A linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. In digital circuits a shift register is a group of registers set up in a linear fashion which have their inputs and outputs connected together in such a way that the data is shifted down the line when the circuit is activated. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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. Exclusive or (usual symbol XOR occasionally XNAD <exclusive nand> or EOR), which is sometimes called exclusive disjunction, is a logical operator that results in true if one of the operands, but not both of them, is true. ...

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. A pseudorandom process is a process that appears random but is not. ... // PN SEQUENCES --Rdschwarz 15:28, 20 August 2005 (UTC) Spread Spectrum technology is being applied to many areas of modern communications such as Wireless Lans, Cellular Telephones, Global Positioning System (GPS), and Very Small Aperture Satellite Terminals (VSAT) just to name a few. ...

How it works

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].

• The outputs that influence the input are called taps.
• A maximal LFSR produces an n-sequence (i.e. cycles through all possible states within the shift register), 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. In telecommunication, an n-sequence is a pseudorandom binary sequence of n bits that is the output of a linear shift register, such that, if the shift register is set to any nonzero state and then cycled, a pseudorandom binary sequence of a maximum of n = 2m -1 bits will... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... A Gray code is a binary numeral system where two successive values differ in only one digit. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...

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 In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

The 'one' in the polynomial does not correspond to a tap. The powers of the terms represent the tapped bits, counting from the left.

• 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
• The tap values in a maximal LFSR will be relatively prime
• 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], then the corresponding 'mirror' sequence is [n,n-A,n-B,n-C]. So the tap sequence [32,3,2,1] has as its counterpart [32,31,30,29]. Both give a maximal sequence.

A primitive polynomial may refer to one of two concepts: A polynomial over a unique factorization domain (such as the integers) whose greatest common divisor of its coefficients is one. ... In mathematics, the parity of an object refers to whether is is even or odd. ... In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... LFSR animation, polynomial . ...

Output Stream Properties

• Ones and zeroes occur in 'runs'. The output stream 000000, for example consists of five runs of lengths 1,2,1,1,2, in order. In one period of a maximal LFSR, 2^{n − 1} runs occurs (for example, a six bit LFSR will have 32 runs). Exactly 1 / 2 of these runs will be one bit long, 1 / 4 will be two bits long, up to a single run of zeroes n − 1 bits long, and a single run of ones n 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 term deterministic may refer to: the more general notion of determinism from philosophy, see determinism a type of algorithm as discussed in computer science, see deterministic algorithm scientific determinism as used by Karl Popper and Stephen Hawking deterministic system in mathematics deterministic system in philosophy deterministic finite state machine...

Applications

LFSR's 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. In telecommunication, the term direct-sequence spread spectrum has the following meanings: A system (a) for generating spread-spectrum transmissions by phase-modulating a sine wave pseudorandomly with a continuous string of pseudonoise code symbols, each of duration much smaller than a bit and (b) that may be time-gated...

The Global Positioning System uses a LFSR to rapidly transmit a sequence that indicates high-precision relative time offsets. GPS satellite in orbit, image courtesy of NASA GPS redirects here. ...

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. Self-retracting pocket tape measure A tape measure or measuring tape is a ribbon of cloth, plastic, or metal with linear-measure markings, often in both imperial and metric units. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... This article is about the term counter used in electronics and computing. ...

• LFSRs are faster than the natural binary counters or Gray code counters.
• Given an output sequence you can construct a LFSR of minimal size by using the Berlekamp-Massey algorithm.

The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... A Gray code is a binary numeral system where two successive values differ in only one digit. ... The Berlekamp-Massey algorithm is a algorithm for finding the shortest linear feedback shift register (LFSR) for a given output sequence. ...

Galois LFSRs

A Galois LFSR, or a LFSR in Galois configuration is a variation on typical LFSR design. Galois at the age of fifteen from the pencil of a classmate. ...

In Galois configuration, when the system is clocked, bits that are not taps are shifted as normal. The taps, on the other hand, are XOR'd with the new output, which also becomes the new input.

• Galois LFSRs do not concatenate every tap to produce the new input (the XOR'ing is done within the LFSR and no XOR's 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.

Image File history File links Galois_LFSR.png Licensing I, the creator of this work, hereby release it into the public domain. ...

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. A pseudorandom number generator (PRNG) is an algorithm that generates a sequence of numbers, the elements of which are approximately independent of each other. ... The operation of A5/1, a LFSR-based stream cipher used to encrypt mobile phone conversations. ... In engineering, electromechanics combines electromagnetism and mechanics. ... The field of electronics is the study and use of systems that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. ... An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The word linear comes from the Latin word linearis, which means created by lines. ... Cryptanalysis (from the Greek kryptós, hidden, and analýein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ...

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.

Important LFSR-based stream ciphers include A5/1, A5/2, E0 and the shrinking generator. To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... This article is about the unit of information; for other uses, see Bit (disambiguation). ... In information processing, a state is the complete set of properties (for example, its energy level, etc. ... A5/1 is a stream cipher used to provide over-the-air voice privacy in the GSM cellular telephone standard. ... A5/1 is a stream cipher used to provide over-the-air voice privacy in the GSM cellular telephone standard. ... This article needs translation. ... In cryptography, the shrinking generator is a form of pseudorandom number generator intended to be used in a stream cipher. ...

Uses in digital broadcasting

In order to keep digital transmissions from forming latent energy patterns that may disrupt other digital or analog transmissions -- linear feedback registers are used to create more randomness in the outgoing digital bitstream.

Digital broadcasting systems that use linear feedback registers

• NICAM (digital audio system for television)
• ATSC (HDTV transmission system -- North America)
• DVB-T (HDTV transmission system -- Europe, Australasia)

NICAM (known also as NICAM 728, after the 728 kbit/s bitstream it is sent over), Near Instantaneous Companded Audio Multiplex, is a format for digital sound over television. ... The Advanced Television Systems Committee (ATSC) is the group that helped to develop the new digital television standard for the United States, also adopted by Canada, Mexico, and South Korea and being considered by other countries. ... DVB-T stands for Digital Video Broadcasting - Terrestrial and it is the DVB European consortium standard for the broadcast transmission of digital terrestrial television. ...

See also

• Pinwheel
• Mersenne twister
• PN Sequences

The German Lorenz cipher machine contained 12 pinwheels, with a total of 501 pins . In cryptography, a pinwheel was a device for producing a short pseudorandom sequence of bits (determined by the machines initial settings), as a component in a cipher machine. ... The Mersenne twister is a pseudorandom number generator developed in 1997 by Makoto Matsumoto (松本 眞) and Takuji Nishimura (西村 拓士). It provides for fast generation of very high quality random numbers, having been designed specifically to rectify many of the flaws found in older algorithms. ... // PN SEQUENCES --Rdschwarz 15:28, 20 August 2005 (UTC) Spread Spectrum technology is being applied to many areas of modern communications such as Wireless Lans, Cellular Telephones, Global Positioning System (GPS), and Very Small Aperture Satellite Terminals (VSAT) just to name a few. ...

External links

• http://www.maxim-ic.com/appnotes.cfm/appnote_number/1743
• 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