Linear Feedback Shift Register (LFSR) Calculator

About the Linear Feedback Shift Register (LFSR) Calculator

A Linear Feedback Shift Register (LFSR) is a fundamental building block in digital systems, cryptography, and communications engineering. It consists of a shift register whose input bit is computed as a linear function (typically XOR) of selected register bits called tap positions. Despite its simple hardware implementation, an LFSR can generate long pseudo-random sequences with excellent statistical properties.

This LFSR calculator lets you configure the register length (number of bits), specify tap positions that define the feedback polynomial, and set the initial seed state. The tool generates the complete bit sequence, analyzes the period length, determines whether the configuration produces a maximal-length sequence (m-sequence), and displays the characteristic polynomial.

Maximal-length LFSRs produce sequences of period 2ⁿ−1, cycling through every possible non-zero state before repeating. These sequences find applications in spread-spectrum communications (CDMA), error detection (CRC), built-in self-test circuits, stream ciphers, and Monte Carlo simulations. The feedback polynomial must be primitive over GF(2) to achieve maximal length.

Use the presets to explore classic configurations like the 4-bit x⁴+x+1 polynomial or the 16-bit CRC polynomial. The state table shows each clock cycle's register contents and output bit, while the period visualization helps you quickly assess whether your polynomial choice achieves full-period generation.

When This Page Helps

Linear Feedback Shift Register (LFSR) Calculator helps you solve linear feedback shift register (lfsr) problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Register Length (bits), Tap Positions, Seed (binary) once and immediately inspect Feedback Polynomial, Period, Max Period (2ⁿ−1) to validate your work.

How to Use the Inputs

1. Enter Register Length (bits) and Tap Positions in the input fields.
2. Select the mode, method, or precision options that match your linear feedback shift register (lfsr) problem.
3. Read Feedback Polynomial first, then use Period to confirm your setup is correct.
4. Try a preset such as "LFSR-3 (x³+x+1)" to test a known case quickly.

Example Calculation

Tips & Best Practices

• Use known primitive polynomials (available in the reference table) to guarantee maximal-length sequences
• The seed must contain at least one 1-bit — all zeros is a locked state for XOR feedback
• For n-bit LFSR, maximal period is 2ⁿ−1; if your period is shorter, the polynomial is not primitive
• Tap positions correspond to polynomial exponents — taps 4,1 means x⁴+x+1
• LFSRs with the same polynomial but different seeds produce the same sequence shifted in phase
• CRC-16 and CRC-32 use specific LFSR polynomials for error detection

How This Linear Feedback Shift Register (LFSR) Calculator Works

This calculator takes Register Length (bits), Tap Positions, Seed (binary), Steps to Show and applies the relevant linear feedback shift register (lfsr) relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.

Interpreting Results

Start with the primary output, then use Feedback Polynomial, Period, Max Period (2ⁿ−1), Maximal Length? to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.

Study Strategy

Frequently Asked Questions

• What is a linear feedback shift register?

  An LFSR is a shift register that uses XOR of selected bit positions (taps) to compute the feedback input bit. Each clock cycle shifts the register right and inserts the feedback bit at the left, producing a pseudo-random output bit sequence.

• What makes an LFSR maximal-length?

  An LFSR achieves maximal length (period = 2ⁿ−1) when its feedback polynomial is primitive over GF(2). This means it generates every possible non-zero n-bit pattern before repeating.

• How are tap positions related to the polynomial?

  Each tap position corresponds to an exponent in the characteristic polynomial. Taps at positions 4 and 1 define the polynomial x⁴+x+1. The constant term (x⁰=1) is always implicitly included.

• Why is the all-zeros state forbidden?

  Since feedback uses XOR (a linear function), an all-zero register produces a zero feedback bit, keeping the register stuck at all zeros forever. At least one bit must be 1 in the seed.

• Where are LFSRs used in practice?

  LFSRs are used in pseudo-random number generation, CRC error detection, scrambling in digital communications, spread-spectrum systems (GPS, CDMA), stream ciphers, and hardware testing (BIST).

• What is the difference between Fibonacci and Galois LFSR?

  In a Fibonacci LFSR, tapped bits are XORed to produce the input bit. In a Galois LFSR, the feedback is applied at each tap position within the register. Both produce the same sequence (with different state representations) for the same polynomial.