$2$ -Adic Complexity of Binary $m$ -Sequences

Although 2 -adic complexity was proposed more than ten years ago, even for m -sequences which are thought of as the most important linear recurring sequences, no theoretical results about their 2-adic complexity has been presented. In this paper, it is shown that for a binary m-sequence, its 2-adic complexity attains the maximum, which implies that no feedback with carry shift registers (FCSRs) with connection integer less than 2 ^{2 n -1} - 1 can generate m-sequences of order n .