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LFSR ā€” Pseudo-Random Sequences

Linear Feedback Shift Registers generate deterministic but seemingly random binary sequences. Max period 2^nāˆ’1; used in CRC, stream ciphers, and scrambling.

Concept Fundamentals
2āæāˆ’1
Max Period
XOR taps
Feedback
CRC, Crypto
Use
n bits
State
Generate SequenceConfigure LFSR

Why This Mathematical Concept Matters

Why: LFSRs produce long pseudo-random sequences from minimal hardware. Essential for CRC checksums, stream ciphers, and test pattern generation.

How: Shift bits right; XOR feedback taps into the new leftmost bit. Maximal-length LFSRs use primitive polynomials.

  • ā—Max period 2^nāˆ’1 when feedback polynomial is primitive.
  • ā—Used in CRC-16, CRC-32, and many stream ciphers.
  • ā—All-zero state is a fixed point; avoid it in practice.
Sources:Wikipedia ā€” LFSRNIST ā€” Random Number Generation
šŸ”
CRYPTOGRAPHYPseudorandom / Stream Ciphers

LFSR ā€” Linear Feedback Shift Register

Generate maximal-length sequences. Feedback polynomials, period detection. Used in CRC, A5/1, stream ciphers.

Configure LFSR Parameters

lfsr.sh
CALCULATED
$ lfsr --length=4 --taps=4,1
Period
15
Maximal
Yes

LFSR Output Sequence (first 30 bits)

Generated Sequence

Output Sequence

000111101011001

Sequence Period

15

Maximal Period

15/15 (Maximal)

Step-by-Step Calculation

Understanding Linear Feedback Shift Registers (LFSR)

A Linear Feedback Shift Register (LFSR) works by shifting bits in a register and calculating a new bit based on feedback taps:

Initial register state:

Register=1000\text{Register} = 1000

Feedback polynomial uses these tap positions (bit positions that feed into the XOR operation):

Taps=[4,1]\text{Taps} = [4, 1]

Generating the LFSR Sequence

Step 1: Output bit is 0

Register=1000Output Bit=0Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 1000 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 2: Output bit is 0

Register=1100Output Bit=0Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 1100 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 3: Output bit is 0

Register=1110Output Bit=0Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 1110 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 4: Output bit is 1

Register=1111Output Bit=1Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 1111 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 5: Output bit is 1

Register=0111Output Bit=1Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 0111 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 6: Output bit is 1

Register=1011Output Bit=1Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 1011 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 7: Output bit is 1

Register=0101Output Bit=1Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 0101 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 8: Output bit is 0

Register=1010Output Bit=0Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 1010 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 9: Output bit is 1

Register=1101Output Bit=1Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 1101 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 10: Output bit is 0

Register=0110Output Bit=0Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 0110 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 11: Output bit is 1

Register=0011Output Bit=1Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 0011 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Step 12: Output bit is 1

Register=1001Output Bit=1Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 1001 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 13: Output bit is 0

Register=0100Output Bit=0Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 0100 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 14: Output bit is 0

Register=0010Output Bit=0Feedback Bit=bit0āŠ•bit3=0\begin{align} \text{Register} &= 0010 \\ \text{Output Bit} &= 0 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 0 \end{align}

Step 15: Output bit is 1

Register=0001Output Bit=1Feedback Bit=bit0āŠ•bit3=1\begin{align} \text{Register} &= 0001 \\ \text{Output Bit} &= 1 \\ \text{Feedback Bit} &= \text{bit}_{0} \oplus \text{bit}_{3} = 1 \end{align}

Period detected! The sequence repeats after 15 bits.

State 1000 was previously seen at position 0\text{State } 1000 \text{ was previously seen at position } 0

Summary

The maximum theoretical period for a 4-bit LFSR is 2^4 - 1 = 15 bits.

The actual period of this LFSR configuration is 15 bits, which is equal to the maximum possible period.

This confirms that the chosen feedback polynomial is primitive (maximal length).

Mathematical Visualization

LFSR State Diagram

This visualization shows the current register state with feedback taps highlighted in blue. Taps are used to generate the feedback bit through XOR operations.

Original Binary Value

7
6
5
4
3
2
1
0

One's Complement

0
1
0
1
0
1
0
1
7
6
5
4
3
2
1
0

How to use: Click on any bit in the top row to toggle its value. Then click "Show Conversion" to see how each bit is flipped in the one's complement.

One's complement rule: Every 0 becomes 1, and every 1 becomes 0. This is also known as a bitwise NOT operation.

Current Register: 1000

Feedback Taps: 4, 1

Applications

Cryptography

Error Detection

Communications

Random Number Generation

āš ļøFor educational and informational purposes only. Verify with a qualified professional.

šŸ§® Fascinating Math Facts

šŸ”€

LFSRs produce maximal period 2^nāˆ’1 when the feedback polynomial is primitive

ā€” Number Theory

šŸ”

Used in CRC checksums, stream ciphers (A5/1), and hardware random number generators

ā€” Cryptography

What is a Linear Feedback Shift Register (LFSR)?

A Linear Feedback Shift Register (LFSR) is a shift register whose input bit is a linear function of its previous state. The most common type is implemented as a shift register whose input bit is driven by the XOR (exclusive or) of some bits of the overall shift register value.

LFSRs are used in many applications, including digital counters, pseudo-random number generators, cryptography, and digital communications. They can generate sequences with good statistical properties and long periods before repetition.

Key Concepts:

  • Register: A sequence of bits that stores the current state
  • Tap: A position in the register that contributes to the feedback function
  • Feedback Polynomial: Defines which register bits are used to create the next bit
  • Period: Number of bits generated before the sequence repeats

LFSR Diagram

1
Bit 3
0
Bit 2
0
Bit 1
0
Bit 0
XOR
Feedback (Taps: 4,1)

A 4-bit LFSR with feedback polynomial x4 + x + 1
(Taps at bits 4 and 1)

How Linear Feedback Shift Registers Work

An LFSR consists of a series of flip-flops connected as a shift register, with feedback logic that uses XOR gates. The basic operation follows these steps:

  1. Shift Operation: During each clock cycle, bits in the register shift one position to the right.
  2. Output Bit: The rightmost bit becomes the output bit of the sequence.
  3. Feedback Calculation: Selected bits (taps) are XORed together to create a new bit.
  4. Feedback Insertion: The new feedback bit is inserted at the leftmost position of the register.

Mathematical Properties of LFSRs

Maximal Length Sequences

An LFSR of length n can generate a sequence with a maximum period of 2n-1 bits before repeating. This occurs only when the feedback polynomial is primitive.

Maximum Period:

Pmax=2nāˆ’1P_{max} = 2^n - 1

where n is the register length

When an LFSR generates a sequence with the maximum possible period, it's called a maximal-length LFSR or m-sequence.

Period vs Register Length

Register LengthMax Period
37
415
531
8255
1665,535

Applications of Linear Feedback Shift Registers

Cryptography

In cryptography, LFSRs serve as building blocks for stream ciphers. The A5/1 algorithm used in GSM mobile phones and the E0 cipher in Bluetooth both utilize LFSRs. However, to enhance security, they are typically combined with non-linear components to resist cryptanalysis.

Error Detection

Cyclic Redundancy Check (CRC) codes are implemented using LFSRs. They provide efficient checksums for detecting errors in data transmission and storage. CRC-8, CRC-16, and CRC-32 are common standards used in various protocols and file formats.

Digital Communications

LFSRs are used in digital communications for scrambling data to avoid long sequences of identical bits that could cause synchronization problems.

Pseudo-Random Number Generation

LFSRs generate pseudo-random sequences for simulations, gaming, and testing. Their statistical properties make them suitable for many applications requiring random data.

How to Use This LFSR Calculator

This calculator allows you to generate and analyze sequences from Linear Feedback Shift Registers with different configurations.

  1. Set the Register Length: Enter the number of bits in your LFSR (1-20).
  2. Specify Initial State: Enter a binary string (e.g., "1000") that represents the starting state of your register.
  3. Choose Polynomial Representation: Select from Tap Positions, Powers of x, or Coefficients based on how you want to specify the feedback polynomial.
  4. Enter Polynomial Value: Specify the feedback polynomial using the chosen representation format.
  5. Set Bits to Generate: Specify how many bits of the sequence you want to generate.

šŸ” Tips for Accurate Results

  • To check if a polynomial is primitive (produces maximal-length sequence), generate enough bits to detect the period (at least 2^n bits for an n-bit register).
  • For cryptographic applications, using a maximal-length LFSR is essential.
  • The initial state must not be all zeros, or the LFSR will stay in that state forever.
  • The tap positions are 1-indexed (first bit is position 1, not 0).
  • For power representation, include the highest power and constant term (e.g., for x^4 + x + 1, enter "4,1,0").

Understanding Polynomial Representations

Example PolynomialTap PositionsPowers of xCoefficients
x4 + x + 14,14,1,01,0,0,1,1
x8 + x4 + x3 + x2 + 18,4,3,28,4,3,2,01,0,0,0,1,1,1,0,1
x5 + x2 + 15,25,2,01,0,0,1,0,1

Known Maximal-Length LFSR Tap Configurations

The table below shows selected primitive polynomials that produce maximal-length sequences for different register sizes. Try these configurations to generate sequences with the maximum possible period.

Register Length (n)Tap PositionsPolynomial (Powers of x)Period (2āæ-1)
33,23,1,07
44,14,1,015
55,25,2,031
66,16,1,063
77,17,1,0127
88,4,3,28,4,3,2,0255
99,49,4,0511
1010,310,3,01,023

Note: There are multiple primitive polynomials for each register length. The ones listed here are commonly used in practice.

LFSR Formulas and Related Concepts

Characteristic Polynomial

C(x)=1+c1x+c2x2+ā€¦+cnxnC(x) = 1 + c_1x + c_2x^2 + \ldots + c_nx^n

The characteristic polynomial defines the feedback taps of an LFSR. The coefficient ci is 1 if the i-th tap is used in the feedback, and 0 otherwise. For maximal length sequences, this polynomial must be primitive.

State Transition

si+n=ā؁j=1ncjā‹…si+nāˆ’js_{i+n} = \bigoplus_{j=1}^{n} c_j \cdot s_{i+n-j}

This formula computes the next bit of the sequence (si+n) based on the previous n bits and the characteristic polynomial coefficients. The symbol āŠ• represents the XOR operation.

Linear Complexity

The linear complexity of a sequence is the length of the shortest LFSR that can generate it. For cryptographic applications, high linear complexity is desirable. A maximal-length n-bit LFSR produces a sequence with linear complexity n.

Berlekamp-Massey Algorithm

This algorithm can find the shortest LFSR that generates a given sequence. It's important in cryptanalysis for breaking LFSR-based stream ciphers when enough plaintext-ciphertext is known. The algorithm works by iteratively refining an LFSR until it matches the observed sequence.

Related Concepts

Galois Fields

LFSRs operate over the Galois Field GF(2), where addition is equivalent to XOR and multiplication is equivalent to AND. Understanding finite fields is crucial for analyzing LFSR properties.

Stream Ciphers

LFSRs are fundamental building blocks for many stream ciphers. Modern designs like A5/1, E0, and SNOW combine multiple LFSRs with non-linear components to increase security.

Pseudorandom Number Generators

LFSRs are used in PRNGs for applications requiring fast, statistically sound random sequences. They're particularly efficient in hardware implementations.

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