A
linear feedback shift register (LFSR) is a
shift registerIn digital circuits, a shift register is a cascade of flip flops, sharing the same clock, which has the output of any one but the last flip-flop connected to the "data" input of the next one in the chain, resulting in a circuit that shifts by one position the one-dimensional "bit array" stored in...
whose input bit is a
linearIn mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself...
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 stream 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, an 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 numbersA pseudorandom process is a process that appears to be random but it is not. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process...
,
pseudo-noise sequencesIn cryptography, pseudorandom noise is a signal similar to noise which satisfies one or more of the standard tests for statistical randomness....
, fast digital counters, and whitening sequences. Both hardware and software implementations of LFSRs are common.
Fibonacci LFSRs
The bit positions that affect the next state are called the taps. In the diagram the taps are [16,14,13,11]. The rightmost bit of the LFSR is called the output bit. The taps are XOR'd sequentially with the output bit and then fed back into the leftmost bit. The sequence of bits in the rightmost position is called the output stream.
- The bits in the LFSR state which influence the input are called taps (white in the diagram).
- A maximum-length LFSR produces an m-sequence
A maximum length sequence is a type of pseudorandom binary sequence.They are bit sequences 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 shift registers A maximum length sequence...
(i.e. it cycles through all possible 2n − 1 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.
- As an alternative to the XOR based feedback in a standard LFSR, one can also use XNOR. A state with all ones is illegal when using an XNOR feedback, in the same way as a state with all zeroes is illegal when using XOR. This state is considered illegal because the counter would remain "locked-up" in this state.
The sequence of numbers generated by an LFSR can be considered a
binary numeral systemThe binary numeral system, or base-2 number system represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
just as valid as
Gray codeThe reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit....
or the natural binary code.
The arrangement of taps for feedback in an LFSR can be expressed in
finite field arithmeticArithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field....
as a
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
modIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...
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 shown), the feedback polynomial is
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.
Tables of
primitive polynomialIn field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF...
s from which maximum-length LFSRs can be constructed are given below and in the references.
- The LFSR will only be maximum-length if the number of taps is even
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
; just 2 or 4 taps can suffice even for extremely long sequences.
- The set of taps must be relatively prime, and share no common divisor to all taps.
- There can be more than one maximum-length tap sequence for a given LFSR length
- Once one maximum-length 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 x0 = 1 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 maximum-length sequence.
Some example C/C++ code is below (assuming 16-bit
shorts):
unsigned short lfsr = 0xACE1u;
unsigned bit;
unsigned period = 0;
do
{
/* taps: 16 14 13 11; characteristic polynomial: x^16 + x^14 + x^13 + x^11 + 1 */
bit = ((lfsr >> 0) ^ (lfsr >> 2) ^ (lfsr >> 3) ^ (lfsr >> 5) ) & 1;
lfsr = (lfsr >> 1) | (bit << 15);
++period;
} while(lfsr != 0xACE1u);
The above code assumes the right most bit is actually bit 1, not bit 16.
As well as
Fibonacci, this LFSR configuration is also known as
Standard,
many-to-one or
external XOR gates. It has an alternative configuration.
Galois LFSRs
Named after the French mathematician
Évariste GaloisÉvariste Galois 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...
, an LFSR in Galois configuration, which is also known as
modular,
internal XORs as well as
one-to-many LFSR, is an alternate structure that can generate the same output stream as a conventional LFSR. In the Galois configuration, when the system is clocked, bits that are not taps are shifted one position to the right unchanged. The taps, on the other hand, are XOR'd with the output bit before they are stored in the next position. The new output bit is the next input bit. The effect of this is that when the output bit is zero all the bits in the register shift to the right unchanged, and the input bit becomes zero. When the output bit is one, the bits in the tap positions all flip (if they are 0, they become 1, and if they are 1, they become 0), and then the entire register is shifted to the right and the input bit becomes 1.
To generate the same output stream, the order of the taps is the
counterpart (see above) of the order for the conventional LFSR, otherwise the stream will be in reverse. Note that the internal state of the LFSR is not necessarily the same. The Galois register shown has the same output stream 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 a code example of a 32-bit maximal period Galois LFSR that is valid in
CC is a general-purpose computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
and
C++C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as a middle-level language, as it comprises a combination of both high-level and low-level language features...
, (assuming that
unsigned int has 32 bit precision):
unsigned lfsr = 1;
unsigned period = 0;
do
{
/* taps: 32 31 29 1; characteristic polynomial: x^32 + x^31 + x^29 + x + 1 */
lfsr = (lfsr >> 1) ^ (unsigned int)(0 - (lfsr & 1u) & 0xd0000001u);
++period;
} while(lfsr != 1u);
And here is the code for the 16 bit example in the figure (Assuming 16-bit
shorts)
unsigned short lfsr = 0xACE1u;
unsigned period = 0;
do
{
/* taps: 16 14 13 11; characteristic polynomial: x^16 + x^14 + x^13 + x^11 + 1 */
lfsr = (lfsr >> 1) ^ (-(lfsr & 1u) & 0xB400u);
++period;
} while(lfsr != 0xACE1u);
These code examples create a toggle mask to apply to the shifted value using the XOR operator. The mask is created by first removing all but the least significant bit (the output bit) of the current value. This value is then negated (
two's complementThe two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two...
negation), which creates a value of either all 0s or all 1s, if the output bit is 0 or 1, respectively. By ANDing the result with the tap-value (e.g., 0xB400 in the second example) before applying it as the toggle mask, it acts functionally as a conditional to either apply or not apply the toggle mask based on the output bit. A more explicit but significantly less efficient code example is shown below.
unsigned short lfsr = 0xACE1u;
unsigned period = 0;
do {
unsigned lsb = lfsr & 1; //Get lsb (i.e., the output bit).
lfsr >>= 1; //Shift register
if(lsb 1) //Only apply toggle mask if output bit is 1.
lfsr ^= 0xB400u; //apply toggle mask, value has 1 at bits corresponding
// to taps, 0 else where.
++period;
} while(lfsr != 0xACE1u);
Non-binary Galois LFSR
Binary Galois LFSRs like the ones shown above can be generalized to any
q-ary alphabet {0, 1, ... ,
q − 1} (e.g., for binary,
q is equal to two, and the alphabet is simply {0, 1}). In this case, the exclusive-or component is generalized to addition
moduloIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...
-
q (note that XOR is addition modulo 2), and the feedback bit (output bit) is multiplied (modulo-
q) by a
q-ary value which is constant for each specific tap point. Note that this is also a generalization of the binary case, where the feedback is multiplied by either 0 (no feedback, i.e., no tap) or 1 (feedback is present). Given an appropriate tap configuration, such LFSRs can be used to generate
Galois fieldsIn abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
for arbitrary values of
q.
Some polynomials for maximal LFSRs
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, 2n − 1 runs occur (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 output streams are deterministic. If you know the present state, you can predict the next state. This is not possible with truly random events.
- The output stream is reversible; an LFSR with mirrored taps will cycle through the states in reverse order.
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 spectrumIn telecommunications, direct-sequence spread spectrum 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...
radio. LFSRs have also been used for generating an approximation of
white noiseWhite noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
in various
programmable sound generatorA Programmable Sound Generator is a sound chip that generates sound waves by synthesizing multiple basic waveforms, and often some kind of noise generator, and combining and mixing these waveforms into a complex waveform, then shaping the amplitude of the resulting waveform using...
s.
The
Global Positioning SystemThe Global Positioning System is a U.S. space-based global navigation satellite system. It provides reliable positioning, navigation, and timing services to worldwide users on a continuous basis in all weather, day and night, anywhere on or near the Earth.GPS is made up of three parts: between 24...
uses an LFSR to rapidly transmit a sequence that indicates high-precision relative time offsets.
Uses as counters
The repeating sequence of states of an LFSR allows it to be used as a clock divider, or as a counter when a non-binary sequence is acceptable as is often the case where computer index or framing locations need to be machine-readable. LFSR
counterIn digital logic and computing, a counter is a device which stores the number of times a particular event or process has occurred, often in relationship to a clock signal...
s have simpler feedback logic than natural binary counters or
Gray codeThe reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit....
counters, and therefore can operate at higher clock rates. However it is necessary to ensure that the LFSR never enters an all-zeros state, for example by presetting it at start-up to any other state in the sequence.
The table of primitive polynomials shows how LFSRs can be arranged in Fibonacci or Galois form to give maximal periods. One can obtain any other period by adding to an LFSR that has a longer period some logic that shortens the sequence by skipping some states.
Uses in cryptography
LFSRs have long been used as pseudo-random number generators for use in
stream cipherIn cryptography, a stream cipher is a symmetric key cipher where plaintext bits are combined with a pseudorandom cipher bit stream , typically by an exclusive-or operation. In a stream cipher the plaintext digits are encrypted one at a time, and the transformation of successive digits varies...
s (especially in
militaryA military is an organization authorized by its nation to use force, usually including use of weapons, in defending its country by combating actual or perceived threats. As an adjective the term "military" is also used to refer to any property or aspect of a military...
cryptographyCryptography is the practice and study of hiding information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering...
), due to the ease of construction from simple electromechanical or electronic circuits, long
periodsIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π...
, and very uniformly
distributedIn probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval...
output streams. However, an LFSR is a linear system, leading to fairly easy
cryptanalysisCryptanalysis 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. Typically, this involves knowing how the system works and finding a secret key...
. For example, given a stretch of known plaintext and corresponding ciphertext, an attacker can intercept and recover a stretch of LFSR output stream used in the system described, and from that stretch of the output stream can construct an LFSR of minimal size that simulates the intended receiver by using the
Berlekamp-Massey algorithmThe Berlekamp–Massey algorithm is an algorithm for finding the shortest linear feedback shift register for a given output sequence. Equivalently, it is an algorithm for finding the minimal polynomial of a linearly recurrent sequence....
. This LFSR can then be fed the intercepted stretch of output stream to recover the remaining plaintext.
Three general methods are employed to reduce this problem in LFSR-based stream ciphers:
- Non-linear combination of several bit
In computing and telecommunications a bit is a basic unit of information storage and communication . It is the maximum amount of information that can be stored by a device or other physical system that can normally exist in only two distinct states...
s from the LFSR stateIn computer science and automata theory, a state is a unique configuration of information in a program or machine. It is a concept that occasionally extends into some forms of systems programming such as lexers and parsers....
;
- Non-linear combination of the output bits of two or more LFSRs (see also: shrinking generator
In cryptography, the shrinking generator is a form of pseudorandom number generator intended to be used in a stream cipher. It was published in Crypto 1993 by Don Coppersmith, Hugo Krawczyk, and Yishay Mansour....
); or
- Irregular clocking of the LFSR, as in the alternating step generator
In cryptography, an alternating step generator is a cryptographic pseudorandom number generator intended to be used in a stream cipher. The design was published in 1987 by C. G. Günther...
.
Important LFSR-based stream ciphers include
A5/1A5/1 is a stream cipher used to provide over-the-air communication privacy in the GSM cellular telephone standard. It was initially kept secret, but became public knowledge through leaks and reverse engineering. A number of serious weaknesses in the cipher have been identified.-History and...
and
A5/2A5/2 is a stream cipher used to provide voice privacy in the GSM cellular telephone protocol.The cipher is based around a combination of four linear feedback shift registers with irregular clocking and a non-linear combiner.In 1999, Ian Goldberg and David A...
, used in GSM cell phones,
E0E0 is a stream cipher used in the Bluetooth protocol. It generates a sequence of pseudorandom numbers and combines it with the data using the XOR operator. The key length may vary, but is generally 128 bits....
, used in
BluetoothBluetooth is an open wireless protocol for exchanging data over short distances from fixed and mobile devices, creating personal area networks . It was originally conceived as a wireless alternative to RS232 data cables...
, and the
shrinking generatorIn cryptography, the shrinking generator is a form of pseudorandom number generator intended to be used in a stream cipher. It was published in Crypto 1993 by Don Coppersmith, Hugo Krawczyk, and Yishay Mansour....
. The A5/2 cipher has been broken and both A5/1 and E0 have serious weaknesses.
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 spectrumIn telecommunications, direct-sequence spread spectrum 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...
.
Neither scheme should be confused with
encryptionIn cryptography, encryption is the process of transforming information using an algorithm to make it unreadable to anyone except those possessing special knowledge, usually referred to as a key. The result of the process is encrypted information...
or encipherment; scrambling and spreading with LFSRs do
not protect the information from eavesdropping. They are instead used to produce equivalent streams that possess convenient engineering properties to allow for robust and efficient modulation and demodulation.
Digital broadcasting systems that use linear feedback registers:
- ATSC Standards (HDTV transmission system – North America)
- DAB
Digital Audio Broadcasting , is a digital radio technology for broadcasting radio stations, used in several countries, particularly in Europe. As of 2006, approximately 1,000 stations worldwide broadcast in the DAB format....
(Digital audio broadcastingDigital Audio Broadcasting , is a digital radio technology for broadcasting radio stations, used in several countries, particularly in Europe. As of 2006, approximately 1,000 stations worldwide broadcast in the DAB format....
system -- for radio)
- DVB-T
DVB-T is an abbreviation for Digital Video Broadcasting — Terrestrial; it is the DVB European-based consortium standard for the broadcast transmission of digital terrestrial television that was first broadcast in the UK in 1997...
(HDTV transmission system – Europe, Australasia)
- NICAM
NICAM stands for Near Instantaneous Companded Audio Multiplex. It is an early form of lossy compression for digital audio. It was originally developed in the early 1970s for point-to-point links within broadcasting networks...
(digital audio system for television)
Other digital communications systems using LFSRs:
- IBS (INTELSAT business service)
- IDR (Intermediate Data Rate service)
- SDI
Serial digital interface refers to a family of video interfaces standardized by SMPTE. For example, ITU-R BT.656 and SMPTE 259M define digital video interfaces used for broadcast-grade video...
(Serial Digital Interface transmission)
- Data transfer over PSTN (according to the ITU-T
The Telecommunication Standardization Sector coordinates standards for telecommunications on behalf of the International Telecommunication Union and is based in Geneva, Switzerland....
V-series recommendations)
- CDMA (Code Division Multiple Access) cellular telephony
- 100BASE-T2 "fast" Ethernet scrambles bits using a LFSR
- 1000BASE-T Ethernet, the most common form of Gigabit Ethernet, scrambles bits using a LFSR
See also
- Pinwheel
In cryptography, a pinwheel was a device for producing a short pseudorandom sequence of bits , as a component in a cipher machine. A pinwheel consisted of a rotating wheel with a certain number of positions on its periphery. Each position had a "pin" or "lug" which could be either "set" or "unset"...
- Mersenne twister
The Mersenne twister is a pseudorandom number generator developed in 1997 by and that is based on a matrix linear recurrence over a finite binary field...
- Maximum length sequence
A maximum length sequence is a type of pseudorandom binary sequence.They are bit sequences 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 shift registers A maximum length sequence...
External links