Generate pseudo-random bit sequences using linear feedback shift registers. Configure register length, tap positions, and seed to analyze LFSR period, feedback polynomial, and sequence properties.
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.
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.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Output bit = register LSB; Feedback bit = XOR of tapped positions; New state = [feedback, reg[0], …, reg[n−2]]; Period ≤ 2ⁿ − 1 for n-bit LFSR. Maximal when characteristic polynomial is primitive over GF(2).
Result: Feedback Polynomial shown by the calculator
Using the preset "LFSR-3 (x³+x+1)", the calculator evaluates the linear feedback shift register (lfsr) setup, applies the selected algebra rules, and reports Feedback Polynomial with supporting checks so you can verify each transformation.
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.
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.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
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.
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.
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.
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.
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). Use this as a practical reminder before finalizing the result.
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.