5357:(BIST) techniques, storing all the circuit outputs on chip is not possible, but the circuit output can be compressed to form a signature that will later be compared to the golden signature (of the good circuit) to detect faults. Since this compression is lossy, there is always a possibility that a faulty output also generates the same signature as the golden signature and the faults cannot be detected. This condition is called error masking or aliasing. BIST is accomplished with a multiple-input signature register (MISR or MSR), which is a type of LFSR. A standard LFSR has a single XOR or XNOR gate, where the input of the gate is connected to several "taps" and the output is connected to the input of the first flip-flop. A MISR has the same structure, but the input to every flip-flop is fed through an XOR/XNOR gate. For example, a 4-bit MISR has a 4-bit parallel output and a 4-bit parallel input. The input of the first flip-flop is XOR/XNORd with parallel input bit zero and the "taps". Every other flip-flop input is XOR/XNORd with the preceding flip-flop output and the corresponding parallel input bit. Consequently, the next state of the MISR depends on the last several states opposed to just the current state. Therefore, a MISR will always generate the same golden signature given that the input sequence is the same every time. Recent applications are proposing set-reset flip-flops as "taps" of the LFSR. This allows the BIST system to optimise storage, since set-reset flip-flops can save the initial seed to generate the whole stream of bits from the LFSR. Nevertheless, this requires changes in the architecture of BIST, is an option for specific applications.
2736:
2084:
1145:, is an alternate structure that can generate the same output stream as a conventional LFSR (but offset in time). 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 XORed 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.
2731:{\displaystyle {\begin{pmatrix}a_{k}\\a_{k+1}\\a_{k+2}\\\vdots \\a_{k+n-1}\end{pmatrix}}={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &0\\0&0&\cdots &0&1\\c_{0}&c_{1}&\cdots &\cdots &c_{n-1}\end{pmatrix}}{\begin{pmatrix}a_{k-1}\\a_{k}\\a_{k+1}\\\vdots \\a_{k+n-2}\end{pmatrix}}={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &0\\0&0&\cdots &0&1\\c_{0}&c_{1}&\cdots &\cdots &c_{n-1}\end{pmatrix}}^{k}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\end{pmatrix}}}
260:, but it results in an equivalent polynomial counter whose state is the complement of the state of an LFSR. 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. This method can be advantageous in hardware LFSRs using flip-flops that start in a zero state, as it does not start in a lockup state, meaning that the register does not need to be seeded in order to begin operation.
3369:
217:
5152:, 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.
237:. In the diagram the taps are . The rightmost bit of the LFSR is called the output bit, which is always also a tap. To obtain the next state, the tap bits are XOR-ed sequentially; then, all bits are shifted one place to the right, with the rightmost bit being discarded, and that result of XOR-ing the tap bits is fed back into the now-vacant leftmost bit. To obtain the pseudorandom output stream, read the rightmost bit after each state transition.
5281:
34:
3070:
2946:
1121:
200:
1152:(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 Fibonacci register in the first section. A time offset exists between the streams, so a different startpoint will be needed to get the same output each cycle.
191:, used to provide a quick check against transmission errors, are closely related to those of an LFSR. In general, the arithmetics behind LFSRs makes them very elegant as an object to study and implement. One can produce relatively complex logics with simple building blocks. However, other methods, that are less elegant but perform better, should be considered as well.
3364:{\displaystyle {\begin{pmatrix}c_{0}&1&0&\cdots &0\\c_{1}&0&1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &0\\c_{n-2}&0&\cdots &0&1\\c_{n-1}&0&\cdots &\cdots &0\end{pmatrix}}^{k}{\begin{pmatrix}a'_{0}\\a'_{1}\\a'_{2}\\\vdots \\a'_{n-1}\end{pmatrix}}}
2747:
221:
220:
224:
223:
219:
225:
5376:
To prevent short repeating sequences (e.g., runs of 0s or 1s) from forming spectral lines that may complicate symbol tracking at the receiver or interfere with other transmissions, the data bit sequence is combined with the output of a linear-feedback register before modulation and transmission. This
5081:
Ones and zeroes occur in "runs". The output stream 1110010, for example, consists of four runs of lengths 3, 2, 1, 1, in order. In one period of a maximal LFSR, 2 runs occur (in the example above, the 3-bit LFSR has 4 runs). Exactly half of these runs are one bit long, a quarter are two bits long, up
5101:. If the present state and the positions of the XOR gates in the LFSR are known, the next state can be predicted. This is not possible with truly random events. With maximal-length LFSRs, it is much easier to compute the next state, as there are only an easily limited number of them for each length.
1124:
A 16-bit Galois LFSR. The register numbers above correspond to the same primitive polynomial as the
Fibonacci example but are counted in reverse to the shifting direction. This register also cycles through the maximal number of 65535 states excluding the all-zeroes state. The state ACE1 hex shown
160:
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
1715:-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
1156:
Galois LFSRs do not concatenate every tap to produce the new input (the XORing 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,
222:
2941:{\displaystyle {\begin{pmatrix}c_{0}&1&0&\cdots &0\\c_{1}&0&1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &0\\c_{n-2}&0&\cdots &0&1\\c_{n-1}&0&\cdots &\cdots &0\end{pmatrix}}}
286:
2. This means that the coefficients of the polynomial must be 1s or 0s. This is called the feedback polynomial or reciprocal characteristic polynomial. For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is
6251: (archived October 1, 2018) – LFSR theory and implementation, maximal length sequences, and comprehensive feedback tables for lengths from 7 to 16,777,215 (3 to 24 stages), and partial tables for lengths up to 4,294,967,295 (25 to 32 stages).
5192:, 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
1745:, linear feedback shift registers can be implemented using XOR and Shift operations. This approach lends itself to fast execution in software because these operations typically map efficiently into modern processor instructions.
5556:. All current systems use LFSR outputs to generate some or all of their ranging codes (as the chipping code for CDMA or DSSS) or to modulate the carrier without data (like GPS L2 CL ranging code). GLONASS also uses
3059:
207:
LFSR. The feedback tap numbers shown correspond to a primitive polynomial in the table, so the register cycles through the maximum number of 65535 states excluding the all-zeroes state. The state shown, 0xACE1
6014:
A. Poorghanad, A. Sadr, A. Kashanipour" Generating High
Quality Pseudo Random Number Using Evolutionary Methods", IEEE Congress on Computational Intelligence and Security, vol. 9, pp. 331-335, May, 2008
218:
5424:
protect the information from eavesdropping. They are instead used to produce equivalent streams that possess convenient engineering properties to allow robust and efficient modulation and demodulation.
2056:
402:
There can be more than one maximum-length tap sequence for a given LFSR length. Also, once one maximum-length tap sequence has been found, another automatically follows. If the tap sequence in an
245:(i.e., it cycles through all possible 2 − 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.
369:, 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 output tap respectively.
360:
5045:
4714:
4505:
4357:
4270:
4183:
1160:
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.
6244:
3913:
1968:
4958:
4897:
4836:
4775:
4627:
4566:
4418:
4096:
4035:
3974:
3826:
3765:
3704:
3643:
3582:
3521:
5333:
LFSRs are used in circuit testing for test-pattern generation (for exhaustive testing, pseudo-random testing or pseudo-exhaustive testing) and for signature analysis.
3475:
6185:
1528:
State and resulting bits can also be combined and computed in parallel. The following function calculates the next 64 bits using 63-bit polynomial x⁶³ + x⁶² + 1:
2076:
6028:
5144:
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
229:
A Fibonacci 31 bit linear feedback shift register with taps at positions 28 and 31, giving it a maximum cycle and period at this speed of nearly 6.7 years.
6162:
Martínez LH, Khursheed S, Reddy SM. LFSR generation for high test coverage and low hardware overhead. IET Computers & Digital
Techniques. 2019 Aug 21.
5120:
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
5345:-input circuit. Maximal-length LFSRs and weighted LFSRs are widely used as pseudo-random test-pattern generators for pseudo-random test applications.
410: = 1 term, then the corresponding "mirror" sequence is . So the tap sequence has as its counterpart . Both give a maximum-length sequence.
5298:
51:
5094:
for a truly random sequence. However, the probability of finding exactly this distribution in a sample of a truly random sequence is rather low.
6140:
6341:
98:
157:(XOR). Thus, an LFSR is most often a shift register whose input bit is driven by the XOR of some bits of the overall shift register value.
70:
2957:
5736:
6219:
1112:
Where a register of 16 bits is used and the xor tap at the fourth, 13th, 15th and sixteenth bit establishes a maximum sequence length.
77:
6586:
3419:
1508:
which may produce more efficient code on some compilers. In addition, the left-shifting variant may produce even better code, as the
5584:
driven by a 9-stage LFSR to increase the accuracy of received time and the robustness of the data stream in the presence of noise.
6720:
6260:
1981:
84:
6281:
6082:
Lu, Yi; Willi Meier; Serge
Vaudenay (2005). "The Conditional Correlation Attack: A Practical Attack on Bluetooth Encryption".
6111:
5935:
5790:
5397:
or a similar modulation method. The resulting signal has a higher bandwidth than the data, and therefore this is a method of
66:
5341:
Complete LFSR are commonly used as pattern generators for exhaustive testing, since they cover all possible inputs for an
399:
Tables of primitive polynomials from which maximum-length LFSRs can be constructed are given below and in the references.
3405:
3397:
373:
162:
5405:; when used to distinguish several signals transmitted in the same channel at the same time and frequency, it is called
6271:
5960:
5557:
6266:
6715:
6648:
6334:
6059:
5711:
5484:
5320:
117:
5893:
5104:
The output stream is reversible; an LFSR with mirrored taps will cycle through the output sequence in reverse order.
293:
6291:
4978:
4647:
4438:
4290:
4203:
4116:
5853:
5815:
5402:
5121:
1941:
Binary LFSRs of both
Fibonacci and Galois configurations can be expressed as linear functions using matrices in
6633:
5644:
5608:
5302:
5161:
3846:
55:
20:
365:
The "one" in the polynomial does not correspond to a tap – it corresponds to the input to the first bit (i.e.
5624:
5494:
5406:
5264:
1699:= 2, and the alphabet is simply {0, 1}). In this case, the exclusive-or component is generalized to addition
6214:
6003:
5688:
91:
6327:
5385:. When the LFSR runs considerably faster than the symbol stream, the LFSR-generated bit sequence is called
5129:
6314:
6016:
5196:. This LFSR can then be fed the intercepted stretch of output stream to recover the remaining plaintext.
5064:
an extended list of tap counters up to 168 bit. Tables of maximum length polynomials are available from
3409:
1835:// 7,9,13 triplet from http://www.retroprogramming.com/2017/07/xorshift-pseudorandom-numbers-in-z80.html
6710:
6705:
6617:
6476:
6187:
Time dissemination via the LF transmitter DCF77 using a pseudo-random phase-shift keying of the carrier
5782:
5442:
5438:
5229:
5193:
392:
6088:. Lecture Notes in Computer Science. Vol. 3621. Santa Barbara, California, USA. pp. 97–117.
1944:
5549:
5394:
4917:
4856:
4795:
4734:
4586:
4525:
4377:
6220:
https://web.archive.org/web/20161007061934/http://courses.cse.tamu.edu/csce680/walker/lfsr_table.pdf
6133:
6094:
4055:
3994:
6612:
5573:
systems to generate pseudo-random noise to raise the noise floor of a target communication system.
5477:
5211:
5181:
3933:
3785:
3724:
3663:
3602:
3541:
1752:
code example for a 16-bit maximal-period
Xorshift LFSR using the 7,9,13 triplet from John Metcalf:
1749:
1165:
414:
6083:
5603:
5593:
5291:
276:
242:
188:
166:
44:
6089:
5222:
5189:
3487:
3400:) for shift-register lengths up to 24. The formalism for maximum-length LFSRs was developed by
5506:
1707:(note that XOR is addition modulo 2), and the feedback bit (output bit) is multiplied (modulo-
6684:
6658:
6511:
3447:
265:
253:
6679:
5545:
5539:
5522:
1509:
385:
8:
6607:
5977:
5401:
communication. When used only for the spread-spectrum property, this technique is called
5377:
scrambling is removed at the receiver after demodulation. When the LFSR runs at the same
5257:
5218:
5173:
264:
The sequence of numbers generated by an LFSR or its XNOR counterpart can be considered a
181:
5500:
5199:
Three general methods are employed to reduce this problem in LFSR-based stream ciphers:
6674:
6121:
6051:
5759:
5581:
5354:
2061:
1700:
283:
177:
5755:
1978:
of the characteristic polynomial of the LFSR and denoting the seed as a column vector
1130:
904:
A sample python implementation of a similar (16 bit taps at ) Fibonacci LFSR would be
6107:
5956:
5931:
5786:
5763:
5717:
5707:
5533:
5177:
5169:
5149:
5145:
3401:
6272:
http://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/1999f/Drivers_Ed/lfsr.html
6224:
380:
GF(2). This means that the following conditions are necessary (but not sufficient):
6456:
6099:
6043:
5923:
5872:
5862:
5844:
5824:
5806:
5751:
5598:
3408:
grows exponentially with shift-register length and can be calculated exactly using
1975:
1742:
1738:
372:
The LFSR is maximal-length if and only if the corresponding feedback polynomial is
173:
6286:
6055:
6576:
6571:
6546:
6420:
6254:
6248:
6163:
5639:
5398:
690:
operation is available, the feedback bit can be computed more efficiently as the
683:
147:
6374:
5927:
5664:
5260:. The A5/2 cipher has been broken and both A5/1 and E0 have serious weaknesses.
6638:
6491:
6446:
6276:
5432:
5091:
143:
6047:
5777:
Press, William; Teukolsky, Saul; Vetterling, William; Flannery, Brian (2007).
1168:
code example for the 16-bit maximal-period Galois LFSR example in the figure:
6699:
6591:
6551:
6531:
6521:
6486:
6350:
5665:"Cyclic Redundancy Check Computation: An Implementation Using the TMS320C54x"
5629:
5570:
5185:
5165:
5141:
5098:
6296:
5721:
5184:
output streams. However, an LFSR is a linear system, leading to fairly easy
5061:
5634:
5619:
5417:
5390:
3396:
The following table lists examples of maximal-length feedback polynomials (
1716:
512:/* taps: 16 14 13 11; feedback polynomial: x^16 + x^14 + x^13 + x^11 + 1 */
377:
154:
5867:
5848:
5829:
5810:
5217:
Non-linear combination of the output bits of two or more LFSRs (see also:
5065:
6526:
6384:
6306:
6215:
http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf
6004:
http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf
5701:
5512:
5249:
5125:
3438:
896:. The alternative Galois configuration is described in the next section.
691:
248:
As an alternative to the XOR-based feedback in an LFSR, one can also use
209:
5509:, the most common form of Gigabit Ethernet, scrambles bits using an LFSR
5107:
The value consisting of all zeros cannot appear. Thus an LFSR of length
1687:
Binary Galois LFSRs like the ones shown above can be generalized to any
6653:
6230:
6103:
5413:
5305: in this section. Unsourced material may be challenged and removed.
5203:
5069:
1513:
771:
If a rotation operation is available, the new state can be computed as
280:
257:
6267:
Pseudo-Random Number
Generation Routine for the MAX765x Microprocessor
6029:"Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication"
5877:
5140:
The repeating sequence of states of an LFSR allows it to be used as a
6566:
6496:
6430:
5382:
5371:
5253:
957:# taps: 16 15 13 4; feedback polynomial: x^16 + x^15 + x^13 + x^4 + 1
763:
turns the popcnt into a true parity function, but the bitshift later
269:
204:
131:
6193:. 2nd European Frequency and Time Forum. Neuchâtel. pp. 351–364
5280:
5124:
radio. LFSRs have also been used for generating an approximation of
275:
The arrangement of taps for feedback in an LFSR can be expressed in
33:
6379:
5451:(digital TV transmission system – Europe, Australia, parts of Asia)
5381:
as the transmitted symbol stream, this technique is referred to as
5378:
3054:{\displaystyle a'_{i}=\sum _{i=0}^{j}a_{i-j}c_{n-j},\ 0\leq i<n}
1732:
687:
5360:
184:. Both hardware and software implementations of LFSRs are common.
6425:
6399:
6319:
5779:
Numerical
Recipes: The Art of Scientific Computing, Third Edition
5553:
5528:
5428:
Digital broadcasting systems that use linear-feedback registers:
1148:
To generate the same output stream, the order of the taps is the
395:; i.e., there must be no divisor other than 1 common to all taps.
19:"LFSR" redirects here. For the airport using that ICAO code, see
6301:
5776:
5263:
The linear feedback shift register has a strong relationship to
6516:
6481:
6451:
6415:
6541:
6536:
6227:— Tables of maximum length feedback polynomials for 2-64 bits.
6561:
5613:
5577:
5488:
5454:
5448:
2058:, the state of the register in Fibonacci configuration after
1971:
488:/* Must be 16-bit to allow bit<<15 later in the code */
165:
can produce a sequence of bits that appears random and has a
6153:
RFC 4086 section 6.1.3 "Traditional Pseudo-random
Sequences"
5148:
have simpler feedback logic than natural binary counters or
233:
The bit positions that affect the next state are called the
161:
eventually enter a repeating cycle. However, an LFSR with a
6581:
6556:
6506:
6501:
6369:
6364:
6315:
LFSR and
Intrinsic Generation of Randomness: Notes From NKS
6081:
5542:
Link Layer is making use of LFSR (referred to as whitening)
5517:
5471:
5241:
5237:
5090:
bits long. This distribution almost equals the statistical
3414:
2051:{\displaystyle (a_{0},a_{1},\dots ,a_{n-1})^{\mathrm {T} }}
1120:
249:
199:
6255:
International Telecommunication Union Recommendation O.151
5737:"Random Numbers Generated by Linear Recurrence Modulo Two"
1133:, an LFSR in Galois configuration, which is also known as
6389:
6233:— Code for generating maximal length feedback polynomials
5245:
5207:
153:
The most commonly used linear function of single bits is
3391:
5849:"Note on Marsaglia's Xorshift Random Number Generators"
5894:"16-Bit Xorshift Pseudorandom Numbers in Z80 Assembly"
3388:
These forms generalize naturally to arbitrary fields.
3273:
3080:
2756:
2652:
2472:
2369:
2195:
2093:
5918:
Klein, A. (2013). "Linear Feedback Shift Registers".
4981:
4920:
4859:
4798:
4737:
4650:
4589:
4528:
4441:
4380:
4293:
4206:
4119:
4058:
3997:
3936:
3849:
3788:
3727:
3666:
3605:
3544:
3490:
3450:
3073:
2960:
2750:
2087:
2064:
1984:
1947:
296:
5389:. The chipping code is combined with the data using
694:
of the register with the characteristic polynomial:
6174:
Section 9.5 of the SATA Specification, revision 2.6
5086: − 1 bits long, and a single run of ones
2741:Matrix for the corresponding Galois form is :
1523:
58:. Unsourced material may be challenged and removed.
5497:(Code Division Multiple Access) cellular telephony
5461:Other digital communications systems using LFSRs:
5039:
4952:
4891:
4830:
4769:
4708:
4621:
4560:
4499:
4412:
4351:
4264:
4177:
4090:
4029:
3968:
3907:
3820:
3759:
3698:
3637:
3576:
3515:
3469:
3363:
3053:
2940:
2730:
2070:
2050:
1962:
354:
6302:Simple VHDL coding for Galois and Fibonacci LFSR.
6225:http://users.ece.cmu.edu/~koopman/lfsr/index.html
6027:Barkam, Elad; Biham, Eli; Keller, Nathan (2008),
6026:
5689:Linear Feedback Shift Registers in Virtex Devices
6697:
6307:mlpolygen: A Maximal Length polynomial generator
5706:(2nd ed.). New York: Springer. p. 38.
5703:Random number generation and Monte Carlo methods
5435:(digital TV transmission system – North America)
3064:the top coefficient of the column vector :
5361:Uses in digital broadcasting and communications
5168:, due to the ease of construction from simple
406:-bit LFSR is , where the 0 corresponds to the
355:{\displaystyle x^{16}+x^{14}+x^{13}+x^{11}+1.}
6335:
5999:
5997:
5040:{\displaystyle x^{24}+x^{23}+x^{22}+x^{17}+1}
4709:{\displaystyle x^{19}+x^{18}+x^{17}+x^{14}+1}
6277:http://www.quadibloc.com/crypto/co040801.htm
5236:Important LFSR-based stream ciphers include
5190:known plaintext and corresponding ciphertext
4500:{\displaystyle x^{16}+x^{15}+x^{13}+x^{4}+1}
4352:{\displaystyle x^{14}+x^{13}+x^{12}+x^{2}+1}
4265:{\displaystyle x^{13}+x^{12}+x^{11}+x^{8}+1}
4178:{\displaystyle x^{12}+x^{11}+x^{10}+x^{4}+1}
5955:. Laguna Hills, Calif.: Aegean Park Press.
5580:, in addition to amplitude keying, employs
5075:
6342:
6328:
5994:
5734:
5336:
5270:
5228:Irregular clocking of the LFSR, as in the
3404:in his 1967 book. The number of different
6282:Simple explanation of LFSRs for Engineers
6093:
5876:
5866:
5828:
5805:
5420:; scrambling and spreading with LFSRs do
5321:Learn how and when to remove this message
3908:{\displaystyle x^{8}+x^{6}+x^{5}+x^{4}+1}
1950:
1682:
884:This LFSR configuration is also known as
172:Applications of LFSRs include generating
118:Learn how and when to remove this message
6177:
5799:
5111:cannot be used to generate all 2 values.
1796:/* Any nonzero start state will work. */
1212:/* Any nonzero start state will work. */
1119:
461:/* Any nonzero start state will work. */
215:
198:
6139:CS1 maint: location missing publisher (
5891:
5480:(Serial Digital Interface transmission)
5412:Neither scheme should be confused with
5155:
5066:http://users.ece.cmu.edu/~koopman/lfsr/
6698:
6183:
5950:
5699:
6323:
5975:
5917:
5843:
5457:(digital audio system for television)
5348:
3392:Example polynomials for maximal LFSRs
1356:/* Get MSB (i.e., the output bit). */
1275:/* Get LSB (i.e., the output bit). */
1129:Named after the French mathematician
6231:https://github.com/hayguen/mlpolygen
6085:Advances in Cryptology – CRYPTO 2005
5922:. London: Springer. pp. 17–18.
5837:
5735:Tausworthe, Robert C. (April 1965).
5616:, Non-Linear Feedback Shift Register
5303:adding citations to reliable sources
5274:
5070:https://github.com/hayguen/mlpolygen
899:
56:adding citations to reliable sources
27:
5662:
5135:
1695: − 1} (e.g., for binary,
16:Type of shift register in computing
13:
6349:
6297:An implementation of LFSR in VHDL.
6208:
5558:frequency-division multiple access
5188:. For example, given a stretch of
2042:
1157:increasing the speed of execution.
241:A maximum-length LFSR produces an
194:
14:
6732:
6237:
5756:10.1090/S0025-5718-1965-0184406-1
1726:
5279:
2951:For a suitable initialisation,
1963:{\displaystyle \mathbb {F} _{2}}
1524:Galois LFSR parallel computation
67:"Linear-feedback shift register"
32:
6245:Linear Feedback Shift Registers
6168:
6156:
6147:
6075:
6020:
6008:
5969:
5944:
5854:Journal of Statistical Software
5816:Journal of Statistical Software
5465:Intelsat business service (IBS)
5403:direct-sequence spread spectrum
5290:needs additional citations for
5162:pseudo-random number generators
5122:direct-sequence spread spectrum
5115:
4953:{\displaystyle x^{23}+x^{18}+1}
4892:{\displaystyle x^{22}+x^{21}+1}
4831:{\displaystyle x^{21}+x^{19}+1}
4770:{\displaystyle x^{20}+x^{17}+1}
4622:{\displaystyle x^{18}+x^{11}+1}
4561:{\displaystyle x^{17}+x^{14}+1}
4413:{\displaystyle x^{15}+x^{14}+1}
1936:
1115:
43:needs additional citations for
6721:Pseudorandom number generators
5911:
5892:Metcalf, John (22 July 2017).
5885:
5770:
5728:
5693:
5682:
5670:. Texas Instruments. p. 6
5656:
5609:Analog feedback shift register
5265:linear congruential generators
4091:{\displaystyle x^{11}+x^{9}+1}
4030:{\displaystyle x^{10}+x^{7}+1}
2037:
1985:
1719:for arbitrary prime values of
767:makes higher bits irrelevant.)
136:linear-feedback shift register
1:
5650:
5625:Pseudo-random binary sequence
5564:
5407:code-division multiple access
5365:
5160:LFSRs have long been used as
5130:programmable sound generators
3969:{\displaystyle x^{9}+x^{5}+1}
3821:{\displaystyle x^{7}+x^{6}+1}
3760:{\displaystyle x^{6}+x^{5}+1}
3699:{\displaystyle x^{5}+x^{3}+1}
3638:{\displaystyle x^{4}+x^{3}+1}
3577:{\displaystyle x^{3}+x^{2}+1}
1386:/* If the output bit is 1, */
1305:/* If the output bit is 1, */
1125:will be followed by E270 hex.
212:) will be followed by 0x5670.
180:, fast digital counters, and
163:well-chosen feedback function
6634:block ciphers in stream mode
6184:Hetzel, P. (16 March 1988).
5546:Satellite navigation systems
5536:scrambles bits using an LFSR
5503:scrambles bits using an LFSR
5468:Intermediate data rate (IDR)
5068:and can be generated by the
272:or the natural binary code.
7:
5951:Golomb, Solomon W. (1967).
5928:10.1007/978-1-4471-5079-4_2
5587:
5225:to introduce non-linearity.
10:
6737:
6618:alternating step generator
5783:Cambridge University Press
5744:Mathematics of Computation
5645:Berlekamp–Massey algorithm
5501:100BASE-T2 "fast" Ethernet
5443:Digital Audio Broadcasting
5393:before transmitting using
5369:
5230:alternating step generator
5194:Berlekamp-Massey algorithm
5082:to a single run of zeroes
3385:of the original sequence.
1730:
1691:-ary alphabet {0, 1, ...,
21:Reims – Champagne Air Base
18:
6667:
6626:
6600:
6469:
6439:
6408:
6398:
6357:
6263:with length from 2 to 67.
6261:Maximal Length LFSR table
6048:10.1007/s00145-007-9001-y
5700:Gentle, James E. (2003).
5491:V-series recommendations)
5395:binary phase-shift keying
5049:111000010000000000000000
3516:{\displaystyle x^{2}+x+1}
1479:
1452:
1401:/* apply toggle mask. */
1320:/* apply toggle mask. */
831:
775:
727:
698:
6716:Cryptographic algorithms
6613:self-shrinking generator
5953:Shift register sequences
5097:LFSR output streams are
5076:Output-stream properties
4962:10000100000000000000000
3410:Euler's totient function
1754:
1741:and further analysed by
1530:
1170:
906:
419:
5604:Maximum length sequence
5576:The German time signal
5569:LFSRs are also used in
5337:Test-pattern generation
5271:Uses in circuit testing
5206:combination of several
4901:1100000000000000000000
3470:{\displaystyle 2^{n}-1}
1478:can also be written as
277:finite field arithmetic
189:cyclic redundancy check
150:of its previous state.
5978:"Primitive Polynomial"
5223:Evolutionary algorithm
5041:
4954:
4893:
4840:101000000000000000000
4832:
4771:
4710:
4623:
4562:
4501:
4414:
4353:
4266:
4179:
4092:
4031:
3970:
3909:
3822:
3761:
3700:
3639:
3578:
3517:
3471:
3365:
3055:
2997:
2942:
2732:
2072:
2052:
1964:
1683:Non-binary Galois LFSR
1126:
384:The number of taps is
356:
252:. This function is an
230:
213:
178:pseudo-noise sequences
6685:stream cipher attacks
6036:Journal of Cryptology
5982:mathworld.wolfram.com
5868:10.18637/jss.v011.i05
5830:10.18637/jss.v008.i14
5180:, and very uniformly
5042:
4955:
4894:
4833:
4779:10010000000000000000
4772:
4711:
4624:
4563:
4502:
4415:
4354:
4267:
4180:
4093:
4032:
3971:
3910:
3823:
3762:
3701:
3640:
3579:
3518:
3472:
3406:primitive polynomials
3398:primitive polynomials
3366:
3056:
2977:
2943:
2733:
2073:
2053:
1965:
1516:from the addition of
1123:
357:
266:binary numeral system
228:
202:
187:The mathematics of a
174:pseudo-random numbers
146:whose input bit is a
6680:correlation immunity
5540:Bluetooth Low Energy
5523:Serial Attached SCSI
5299:improve this article
5156:Uses in cryptography
4979:
4918:
4857:
4796:
4735:
4718:1110010000000000000
4648:
4587:
4526:
4439:
4378:
4291:
4204:
4117:
4056:
3995:
3934:
3847:
3786:
3725:
3664:
3603:
3542:
3488:
3448:
3431:Feedback polynomial
3071:
2958:
2748:
2085:
2062:
1982:
1945:
1371:/* Shift register */
1290:/* Shift register */
294:
52:improve this article
6608:shrinking generator
6358:Widely used ciphers
6292:General LFSR Theory
5976:Weisstein, Eric W.
5560:combined with DSSS.
5507:1000BASE-T Ethernet
5483:Data transfer over
5445:system – for radio)
5258:shrinking generator
5219:shrinking generator
5174:electronic circuits
4631:100000010000000000
3352:
3322:
3305:
3288:
2973:
391:The set of taps is
182:whitening sequences
6675:correlation attack
6104:10.1007/11535218_7
5663:Geremia, Patrick.
5582:phase-shift keying
5487:(according to the
5355:built-in self-test
5349:Signature analysis
5150:Gray-code counters
5037:
4950:
4889:
4828:
4767:
4706:
4619:
4570:10010000000000000
4558:
4497:
4410:
4349:
4262:
4175:
4088:
4027:
3966:
3905:
3818:
3757:
3696:
3635:
3574:
3513:
3467:
3361:
3355:
3334:
3310:
3293:
3276:
3256:
3051:
2961:
2938:
2932:
2728:
2722:
2635:
2457:
2358:
2181:
2078:steps is given by
2068:
2048:
1960:
1127:
894:external XOR gates
352:
231:
214:
6711:Digital registers
6706:Binary arithmetic
6693:
6692:
6465:
6464:
6113:978-3-540-28114-6
5937:978-1-4471-5079-4
5898:Retro Programming
5845:Brent, Richard P.
5807:Marsaglia, George
5792:978-0-521-88407-5
5331:
5330:
5323:
5170:electromechanical
5092:expectation value
5059:
5058:
4509:1101000000001000
3402:Solomon W. Golomb
3035:
2071:{\displaystyle k}
900:Example in Python
828:, or equivalently
724:, or equivalently
268:just as valid as
256:, not strictly a
226:
128:
127:
120:
102:
6728:
6406:
6405:
6344:
6337:
6330:
6321:
6320:
6203:
6202:
6200:
6198:
6192:
6181:
6175:
6172:
6166:
6160:
6154:
6151:
6145:
6144:
6137:
6131:
6127:
6125:
6117:
6097:
6079:
6073:
6072:
6071:
6070:
6064:
6058:, archived from
6033:
6024:
6018:
6012:
6006:
6001:
5992:
5991:
5989:
5988:
5973:
5967:
5966:
5948:
5942:
5941:
5915:
5909:
5908:
5906:
5904:
5889:
5883:
5882:
5880:
5870:
5841:
5835:
5834:
5832:
5803:
5797:
5796:
5774:
5768:
5767:
5741:
5732:
5726:
5725:
5697:
5691:
5686:
5680:
5679:
5677:
5675:
5669:
5660:
5599:Mersenne twister
5326:
5319:
5315:
5312:
5306:
5283:
5275:
5136:Uses as counters
5062:Xilinx published
5046:
5044:
5043:
5038:
5030:
5029:
5017:
5016:
5004:
5003:
4991:
4990:
4959:
4957:
4956:
4951:
4943:
4942:
4930:
4929:
4898:
4896:
4895:
4890:
4882:
4881:
4869:
4868:
4837:
4835:
4834:
4829:
4821:
4820:
4808:
4807:
4776:
4774:
4773:
4768:
4760:
4759:
4747:
4746:
4715:
4713:
4712:
4707:
4699:
4698:
4686:
4685:
4673:
4672:
4660:
4659:
4628:
4626:
4625:
4620:
4612:
4611:
4599:
4598:
4567:
4565:
4564:
4559:
4551:
4550:
4538:
4537:
4506:
4504:
4503:
4498:
4490:
4489:
4477:
4476:
4464:
4463:
4451:
4450:
4422:110000000000000
4419:
4417:
4416:
4411:
4403:
4402:
4390:
4389:
4358:
4356:
4355:
4350:
4342:
4341:
4329:
4328:
4316:
4315:
4303:
4302:
4271:
4269:
4268:
4263:
4255:
4254:
4242:
4241:
4229:
4228:
4216:
4215:
4184:
4182:
4181:
4176:
4168:
4167:
4155:
4154:
4142:
4141:
4129:
4128:
4097:
4095:
4094:
4089:
4081:
4080:
4068:
4067:
4036:
4034:
4033:
4028:
4020:
4019:
4007:
4006:
3975:
3973:
3972:
3967:
3959:
3958:
3946:
3945:
3914:
3912:
3911:
3906:
3898:
3897:
3885:
3884:
3872:
3871:
3859:
3858:
3827:
3825:
3824:
3819:
3811:
3810:
3798:
3797:
3766:
3764:
3763:
3758:
3750:
3749:
3737:
3736:
3705:
3703:
3702:
3697:
3689:
3688:
3676:
3675:
3644:
3642:
3641:
3636:
3628:
3627:
3615:
3614:
3583:
3581:
3580:
3575:
3567:
3566:
3554:
3553:
3522:
3520:
3519:
3514:
3500:
3499:
3476:
3474:
3473:
3468:
3460:
3459:
3425:
3424:
3417:
3384:
3370:
3368:
3367:
3362:
3360:
3359:
3348:
3318:
3301:
3284:
3267:
3266:
3261:
3260:
3233:
3232:
3193:
3192:
3126:
3125:
3092:
3091:
3060:
3058:
3057:
3052:
3033:
3029:
3028:
3013:
3012:
2996:
2991:
2969:
2947:
2945:
2944:
2939:
2937:
2936:
2909:
2908:
2869:
2868:
2802:
2801:
2768:
2767:
2737:
2735:
2734:
2729:
2727:
2726:
2719:
2718:
2692:
2691:
2678:
2677:
2664:
2663:
2646:
2645:
2640:
2639:
2632:
2631:
2604:
2603:
2592:
2591:
2462:
2461:
2454:
2453:
2421:
2420:
2401:
2400:
2387:
2386:
2363:
2362:
2355:
2354:
2327:
2326:
2315:
2314:
2186:
2185:
2178:
2177:
2145:
2144:
2125:
2124:
2105:
2104:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2049:
2047:
2046:
2045:
2035:
2034:
2010:
2009:
1997:
1996:
1976:companion matrix
1969:
1967:
1966:
1961:
1959:
1958:
1953:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1760:<stdint.h>
1758:
1743:Richard P. Brent
1739:George Marsaglia
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1536:<stdint.h>
1534:
1519:
1507:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1477:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1176:<stdint.h>
1174:
1143:one-to-many LFSR
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
880:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
827:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
766:
762:
758:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
723:
722:
719:
716:
713:
710:
707:
704:
701:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
425:<stdint.h>
423:
393:setwise co-prime
361:
359:
358:
353:
345:
344:
332:
331:
319:
318:
306:
305:
227:
123:
116:
112:
109:
103:
101:
60:
36:
28:
6736:
6735:
6731:
6730:
6729:
6727:
6726:
6725:
6696:
6695:
6694:
6689:
6663:
6622:
6596:
6461:
6435:
6394:
6353:
6348:
6311:
6249:Wayback Machine
6240:
6211:
6209:Further reading
6206:
6196:
6194:
6190:
6182:
6178:
6173:
6169:
6161:
6157:
6152:
6148:
6138:
6129:
6128:
6119:
6118:
6114:
6095:10.1.1.323.9416
6080:
6076:
6068:
6066:
6062:
6031:
6025:
6021:
6013:
6009:
6002:
5995:
5986:
5984:
5974:
5970:
5963:
5949:
5945:
5938:
5916:
5912:
5902:
5900:
5890:
5886:
5847:(August 2004).
5842:
5838:
5811:"Xorshift RNGs"
5804:
5800:
5793:
5785:. p. 386.
5775:
5771:
5750:(90): 201–209.
5739:
5733:
5729:
5714:
5698:
5694:
5687:
5683:
5673:
5671:
5667:
5661:
5657:
5653:
5640:Kasami sequence
5590:
5567:
5399:spread-spectrum
5374:
5368:
5363:
5351:
5339:
5327:
5316:
5310:
5307:
5296:
5284:
5273:
5158:
5138:
5118:
5078:
5025:
5021:
5012:
5008:
4999:
4995:
4986:
4982:
4980:
4977:
4976:
4938:
4934:
4925:
4921:
4919:
4916:
4915:
4877:
4873:
4864:
4860:
4858:
4855:
4854:
4816:
4812:
4803:
4799:
4797:
4794:
4793:
4755:
4751:
4742:
4738:
4736:
4733:
4732:
4694:
4690:
4681:
4677:
4668:
4664:
4655:
4651:
4649:
4646:
4645:
4607:
4603:
4594:
4590:
4588:
4585:
4584:
4546:
4542:
4533:
4529:
4527:
4524:
4523:
4485:
4481:
4472:
4468:
4459:
4455:
4446:
4442:
4440:
4437:
4436:
4398:
4394:
4385:
4381:
4379:
4376:
4375:
4361:11100000000010
4337:
4333:
4324:
4320:
4311:
4307:
4298:
4294:
4292:
4289:
4288:
4250:
4246:
4237:
4233:
4224:
4220:
4211:
4207:
4205:
4202:
4201:
4163:
4159:
4150:
4146:
4137:
4133:
4124:
4120:
4118:
4115:
4114:
4076:
4072:
4063:
4059:
4057:
4054:
4053:
4015:
4011:
4002:
3998:
3996:
3993:
3992:
3954:
3950:
3941:
3937:
3935:
3932:
3931:
3893:
3889:
3880:
3876:
3867:
3863:
3854:
3850:
3848:
3845:
3844:
3806:
3802:
3793:
3789:
3787:
3784:
3783:
3745:
3741:
3732:
3728:
3726:
3723:
3722:
3684:
3680:
3671:
3667:
3665:
3662:
3661:
3623:
3619:
3610:
3606:
3604:
3601:
3600:
3562:
3558:
3549:
3545:
3543:
3540:
3539:
3495:
3491:
3489:
3486:
3485:
3455:
3451:
3449:
3446:
3445:
3413:
3394:
3383:
3375:
3374:gives the term
3354:
3353:
3338:
3331:
3330:
3324:
3323:
3314:
3307:
3306:
3297:
3290:
3289:
3280:
3269:
3268:
3262:
3255:
3254:
3249:
3244:
3239:
3234:
3222:
3218:
3215:
3214:
3209:
3204:
3199:
3194:
3182:
3178:
3175:
3174:
3169:
3164:
3159:
3154:
3148:
3147:
3142:
3137:
3132:
3127:
3121:
3117:
3114:
3113:
3108:
3103:
3098:
3093:
3087:
3083:
3076:
3075:
3074:
3072:
3069:
3068:
3018:
3014:
3002:
2998:
2992:
2981:
2965:
2959:
2956:
2955:
2931:
2930:
2925:
2920:
2915:
2910:
2898:
2894:
2891:
2890:
2885:
2880:
2875:
2870:
2858:
2854:
2851:
2850:
2845:
2840:
2835:
2830:
2824:
2823:
2818:
2813:
2808:
2803:
2797:
2793:
2790:
2789:
2784:
2779:
2774:
2769:
2763:
2759:
2752:
2751:
2749:
2746:
2745:
2721:
2720:
2708:
2704:
2701:
2700:
2694:
2693:
2687:
2683:
2680:
2679:
2673:
2669:
2666:
2665:
2659:
2655:
2648:
2647:
2641:
2634:
2633:
2621:
2617:
2615:
2610:
2605:
2599:
2595:
2593:
2587:
2583:
2580:
2579:
2574:
2569:
2564:
2559:
2553:
2552:
2547:
2542:
2537:
2532:
2526:
2525:
2520:
2515:
2510:
2505:
2499:
2498:
2493:
2488:
2483:
2478:
2468:
2467:
2466:
2456:
2455:
2437:
2433:
2430:
2429:
2423:
2422:
2410:
2406:
2403:
2402:
2396:
2392:
2389:
2388:
2376:
2372:
2365:
2364:
2357:
2356:
2344:
2340:
2338:
2333:
2328:
2322:
2318:
2316:
2310:
2306:
2303:
2302:
2297:
2292:
2287:
2282:
2276:
2275:
2270:
2265:
2260:
2255:
2249:
2248:
2243:
2238:
2233:
2228:
2222:
2221:
2216:
2211:
2206:
2201:
2191:
2190:
2180:
2179:
2161:
2157:
2154:
2153:
2147:
2146:
2134:
2130:
2127:
2126:
2114:
2110:
2107:
2106:
2100:
2096:
2089:
2088:
2086:
2083:
2082:
2063:
2060:
2059:
2041:
2040:
2036:
2024:
2020:
2005:
2001:
1992:
1988:
1983:
1980:
1979:
1954:
1949:
1948:
1946:
1943:
1942:
1939:
1934:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1735:
1729:
1685:
1680:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1526:
1517:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1449:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1131:Évariste Galois
1118:
1110:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
902:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
765:bit << 15
764:
760:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
720:
717:
714:
711:
708:
705:
702:
699:
680:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
340:
336:
327:
323:
314:
310:
301:
297:
295:
292:
291:
216:
197:
195:Fibonacci LFSRs
167:very long cycle
148:linear function
124:
113:
107:
104:
61:
59:
49:
37:
24:
17:
12:
11:
5:
6734:
6724:
6723:
6718:
6713:
6708:
6691:
6690:
6688:
6687:
6682:
6677:
6671:
6669:
6665:
6664:
6662:
6661:
6656:
6651:
6646:
6641:
6639:shift register
6636:
6630:
6628:
6624:
6623:
6621:
6620:
6615:
6610:
6604:
6602:
6598:
6597:
6595:
6594:
6589:
6584:
6579:
6574:
6569:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6524:
6519:
6514:
6509:
6504:
6499:
6494:
6489:
6484:
6479:
6473:
6471:
6467:
6466:
6463:
6462:
6460:
6459:
6454:
6449:
6443:
6441:
6437:
6436:
6434:
6433:
6428:
6423:
6418:
6412:
6410:
6403:
6396:
6395:
6393:
6392:
6387:
6382:
6377:
6372:
6367:
6361:
6359:
6355:
6354:
6351:Stream ciphers
6347:
6346:
6339:
6332:
6324:
6318:
6317:
6310:
6309:
6304:
6299:
6294:
6289:
6287:Feedback terms
6284:
6279:
6274:
6269:
6264:
6258:
6252:
6241:
6239:
6238:External links
6236:
6235:
6234:
6228:
6222:
6217:
6210:
6207:
6205:
6204:
6176:
6167:
6164:UoL repository
6155:
6146:
6130:|journal=
6112:
6074:
6042:(3): 392–429,
6019:
6007:
5993:
5968:
5962:978-0894120480
5961:
5943:
5936:
5920:Stream Ciphers
5910:
5884:
5836:
5798:
5791:
5769:
5727:
5712:
5692:
5681:
5654:
5652:
5649:
5648:
5647:
5642:
5637:
5632:
5627:
5622:
5617:
5611:
5606:
5601:
5596:
5589:
5586:
5566:
5563:
5562:
5561:
5543:
5537:
5531:
5526:
5520:
5515:
5510:
5504:
5498:
5492:
5481:
5475:
5469:
5466:
5459:
5458:
5452:
5446:
5436:
5433:ATSC Standards
5370:Main article:
5367:
5364:
5362:
5359:
5350:
5347:
5338:
5335:
5329:
5328:
5287:
5285:
5278:
5272:
5269:
5234:
5233:
5226:
5215:
5210:from the LFSR
5166:stream ciphers
5157:
5154:
5137:
5134:
5117:
5114:
5113:
5112:
5105:
5102:
5095:
5077:
5074:
5057:
5056:
5053:
5050:
5047:
5036:
5033:
5028:
5024:
5020:
5015:
5011:
5007:
5002:
4998:
4994:
4989:
4985:
4974:
4970:
4969:
4966:
4963:
4960:
4949:
4946:
4941:
4937:
4933:
4928:
4924:
4913:
4909:
4908:
4905:
4902:
4899:
4888:
4885:
4880:
4876:
4872:
4867:
4863:
4852:
4848:
4847:
4844:
4841:
4838:
4827:
4824:
4819:
4815:
4811:
4806:
4802:
4791:
4787:
4786:
4783:
4780:
4777:
4766:
4763:
4758:
4754:
4750:
4745:
4741:
4730:
4726:
4725:
4722:
4719:
4716:
4705:
4702:
4697:
4693:
4689:
4684:
4680:
4676:
4671:
4667:
4663:
4658:
4654:
4643:
4639:
4638:
4635:
4632:
4629:
4618:
4615:
4610:
4606:
4602:
4597:
4593:
4582:
4578:
4577:
4574:
4571:
4568:
4557:
4554:
4549:
4545:
4541:
4536:
4532:
4521:
4517:
4516:
4513:
4510:
4507:
4496:
4493:
4488:
4484:
4480:
4475:
4471:
4467:
4462:
4458:
4454:
4449:
4445:
4434:
4430:
4429:
4426:
4423:
4420:
4409:
4406:
4401:
4397:
4393:
4388:
4384:
4373:
4369:
4368:
4365:
4362:
4359:
4348:
4345:
4340:
4336:
4332:
4327:
4323:
4319:
4314:
4310:
4306:
4301:
4297:
4286:
4282:
4281:
4278:
4275:
4274:1110010000000
4272:
4261:
4258:
4253:
4249:
4245:
4240:
4236:
4232:
4227:
4223:
4219:
4214:
4210:
4199:
4195:
4194:
4191:
4188:
4185:
4174:
4171:
4166:
4162:
4158:
4153:
4149:
4145:
4140:
4136:
4132:
4127:
4123:
4112:
4108:
4107:
4104:
4101:
4098:
4087:
4084:
4079:
4075:
4071:
4066:
4062:
4051:
4047:
4046:
4043:
4040:
4037:
4026:
4023:
4018:
4014:
4010:
4005:
4001:
3990:
3986:
3985:
3982:
3979:
3976:
3965:
3962:
3957:
3953:
3949:
3944:
3940:
3929:
3925:
3924:
3921:
3918:
3915:
3904:
3901:
3896:
3892:
3888:
3883:
3879:
3875:
3870:
3866:
3862:
3857:
3853:
3842:
3838:
3837:
3834:
3831:
3828:
3817:
3814:
3809:
3805:
3801:
3796:
3792:
3781:
3777:
3776:
3773:
3770:
3767:
3756:
3753:
3748:
3744:
3740:
3735:
3731:
3720:
3716:
3715:
3712:
3709:
3706:
3695:
3692:
3687:
3683:
3679:
3674:
3670:
3659:
3655:
3654:
3651:
3648:
3645:
3634:
3631:
3626:
3622:
3618:
3613:
3609:
3598:
3594:
3593:
3590:
3587:
3584:
3573:
3570:
3565:
3561:
3557:
3552:
3548:
3537:
3533:
3532:
3529:
3526:
3523:
3512:
3509:
3506:
3503:
3498:
3494:
3483:
3479:
3478:
3466:
3463:
3458:
3454:
3442:
3435:
3432:
3429:
3393:
3390:
3379:
3372:
3371:
3358:
3351:
3347:
3344:
3341:
3337:
3333:
3332:
3329:
3326:
3325:
3321:
3317:
3313:
3309:
3308:
3304:
3300:
3296:
3292:
3291:
3287:
3283:
3279:
3275:
3274:
3272:
3265:
3259:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3231:
3228:
3225:
3221:
3217:
3216:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3191:
3188:
3185:
3181:
3177:
3176:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3149:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3124:
3120:
3116:
3115:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3090:
3086:
3082:
3081:
3079:
3062:
3061:
3050:
3047:
3044:
3041:
3038:
3032:
3027:
3024:
3021:
3017:
3011:
3008:
3005:
3001:
2995:
2990:
2987:
2984:
2980:
2976:
2972:
2968:
2964:
2949:
2948:
2935:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2907:
2904:
2901:
2897:
2893:
2892:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2867:
2864:
2861:
2857:
2853:
2852:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2825:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2800:
2796:
2792:
2791:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2766:
2762:
2758:
2757:
2755:
2739:
2738:
2725:
2717:
2714:
2711:
2707:
2703:
2702:
2699:
2696:
2695:
2690:
2686:
2682:
2681:
2676:
2672:
2668:
2667:
2662:
2658:
2654:
2653:
2651:
2644:
2638:
2630:
2627:
2624:
2620:
2616:
2614:
2611:
2609:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2581:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2554:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2527:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2500:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2473:
2471:
2465:
2460:
2452:
2449:
2446:
2443:
2440:
2436:
2432:
2431:
2428:
2425:
2424:
2419:
2416:
2413:
2409:
2405:
2404:
2399:
2395:
2391:
2390:
2385:
2382:
2379:
2375:
2371:
2370:
2368:
2361:
2353:
2350:
2347:
2343:
2339:
2337:
2334:
2332:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2304:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2277:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2250:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2223:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2194:
2189:
2184:
2176:
2173:
2170:
2167:
2164:
2160:
2156:
2155:
2152:
2149:
2148:
2143:
2140:
2137:
2133:
2129:
2128:
2123:
2120:
2117:
2113:
2109:
2108:
2103:
2099:
2095:
2094:
2092:
2067:
2044:
2039:
2033:
2030:
2027:
2023:
2019:
2016:
2013:
2008:
2004:
2000:
1995:
1991:
1987:
1957:
1952:
1938:
1935:
1755:
1731:Main article:
1728:
1727:Xorshift LFSRs
1725:
1684:
1681:
1531:
1525:
1522:
1171:
1162:
1161:
1158:
1117:
1114:
907:
901:
898:
882:
881:
829:
769:
768:
753:/* & 1u */
725:
420:
413:An example in
397:
396:
389:
363:
362:
351:
348:
343:
339:
335:
330:
326:
322:
317:
313:
309:
304:
300:
262:
261:
246:
196:
193:
144:shift register
126:
125:
40:
38:
31:
15:
9:
6:
4:
3:
2:
6733:
6722:
6719:
6717:
6714:
6712:
6709:
6707:
6704:
6703:
6701:
6686:
6683:
6681:
6678:
6676:
6673:
6672:
6670:
6666:
6660:
6657:
6655:
6652:
6650:
6647:
6645:
6642:
6640:
6637:
6635:
6632:
6631:
6629:
6625:
6619:
6616:
6614:
6611:
6609:
6606:
6605:
6603:
6599:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6523:
6520:
6518:
6515:
6513:
6510:
6508:
6505:
6503:
6500:
6498:
6495:
6493:
6490:
6488:
6485:
6483:
6480:
6478:
6475:
6474:
6472:
6470:Other ciphers
6468:
6458:
6455:
6453:
6450:
6448:
6445:
6444:
6442:
6438:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6413:
6411:
6407:
6404:
6401:
6397:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6362:
6360:
6356:
6352:
6345:
6340:
6338:
6333:
6331:
6326:
6325:
6322:
6316:
6313:
6312:
6308:
6305:
6303:
6300:
6298:
6295:
6293:
6290:
6288:
6285:
6283:
6280:
6278:
6275:
6273:
6270:
6268:
6265:
6262:
6259:
6257:(August 1992)
6256:
6253:
6250:
6246:
6243:
6242:
6232:
6229:
6226:
6223:
6221:
6218:
6216:
6213:
6212:
6189:
6188:
6180:
6171:
6165:
6159:
6150:
6142:
6135:
6123:
6115:
6109:
6105:
6101:
6096:
6091:
6087:
6086:
6078:
6065:on 2020-01-25
6061:
6057:
6053:
6049:
6045:
6041:
6037:
6030:
6023:
6017:
6011:
6005:
6000:
5998:
5983:
5979:
5972:
5964:
5958:
5954:
5947:
5939:
5933:
5929:
5925:
5921:
5914:
5899:
5895:
5888:
5879:
5874:
5869:
5864:
5860:
5856:
5855:
5850:
5846:
5840:
5831:
5826:
5822:
5818:
5817:
5812:
5809:(July 2003).
5808:
5802:
5794:
5788:
5784:
5780:
5773:
5765:
5761:
5757:
5753:
5749:
5745:
5738:
5731:
5723:
5719:
5715:
5713:0-387-00178-6
5709:
5705:
5704:
5696:
5690:
5685:
5666:
5659:
5655:
5646:
5643:
5641:
5638:
5636:
5633:
5631:
5630:Gold sequence
5628:
5626:
5623:
5621:
5618:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5591:
5585:
5583:
5579:
5574:
5572:
5571:radio jamming
5559:
5555:
5551:
5547:
5544:
5541:
5538:
5535:
5532:
5530:
5527:
5524:
5521:
5519:
5516:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5486:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5463:
5462:
5456:
5453:
5450:
5447:
5444:
5440:
5437:
5434:
5431:
5430:
5429:
5426:
5423:
5419:
5415:
5410:
5408:
5404:
5400:
5396:
5392:
5388:
5387:chipping code
5384:
5380:
5373:
5358:
5356:
5346:
5344:
5334:
5325:
5322:
5314:
5311:November 2022
5304:
5300:
5294:
5293:
5288:This section
5286:
5282:
5277:
5276:
5268:
5266:
5261:
5259:
5255:
5251:
5248:cell phones,
5247:
5243:
5239:
5231:
5227:
5224:
5220:
5216:
5213:
5209:
5205:
5202:
5201:
5200:
5197:
5195:
5191:
5187:
5186:cryptanalysis
5183:
5179:
5175:
5171:
5167:
5163:
5153:
5151:
5147:
5143:
5142:clock divider
5133:
5131:
5127:
5123:
5110:
5106:
5103:
5100:
5099:deterministic
5096:
5093:
5089:
5085:
5080:
5079:
5073:
5071:
5067:
5063:
5054:
5051:
5048:
5034:
5031:
5026:
5022:
5018:
5013:
5009:
5005:
5000:
4996:
4992:
4987:
4983:
4975:
4972:
4971:
4967:
4964:
4961:
4947:
4944:
4939:
4935:
4931:
4926:
4922:
4914:
4911:
4910:
4906:
4903:
4900:
4886:
4883:
4878:
4874:
4870:
4865:
4861:
4853:
4850:
4849:
4845:
4842:
4839:
4825:
4822:
4817:
4813:
4809:
4804:
4800:
4792:
4789:
4788:
4784:
4781:
4778:
4764:
4761:
4756:
4752:
4748:
4743:
4739:
4731:
4728:
4727:
4723:
4720:
4717:
4703:
4700:
4695:
4691:
4687:
4682:
4678:
4674:
4669:
4665:
4661:
4656:
4652:
4644:
4641:
4640:
4636:
4633:
4630:
4616:
4613:
4608:
4604:
4600:
4595:
4591:
4583:
4580:
4579:
4575:
4572:
4569:
4555:
4552:
4547:
4543:
4539:
4534:
4530:
4522:
4519:
4518:
4514:
4511:
4508:
4494:
4491:
4486:
4482:
4478:
4473:
4469:
4465:
4460:
4456:
4452:
4447:
4443:
4435:
4432:
4431:
4427:
4424:
4421:
4407:
4404:
4399:
4395:
4391:
4386:
4382:
4374:
4371:
4370:
4366:
4363:
4360:
4346:
4343:
4338:
4334:
4330:
4325:
4321:
4317:
4312:
4308:
4304:
4299:
4295:
4287:
4284:
4283:
4279:
4276:
4273:
4259:
4256:
4251:
4247:
4243:
4238:
4234:
4230:
4225:
4221:
4217:
4212:
4208:
4200:
4197:
4196:
4192:
4189:
4187:111000001000
4186:
4172:
4169:
4164:
4160:
4156:
4151:
4147:
4143:
4138:
4134:
4130:
4125:
4121:
4113:
4110:
4109:
4105:
4102:
4099:
4085:
4082:
4077:
4073:
4069:
4064:
4060:
4052:
4049:
4048:
4044:
4041:
4038:
4024:
4021:
4016:
4012:
4008:
4003:
3999:
3991:
3988:
3987:
3983:
3980:
3977:
3963:
3960:
3955:
3951:
3947:
3942:
3938:
3930:
3927:
3926:
3922:
3919:
3916:
3902:
3899:
3894:
3890:
3886:
3881:
3877:
3873:
3868:
3864:
3860:
3855:
3851:
3843:
3840:
3839:
3835:
3832:
3829:
3815:
3812:
3807:
3803:
3799:
3794:
3790:
3782:
3779:
3778:
3774:
3771:
3768:
3754:
3751:
3746:
3742:
3738:
3733:
3729:
3721:
3718:
3717:
3713:
3710:
3707:
3693:
3690:
3685:
3681:
3677:
3672:
3668:
3660:
3657:
3656:
3652:
3649:
3646:
3632:
3629:
3624:
3620:
3616:
3611:
3607:
3599:
3596:
3595:
3591:
3588:
3585:
3571:
3568:
3563:
3559:
3555:
3550:
3546:
3538:
3535:
3534:
3530:
3527:
3524:
3510:
3507:
3504:
3501:
3496:
3492:
3484:
3481:
3480:
3464:
3461:
3456:
3452:
3443:
3440:
3436:
3433:
3430:
3427:
3426:
3423:
3421:
3416:
3411:
3407:
3403:
3399:
3389:
3386:
3382:
3378:
3356:
3349:
3345:
3342:
3339:
3335:
3327:
3319:
3315:
3311:
3302:
3298:
3294:
3285:
3281:
3277:
3270:
3263:
3257:
3251:
3246:
3241:
3236:
3229:
3226:
3223:
3219:
3211:
3206:
3201:
3196:
3189:
3186:
3183:
3179:
3171:
3166:
3161:
3156:
3151:
3144:
3139:
3134:
3129:
3122:
3118:
3110:
3105:
3100:
3095:
3088:
3084:
3077:
3067:
3066:
3065:
3048:
3045:
3042:
3039:
3036:
3030:
3025:
3022:
3019:
3015:
3009:
3006:
3003:
2999:
2993:
2988:
2985:
2982:
2978:
2974:
2970:
2966:
2962:
2954:
2953:
2952:
2933:
2927:
2922:
2917:
2912:
2905:
2902:
2899:
2895:
2887:
2882:
2877:
2872:
2865:
2862:
2859:
2855:
2847:
2842:
2837:
2832:
2827:
2820:
2815:
2810:
2805:
2798:
2794:
2786:
2781:
2776:
2771:
2764:
2760:
2753:
2744:
2743:
2742:
2723:
2715:
2712:
2709:
2705:
2697:
2688:
2684:
2674:
2670:
2660:
2656:
2649:
2642:
2636:
2628:
2625:
2622:
2618:
2612:
2607:
2600:
2596:
2588:
2584:
2576:
2571:
2566:
2561:
2556:
2549:
2544:
2539:
2534:
2529:
2522:
2517:
2512:
2507:
2502:
2495:
2490:
2485:
2480:
2475:
2469:
2463:
2458:
2450:
2447:
2444:
2441:
2438:
2434:
2426:
2417:
2414:
2411:
2407:
2397:
2393:
2383:
2380:
2377:
2373:
2366:
2359:
2351:
2348:
2345:
2341:
2335:
2330:
2323:
2319:
2311:
2307:
2299:
2294:
2289:
2284:
2279:
2272:
2267:
2262:
2257:
2252:
2245:
2240:
2235:
2230:
2225:
2218:
2213:
2208:
2203:
2198:
2192:
2187:
2182:
2174:
2171:
2168:
2165:
2162:
2158:
2150:
2141:
2138:
2135:
2131:
2121:
2118:
2115:
2111:
2101:
2097:
2090:
2081:
2080:
2079:
2065:
2031:
2028:
2025:
2021:
2017:
2014:
2011:
2006:
2002:
1998:
1993:
1989:
1977:
1974:). Using the
1973:
1955:
1766:lfsr_xorshift
1753:
1751:
1746:
1744:
1740:
1734:
1724:
1722:
1718:
1717:Galois fields
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1529:
1521:
1515:
1511:
1169:
1167:
1159:
1155:
1154:
1153:
1151:
1146:
1144:
1140:
1139:internal XORs
1136:
1132:
1122:
1113:
905:
897:
895:
891:
887:
830:
774:
773:
772:
726:
697:
696:
695:
693:
689:
685:
418:
416:
411:
409:
405:
400:
394:
390:
387:
383:
382:
381:
379:
375:
370:
368:
349:
346:
341:
337:
333:
328:
324:
320:
315:
311:
307:
302:
298:
290:
289:
288:
285:
282:
278:
273:
271:
267:
259:
255:
251:
247:
244:
240:
239:
238:
236:
211:
206:
201:
192:
190:
185:
183:
179:
175:
170:
168:
164:
158:
156:
151:
149:
145:
141:
137:
133:
122:
119:
111:
100:
97:
93:
90:
86:
83:
79:
76:
72:
69: –
68:
64:
63:Find sources:
57:
53:
47:
46:
41:This article
39:
35:
30:
29:
26:
22:
6643:
6195:. Retrieved
6186:
6179:
6170:
6158:
6149:
6084:
6077:
6067:, retrieved
6060:the original
6039:
6035:
6022:
6010:
5985:. Retrieved
5981:
5971:
5952:
5946:
5919:
5913:
5901:. Retrieved
5897:
5887:
5858:
5852:
5839:
5820:
5814:
5801:
5778:
5772:
5747:
5743:
5730:
5702:
5695:
5684:
5672:. Retrieved
5658:
5635:JPL sequence
5620:Ring counter
5575:
5568:
5534:IEEE 802.11a
5460:
5427:
5421:
5418:encipherment
5411:
5391:exclusive or
5386:
5375:
5352:
5342:
5340:
5332:
5317:
5308:
5297:Please help
5292:verification
5289:
5262:
5235:
5221:); or using
5198:
5159:
5139:
5119:
5116:Applications
5108:
5087:
5083:
5060:
4100:10100000000
3395:
3387:
3380:
3376:
3373:
3063:
2950:
2740:
1940:
1937:Matrix forms
1747:
1737:As shown by
1736:
1720:
1712:
1708:
1704:
1696:
1692:
1688:
1686:
1527:
1450:
1251:#ifndef LEFT
1163:
1149:
1147:
1142:
1138:
1134:
1128:
1116:Galois LFSRs
1111:
903:
893:
889:
885:
883:
770:
681:
412:
407:
403:
401:
398:
378:Galois field
371:
366:
364:
274:
263:
234:
232:
186:
171:
159:
155:exclusive-or
152:
139:
135:
129:
114:
105:
95:
88:
81:
74:
62:
50:Please help
45:verification
42:
25:
5674:October 16,
5513:PCI Express
5182:distributed
5164:for use in
5128:in various
5126:white noise
5055:16,777,215
4039:1001000000
1916:start_state
1808:start_state
1784:start_state
1748:Below is a
1520:to itself.
1451:The branch
1431:start_state
1224:start_state
1200:start_state
1182:lfsr_galois
1164:Below is a
1150:counterpart
1089:start_state
936:start_state
909:start_state
890:many-to-one
839:rotateright
783:rotateright
692:dot product
662:start_state
473:start_state
449:start_state
210:hexadecimal
6700:Categories
6654:T-function
6601:Generators
6477:Achterbahn
6197:11 October
6069:2019-09-15
5987:2021-04-27
5878:1885/34049
5651:References
5565:Other uses
5414:encryption
5383:scrambling
5366:Scrambling
5256:, and the
5252:, used in
5244:, used in
5204:Non-linear
4968:8,388,607
4907:4,194,303
4846:2,097,151
4785:1,048,575
3978:100010000
3412:(sequence
682:If a fast
417:is below:
281:polynomial
258:linear map
254:affine map
243:m-sequence
108:March 2009
78:newspapers
6567:SOBER-128
6497:KCipher-2
6431:SOSEMANUK
6402:Portfolio
6132:ignored (
6122:cite book
6090:CiteSeerX
5903:5 January
5764:120804149
5525:(SAS/SPL)
5372:Scrambler
5254:Bluetooth
5072:project.
5052:0xE10000
4965:0x420000
4904:0x300000
4843:0x140000
3917:10111000
3462:−
3428:Bits (n)
3343:−
3328:⋮
3247:⋯
3242:⋯
3227:−
3202:⋯
3187:−
3167:⋱
3162:⋱
3157:⋮
3152:⋮
3145:⋮
3140:⋱
3106:⋯
3040:≤
3023:−
3007:−
2979:∑
2923:⋯
2918:⋯
2903:−
2878:⋯
2863:−
2843:⋱
2838:⋱
2833:⋮
2828:⋮
2821:⋮
2816:⋱
2782:⋯
2713:−
2698:⋮
2626:−
2613:⋯
2608:⋯
2567:⋯
2545:⋱
2540:⋱
2535:⋮
2530:⋮
2523:⋮
2518:⋱
2491:⋯
2448:−
2427:⋮
2381:−
2349:−
2336:⋯
2331:⋯
2290:⋯
2268:⋱
2263:⋱
2258:⋮
2253:⋮
2246:⋮
2241:⋱
2214:⋯
2172:−
2151:⋮
2029:−
2015:…
1362:<<=
1281:>>=
376:over the
374:primitive
270:Gray code
205:Fibonacci
203:A 16-bit
132:computing
6440:Hardware
6409:Software
6380:Crypto-1
5722:51534945
5594:Pinwheel
5588:See also
5548:such as
5379:bit rate
5146:counters
4782:0x90000
4724:524,287
4721:0x72000
4637:262,143
4634:0x20400
4576:131,071
4573:0x12000
3830:1100000
3444:Period (
3350:′
3320:′
3303:′
3286:′
2971:′
1883:>>
1865:<<
1847:>>
1814:unsigned
1799:uint16_t
1781:uint16_t
1763:unsigned
1757:#include
1733:Xorshift
1659:>>
1650:<<
1638:<<
1623:<<
1605:>>
1596:<<
1584:<<
1569:<<
1548:uint64_t
1539:uint64_t
1533:#include
1326:unsigned
1254:unsigned
1230:unsigned
1215:uint16_t
1197:uint16_t
1179:unsigned
1173:#include
1062:<<
1044:>>
1017:>>
999:>>
981:>>
918:<<
886:standard
761:& 1u
688:popcount
629:<<
611:>>
581:>>
563:>>
545:>>
527:>>
491:unsigned
479:uint16_t
464:uint16_t
446:uint16_t
431:lfsr_fib
428:unsigned
422:#include
6668:Attacks
6457:Trivium
6426:Salsa20
6400:eSTREAM
6247:at the
5554:GLONASS
5529:USB 3.0
5178:periods
5176:, long
4515:65,535
4512:0xD008
4428:32,767
4425:0x6000
4367:16,383
4364:0x3802
4277:0x1C80
3769:110000
3418:in the
3415:A011260
1790:0xACE1u
1711:) by a
1512:is the
1502:0xB400u
1472:0xB400u
1395:0x002Du
1338:int16_t
1314:0xB400u
1206:0xACE1u
1135:modular
759:. (The
747:0x002Du
718:0x002Du
455:0xACE1u
142:) is a
92:scholar
6627:Theory
6577:Turing
6572:Spritz
6547:Scream
6517:Phelix
6512:Panama
6482:F-FCSR
6452:MICKEY
6421:Rabbit
6416:HC-128
6375:ChaCha
6110:
6092:
6056:459117
6054:
5959:
5934:
5823:(14).
5789:
5762:
5720:
5710:
4280:8,191
4193:4,095
4190:0xE08
4106:2,047
4103:0x500
4045:1,023
4042:0x240
3981:0x110
3708:10100
3437:Taps (
3034:
1925:period
1922:return
1895:period
1817:period
1701:modulo
1668:return
1542:prsg63
1440:period
1437:return
1410:period
1404:#endif
1233:period
1101:period
1071:period
939:period
735:popcnt
706:parity
684:parity
671:period
668:return
641:period
494:period
94:
87:
80:
73:
65:
6649:NLFSR
6562:SOBER
6492:ISAAC
6447:Grain
6191:(PDF)
6063:(PDF)
6052:S2CID
6032:(PDF)
5861:(5).
5760:S2CID
5740:(PDF)
5668:(PDF)
5614:NLFSR
5578:DCF77
5489:ITU-T
5455:NICAM
5449:DVB-T
5212:state
3920:0xB8
3833:0x60
3772:0x30
3711:0x14
3647:1100
3434:Taps
1972:GF(2)
1970:(see
1904:while
1514:carry
1499:&
1419:while
1323:#else
1266:&
1141:, or
1107:break
1095:print
1026:&
948:while
857:&
813:&
792:&
744:&
715:&
650:while
590:&
279:as a
99:JSTOR
85:books
6644:LFSR
6592:WAKE
6587:VMPC
6582:VEST
6557:SNOW
6552:SEAL
6542:RC4A
6537:RC4+
6532:QUAD
6522:Pike
6507:ORYX
6502:MUGI
6487:FISH
6370:A5/2
6365:A5/1
6199:2011
6141:link
6134:help
6108:ISBN
5957:ISBN
5932:ISBN
5905:2022
5787:ISBN
5718:OCLC
5708:ISBN
5676:2016
5552:and
5518:SATA
5495:CDMA
5485:PSTN
5472:HDMI
5242:A5/2
5240:and
5238:A5/1
5208:bits
3984:511
3923:255
3836:127
3650:0xC
3589:0x6
3586:110
3528:0x3
3420:OEIS
3046:<
1910:lfsr
1880:lfsr
1874:lfsr
1862:lfsr
1856:lfsr
1844:lfsr
1838:lfsr
1802:lfsr
1772:void
1671:lfsr
1647:lfsr
1635:lfsr
1620:lfsr
1614:lfsr
1593:lfsr
1581:lfsr
1566:lfsr
1560:lfsr
1551:lfsr
1518:lfsr
1481:lfsr
1466:lfsr
1425:lfsr
1389:lfsr
1359:lfsr
1347:<
1344:lfsr
1308:lfsr
1278:lfsr
1263:lfsr
1218:lfsr
1188:void
1083:lfsr
1041:lfsr
1032:lfsr
1014:lfsr
996:lfsr
978:lfsr
969:lfsr
951:True
930:lfsr
869:lfsr
851:lfsr
833:lfsr
789:lfsr
777:lfsr
741:lfsr
712:lfsr
656:lfsr
608:lfsr
599:lfsr
578:lfsr
560:lfsr
542:lfsr
524:lfsr
467:lfsr
437:void
386:even
250:XNOR
235:taps
140:LFSR
134:, a
71:news
6390:RC4
6100:doi
6044:doi
5924:doi
5873:hdl
5863:doi
5825:doi
5752:doi
5550:GPS
5478:SDI
5474:2.0
5439:DAB
5422:not
5416:or
5353:In
5301:by
5246:GSM
5172:or
4973:24
4912:23
4851:22
4790:21
4729:20
4642:19
4581:18
4520:17
4433:16
4372:15
4285:14
4198:13
4111:12
4050:11
3989:10
3775:63
3714:31
3653:15
3525:11
3439:hex
3422:).
1510:msb
1493:lsb
1460:lsb
1380:msb
1329:msb
1299:lsb
1257:lsb
1059:bit
960:bit
892:or
845:bit
842:(((
810:bit
729:bit
700:bit
686:or
626:bit
515:bit
482:bit
284:mod
130:In
54:by
6702::
6659:IV
6527:Py
6385:E0
6126::
6124:}}
6120:{{
6106:.
6098:.
6050:,
6040:21
6038:,
6034:,
5996:^
5980:.
5930:.
5896:.
5871:.
5859:11
5857:.
5851:.
5819:.
5813:.
5781:.
5758:.
5748:19
5746:.
5742:.
5716:.
5409:.
5267:.
5250:E0
5132:.
5027:17
5014:22
5001:23
4988:24
4940:18
4927:23
4879:21
4866:22
4818:19
4805:21
4757:17
4744:20
4696:14
4683:17
4670:18
4657:19
4609:11
4596:18
4548:14
4535:17
4474:13
4461:15
4448:16
4400:14
4387:15
4326:12
4313:13
4300:14
4239:11
4226:12
4213:13
4152:10
4139:11
4126:12
4065:11
4004:10
3928:9
3841:8
3780:7
3719:6
3658:5
3597:4
3592:7
3536:3
3531:3
3482:2
3477:)
3441:)
1919:);
1913:!=
1892:++
1886:13
1877:^=
1859:^=
1841:^=
1829:do
1723:.
1662:32
1626:32
1608:32
1572:32
1484:^=
1469:^=
1454:if
1434:);
1428:!=
1407:++
1392:^=
1374:if
1311:^=
1293:if
1269:1u
1245:do
1137:,
1086:==
1080:if
1074:+=
1065:15
1023:))
1020:12
921:15
888:,
878:);
860:1u
825:);
819:),
816:1u
798:1u
786:((
721:);
665:);
659:!=
638:++
635:);
632:15
593:1u
587:))
521:((
506:do
350:1.
342:11
329:13
316:14
303:16
176:,
169:.
6343:e
6336:t
6329:v
6201:.
6143:)
6136:)
6116:.
6102::
6046::
5990:.
5965:.
5940:.
5926::
5907:.
5881:.
5875::
5865::
5833:.
5827::
5821:8
5795:.
5766:.
5754::
5724:.
5678:.
5441:(
5343:n
5324:)
5318:(
5313:)
5309:(
5295:.
5232:.
5214:;
5109:n
5088:n
5084:n
5035:1
5032:+
5023:x
5019:+
5010:x
5006:+
4997:x
4993:+
4984:x
4948:1
4945:+
4936:x
4932:+
4923:x
4887:1
4884:+
4875:x
4871:+
4862:x
4826:1
4823:+
4814:x
4810:+
4801:x
4765:1
4762:+
4753:x
4749:+
4740:x
4704:1
4701:+
4692:x
4688:+
4679:x
4675:+
4666:x
4662:+
4653:x
4617:1
4614:+
4605:x
4601:+
4592:x
4556:1
4553:+
4544:x
4540:+
4531:x
4495:1
4492:+
4487:4
4483:x
4479:+
4470:x
4466:+
4457:x
4453:+
4444:x
4408:1
4405:+
4396:x
4392:+
4383:x
4347:1
4344:+
4339:2
4335:x
4331:+
4322:x
4318:+
4309:x
4305:+
4296:x
4260:1
4257:+
4252:8
4248:x
4244:+
4235:x
4231:+
4222:x
4218:+
4209:x
4173:1
4170:+
4165:4
4161:x
4157:+
4148:x
4144:+
4135:x
4131:+
4122:x
4086:1
4083:+
4078:9
4074:x
4070:+
4061:x
4025:1
4022:+
4017:7
4013:x
4009:+
4000:x
3964:1
3961:+
3956:5
3952:x
3948:+
3943:9
3939:x
3903:1
3900:+
3895:4
3891:x
3887:+
3882:5
3878:x
3874:+
3869:6
3865:x
3861:+
3856:8
3852:x
3816:1
3813:+
3808:6
3804:x
3800:+
3795:7
3791:x
3755:1
3752:+
3747:5
3743:x
3739:+
3734:6
3730:x
3694:1
3691:+
3686:3
3682:x
3678:+
3673:5
3669:x
3633:1
3630:+
3625:3
3621:x
3617:+
3612:4
3608:x
3572:1
3569:+
3564:2
3560:x
3556:+
3551:3
3547:x
3511:1
3508:+
3505:x
3502:+
3497:2
3493:x
3465:1
3457:n
3453:2
3381:k
3377:a
3357:)
3346:1
3340:n
3336:a
3316:2
3312:a
3299:1
3295:a
3282:0
3278:a
3271:(
3264:k
3258:)
3252:0
3237:0
3230:1
3224:n
3220:c
3212:1
3207:0
3197:0
3190:2
3184:n
3180:c
3172:0
3135:1
3130:0
3123:1
3119:c
3111:0
3101:0
3096:1
3089:0
3085:c
3078:(
3049:n
3043:i
3037:0
3031:,
3026:j
3020:n
3016:c
3010:j
3004:i
3000:a
2994:j
2989:0
2986:=
2983:i
2975:=
2967:i
2963:a
2934:)
2928:0
2913:0
2906:1
2900:n
2896:c
2888:1
2883:0
2873:0
2866:2
2860:n
2856:c
2848:0
2811:1
2806:0
2799:1
2795:c
2787:0
2777:0
2772:1
2765:0
2761:c
2754:(
2724:)
2716:1
2710:n
2706:a
2689:2
2685:a
2675:1
2671:a
2661:0
2657:a
2650:(
2643:k
2637:)
2629:1
2623:n
2619:c
2601:1
2597:c
2589:0
2585:c
2577:1
2572:0
2562:0
2557:0
2550:0
2513:1
2508:0
2503:0
2496:0
2486:0
2481:1
2476:0
2470:(
2464:=
2459:)
2451:2
2445:n
2442:+
2439:k
2435:a
2418:1
2415:+
2412:k
2408:a
2398:k
2394:a
2384:1
2378:k
2374:a
2367:(
2360:)
2352:1
2346:n
2342:c
2324:1
2320:c
2312:0
2308:c
2300:1
2295:0
2285:0
2280:0
2273:0
2236:1
2231:0
2226:0
2219:0
2209:0
2204:1
2199:0
2193:(
2188:=
2183:)
2175:1
2169:n
2166:+
2163:k
2159:a
2142:2
2139:+
2136:k
2132:a
2122:1
2119:+
2116:k
2112:a
2102:k
2098:a
2091:(
2066:k
2043:T
2038:)
2032:1
2026:n
2022:a
2018:,
2012:,
2007:1
2003:a
1999:,
1994:0
1990:a
1986:(
1956:2
1951:F
1931:}
1928:;
1907:(
1901:}
1898:;
1889:;
1871:;
1868:9
1853:;
1850:7
1832:{
1826:;
1823:0
1820:=
1811:;
1805:=
1793:;
1787:=
1778:{
1775:)
1769:(
1750:C
1721:q
1713:q
1709:q
1705:q
1703:-
1697:q
1693:q
1689:q
1677:}
1674:;
1665:;
1656:)
1653:2
1644:^
1641:1
1632:(
1629:|
1617:=
1611:;
1602:)
1599:2
1590:^
1587:1
1578:(
1575:|
1563:=
1557:{
1554:)
1545:(
1505:;
1496:)
1490:-
1487:(
1475:;
1463:)
1457:(
1446:}
1443:;
1422:(
1416:}
1413:;
1398:;
1383:)
1377:(
1368:;
1365:1
1353:;
1350:0
1341:)
1335:(
1332:=
1317:;
1302:)
1296:(
1287:;
1284:1
1272:;
1260:=
1248:{
1242:;
1239:0
1236:=
1227:;
1221:=
1209:;
1203:=
1194:{
1191:)
1185:(
1166:C
1104:)
1098:(
1092::
1077:1
1068:)
1056:(
1053:|
1050:)
1047:1
1038:(
1035:=
1029:1
1011:(
1008:^
1005:)
1002:3
993:(
990:^
987:)
984:1
975:(
972:^
966:(
963:=
954::
945:0
942:=
933:=
927:1
924:|
915:1
912:=
875:1
872:,
866:^
863:)
854:)
848:^
836:=
822:1
807:(
804:|
801:)
795:~
780:=
756:;
750:)
738:(
732:=
709:(
703:=
677:}
674:;
653:(
647:}
644:;
623:(
620:|
617:)
614:1
605:(
602:=
596:;
584:5
575:(
572:^
569:)
566:3
557:(
554:^
551:)
548:2
539:(
536:^
533:)
530:0
518:=
509:{
503:;
500:0
497:=
485:;
476:;
470:=
458:;
452:=
443:{
440:)
434:(
415:C
408:x
404:n
388:.
367:x
347:+
338:x
334:+
325:x
321:+
312:x
308:+
299:x
208:(
138:(
121:)
115:(
110:)
106:(
96:·
89:·
82:·
75:·
48:.
23:.
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