Berlekamp-Massey Algorithm for Binary LFSRs
The Berlekamp-Massey algorithm is a computationally efficient method used to determine the minimal Linear Feedback Shift Register (LFSR) capable of generating a given binary sequence. In the binary field, also known as Galois Field 2 or \(\text{GF}(2)\), the algorithm iteratively constructs the shortest linear recurrence relation that satisfies the observed bit stream. This article explains the step-by-step mechanics of the binary Berlekamp-Massey algorithm, detailing how it computes discrepancies, updates the connection polynomial using modulo-2 arithmetic, and guarantees the minimal linear complexity for the sequence.
Mathematical Foundations in \(\text{GF}(2)\)
An LFSR generates a sequence of binary digits \(s_0, s_1, s_2, \dots\) based on a feedback polynomial (or connection polynomial) denoted as:
\[C(x) = 1 + c_1 x + c_2 x^2 + \dots + c_L x^L\]
Where: * \(L\) represents the length (linear complexity) of the register. * \(c_i \in \{0, 1\}\) are the binary feedback coefficients. * Arithmetic operations are performed modulo 2: addition corresponds to bitwise XOR (\(\oplus\)), and multiplication corresponds to bitwise AND (\(\cdot\)).
For an LFSR of length \(L\), any bit \(s_k\) (for \(k \ge L\)) is generated by the linear recurrence:
\[s_k = \sum_{i=1}^{L} c_i s_{k-i} \pmod 2\]
Initialization
The algorithm processes the binary sequence \(s_0, s_1, \dots, s_{N-1}\) bit by bit. It initializes five key variables:
- Current Connection Polynomial: \(C(x) = 1\)
- Previous Best Polynomial: \(B(x) = 1\)
- Current Minimal Length: \(L = 0\)
- Step Shift Counter: \(m = 1\)
- Bit Index: \(k = 0\) (iterating up to \(N-1\))
Step-by-Step Execution
For each bit index \(k\) from \(0\) to \(N-1\), the algorithm executes the following steps:
1. Calculate the Discrepancy (\(d\))
The discrepancy \(d \in \{0, 1\}\) measures whether the current polynomial \(C(x)\) correctly predicts the next bit \(s_k\):
\[d = s_k \oplus \bigoplus_{i=1}^{L} c_i s_{k-i}\]
- If \(d = 0\), the current LFSR correctly produces \(s_k\). The algorithm increments the step counter \(m \leftarrow m + 1\) and proceeds to the next bit.
- If \(d = 1\), the current LFSR fails to produce \(s_k\), requiring a polynomial update.
2. Update the Polynomial
When \(d = 1\), a correction term based on the previous best polynomial \(B(x)\) is applied to eliminate the discrepancy:
\[C_{\text{new}}(x) = C(x) \oplus \left( x^m \cdot B(x) \right)\]
Because operations are in \(\text{GF}(2)\), the addition is performed via bitwise XOR of the coefficient vectors.
3. Update Register Length and Auxiliary Variables
The algorithm checks if the register length must increase:
Case A: \(2L \le k\) The current LFSR is too short to generate the sequence up to bit \(k\). The new length becomes: \[L_{\text{new}} = k + 1 - L\] The auxiliary state is updated to store the failing configuration: \[B(x) \leftarrow C(x)\] \[L \leftarrow L_{\text{new}}\] \[m \leftarrow 1\]
Case B: \(2L > k\) The current length \(L\) is sufficient to accommodate the change without increasing register size. The length \(L\) remains unchanged, and the shift counter increments: \[m \leftarrow m + 1\]
Finally, set \(C(x) \leftarrow C_{\text{new}}(x)\) and advance \(k \leftarrow k + 1\).
Termination and Result
Once all \(N\) bits are processed, the algorithm terminates with: * The final polynomial \(C(x)\), which defines the feedback taps of the LFSR. * The integer \(L\), representing the minimal length (linear complexity) of the binary sequence.
If \(N \ge 2L\), the resulting polynomial \(C(x)\) is mathematically guaranteed to be the unique minimal-length LFSR that generates the entire prefix of the sequence.