LFSR Calculator - Linear Feedback Shift Register

A Linear Feedback Shift Register (LFSR) is a shift register whose input bit is a linear function (XOR) of some of its previous state bits. At each clock cycle, the register shifts all bits by one position, the new input bit is computed by XOR-ing the bits at specific positions called feedback taps, and the output is the bit shifted out. The resulting binary sequence appears random but is entirely deterministic, repeating with a period that depends on the register length and the choice of taps. LFSRs are used extensively in digital communications, cryptography, built-in self-test (BIST) circuits, spread-spectrum systems, and pseudo-random number generation. Because an LFSR with n stages can produce a sequence of at most 2ⁿ − 1 bits before repeating (the all-zeros state is excluded), and because this maximum-length sequence can be achieved with carefully chosen primitive polynomial taps, LFSRs are an efficient hardware implementation of pseudo-randomness. There are two common LFSR architectures. In the Fibonacci (external feedback) LFSR, the feedback bit is computed from a fixed set of tap positions, and this bit is fed into the leftmost stage while all other stages shift right. The output is taken from the rightmost stage. In the Galois (internal feedback) LFSR, the feedback bit is XOR-ed directly into multiple intermediate stages rather than being fed back to the input alone. Galois LFSRs are faster in hardware because all XOR operations happen in parallel rather than in series. A maximal-length LFSR — one that cycles through all 2ⁿ − 1 non-zero states before repeating — requires a primitive polynomial as the feedback polynomial. The tap positions correspond to the non-zero coefficients of this polynomial. For a 4-bit LFSR with taps at positions 4 and 3, the primitive polynomial is x⁴ + x³ + 1, and the LFSR cycles through all 15 non-zero states. This calculator lets you explore any combination of register length, seed, tap positions, and LFSR type. You can verify whether your chosen taps produce a maximal-length sequence by checking the period in the results — if the period equals 2ⁿ − 1, the taps form a primitive polynomial. Use the examples provided to see well-known maximal-length configurations. Applications of LFSRs span many domains. In Wi-Fi and Bluetooth, LFSRs provide the spreading codes for frequency hopping and direct-sequence spread spectrum. In storage and communications, CRC (cyclic redundancy check) circuits are implemented as LFSRs. In cryptographic stream ciphers, LFSRs form the core primitive (though modern ciphers combine multiple LFSRs with non-linear components to prevent linear cryptanalysis). In chip testing, LFSR-based signature analysis compresses test response data into a compact signature.