or odd.

The formal definition of an odd number is an integer of the form n=2k +1, where k is an integer. The definition of an even number is n=2k where k is an integer.
..... Click the link for more information.

Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences.
..... Click the link for more information.

tape measure or measuring tape is a ribbon of cloth, plastic, or metal with linear-measure markings, often in both imperial and metric units. Surveyors use tape measures in lengths on the order of hectometres. It is a convenient, common measuring tool.
..... Click the link for more information.

decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have four fingers and a thumb on each hand, giving a total of ten digits over both hands.
..... Click the link for more information.

counter is a device which stores (and sometimes displays) the number of times a particular event or process has occurred, often in relationship to a clock signal. In practice, there are two types of counters:

up counters which increase (increment) in value

..... Click the link for more information.

The Berlekamp-Massey algorithm is an algorithm for finding the shortest linear feedback shift register (LFSR) for a given output sequence. Equivalently, it is an algorithm for finding the minimal polynomial of a linearly recurrent sequence.
..... Click the link for more information.

Évariste Galois (October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem.
..... Click the link for more information.

In telecommunications, direct-sequence spread spectrum (DSSS) is a modulation technique. As with other spread spectrum technologies, the transmitted signal takes up more bandwidth than the information signal that is being modulated.
..... Click the link for more information.

Global Positioning System (GPS) is the only fully functional Global Navigation Satellite System (GNSS). Utilizing a constellation of at least 24 medium Earth orbit satellites that transmit precise microwave signals, the system enables a GPS receiver to determine its
..... Click the link for more information.

^JPN July 15, 1983
^USA October 18, 1985
^CAN February 1986
^EUR September 1, 1986^[1]
^EUR/AUS 1987<ref name="eur_regions" />
..... Click the link for more information.

A pseudorandom number generator (PRNG) is an algorithm to generate a sequence of numbers that approximate the properties of random numbers. The sequence is not truly random in that it is completely determined by a relatively small set of initial values, called the PRNG's
..... Click the link for more information.

stream cipher is a symmetric cipher where plaintext bits are combined with a pseudorandom cipher bit stream (keystream), typically by an exclusive-or (xor) operation. In a stream cipher the plaintext digits are encrypted one at a time, and in which the transformation of successive
..... Click the link for more information.

Military has two broad meanings. In its first sense, it refers to soldiers and soldiering. In its second sense, it refers to armed forces as a whole. Over the years, military units have come in all shapes and sizes.
..... Click the link for more information.

Cryptography (or cryptology; derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write" or λεγειν legein
..... Click the link for more information.

In engineering, electromechanics combines the sciences of electromagnetism of electrical engineering and mechanics. Mechatronics is the discipline of engineering that combines mechanics, electronics and information technology (software engineering).
..... Click the link for more information.

electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches.
..... Click the link for more information.

In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable.

Examples

Everyday examples are seen when the variable is time
..... Click the link for more information.

This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.

Herod_Archelaus

iPedia Free Information Center