LFSR Calculator

How LFSR Calculator Works?

An LFSR, or linear-feedback shift register, is an n-bit state machine. At every clock cycle, it reads the current register state, emits one selected edge bit, calculates a feedback bit using XOR operations, and shifts the register in the chosen direction. The calculator uses the Register Length to define the state width, the Tap Mask to identify which state positions participate in feedback, the Initial Seed as the starting nonzero state, and Clock Cycles to determine how many transitions are simulated.

The feedback bit is the parity of the tapped state bits. If s is the current register value and m is the tap mask, the tapped value is the bitwise AND of these values, and the feedback is the XOR of all selected bits:

For a right shift, the register moves toward lower-order bit positions and the feedback bit enters the most significant position. For a left shift, the register moves toward higher-order positions and the feedback bit enters the least significant position. The state is restricted to the selected register width:

Conceptually, the transitions are:

The Output Bit Position controls whether the least significant bit or most significant bit of the current register is appended to the output sequence on each cycle. The calculator also records every inserted feedback bit, the final register value in decimal and fixed-width binary form, the number of distinct states encountered, and the first repeated-state interval. The theoretical maximum nonzero period is $2^n-1$, although that maximum is reached only when the feedback polynomial and tap arrangement produce a maximal-length sequence.

Practical Use Cases

Digital hardware engineers can use the calculator to verify shift direction, tap placement, inserted feedback bits, and final register states before implementing an LFSR in an FPGA, ASIC, test circuit, or hardware description language.

Electrical and computer engineering students can compare hand-calculated XOR feedback and register transitions with the generated output sequence, making it easier to understand state machines, polynomial feedback, and sequence periods.

Cryptography professionals can inspect candidate tap masks, nonzero-state cycles, and detected periods when studying stream-generation techniques or evaluating the sequence behavior of an LFSR component. An LFSR by itself should not be treated as a complete secure cryptographic system.

Embedded programmers can generate deterministic pseudorandom test sequences, reproduce firmware behavior, and confirm that a chosen seed and tap mask produce the expected state progression.

Communications and testing specialists can examine output and feedback sequences for scrambling, built-in self-test patterns, error simulation, and repeatable digital signal experiments.

Accuracy and Limitations

The calculator performs deterministic bitwise XOR and shift simulation for register lengths from 2 through 32 bits and reports the exact states produced by the selected seed, tap mask, direction, output position, and clock-cycle count. Its detected period is limited to the simulated run and indicates a repeated state only when that repetition occurs within the requested cycles; it is not automatically a proof of the complete cycle length. The theoretical maximum nonzero period is $2^n-1$, but the actual period depends on the tap mask and shift convention. A zero state is excluded by the stated nonzero-seed requirement because XOR-feedback LFSRs can remain locked at zero. Tap masks and seeds should represent valid positions and states within the selected register width; extra high-order bits do not correspond to storage positions in an n-bit register.

Quick Examples