Linear Complexity of Memory Sequence

The linear complexity is an important measure of pseudorandomness for binary sequences. Based on the relationship between the minimal polynomial and linear complexity, as well as certain number-theoretic properties, we have established both upper and lower bounds for the linear complexity of memory sequences. Additionally, we have identified a special class of memory sequences and demonstrated that they share the same cryptographic properties as the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell$</tex>-sequences from which they are derived.