LFSR Calculator

An LFSR consists of a shift register whose input bit is driven by a linear function of its previous state. The most common linear function used for this feedback is exclusive-OR (XOR) or exclusive-NOR (XNOR). The positions that affect the feedback are called taps, defined by a polynomial vector of coefficients. There are two primary configurations: Fibonacci and Galois. In a Fibonacci LFSR, the bits from specific tap positions are combined through a series of XOR gates and fed directly into the leftmost bit. In a Galois LFSR, when the register is clocked, the bits are shifted simultaneously, and the output of the rightmost bit (the output bit) is conditionally XORed back into the tap positions.

Consider a 4-bit Fibonacci LFSR with an initial message state of 1101 and feedback coefficients (taps) corresponding to the polynomial x^4 + x^3 + 1, which translates to coefficients 1101 (where length n = 4). Step 1: The current register holds 1101. Step 2: Identify the tap positions based on the coefficients. For this setup, the feedback bit is calculated by taking the XOR sum of the relevant bits (e.g., bit 4 XOR bit 3 XOR bit 1). Step 3: Shift all bits to the right and place the newly calculated feedback bit into the leftmost position. Repeating this deterministic process generates a repeating cycle of pseudo-random binary numbers until the register eventually returns to its initial state 1101.

When configuring your LFSR, always ensure your coefficient polynomial is primitive if your goal is to achieve the maximum possible period of (2^n) - 1, where n is the register length. Avoid using an all-zero initial state in a standard XOR feedback register, as this results in a permanent lock-state of all zeros. Furthermore, verify that your binary checkpoints and message length match your designated register dimension to prevent truncation or shifting errors.