# LFSR == Linear Feedback Shift Register

## FSR

Feedback shift register

connection polynomial:

C(x) = 1 + C_{1}x + C_{2}x^{2} + \cdots + C_{L}x^{L}


## LFSR

If the feedback function is linear, the FSR is called LFSR

i.e.

S_{L} = C_{1}S_{L-1} + C_{2}S_{L-2} + \cdots + C_{L}S_0{}


is linear.

\begin{align}

& \text{If } C(x) \text{ is primitive and the initial state } \\
& (S_{0}, S_{1}, \cdots , S_{L-1}) \text{ is not zero, } \text{the period is } 2^{L} - 1

\end{align}


A stream cipher constructed by a LFSR alone is not secure.

## Nonlinear combination generators

• Geff Generator (1973)

## Clcok-controlled generators

• Stop-and-Go Generators (1987)
• The Shrinking Generator (1993)
• A5 (the GSM standard)
• E0 (Bluetooth's standard encryption)

## GSM A5/1

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