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Joint Video Compression and Encryption (JVCE) has gained increased attention in the past couple of years to reduce the computational complexity of video compression, as well as provide encryption of multimedia content for web services. In this paper, we present a JVCE framework based on Binary Arithmetic Coding (BAC). We first present an interpretation of BAC in terms of a skewed binary map and then describe 7 other possible chaotic maps which give similar Shannon optimal performance as BAC. We then propose a modification of BAC in which the overall length within the range (0, 1) allocated to each symbol is preserved, but the choice of map used to encode each symbol is based on a key. The encoder, referred to as Chaotic Binary Arithmetic Coder (CBAC), has the effect of scrambling the intervals without making any changes to the width of interval in which the codeword must lie, thereby allowing encryption without sacrificing any coding efficiency. We also present some some security enhancement features to show how they can alleviate the limitations of our technique against known cryptanalysis on BAC-based encryption schemes.