||Both stream ciphers and block ciphers can be used to provide data confidentiality service. Stream ciphers are preferred in many applications, since they can destroy statistical properties of natural languages to some extent. As an important type of stream ciphers, additive synchronous stream ciphers (ASSCs) are simple in structure and could have many security properties. First, we investigate various ASSCs proposed in the past studies and their cryptographic aspects, in order to provide a basic knowledge of ASSCs as well as their state-of-the-art developments. Some common attacks against ASSCs and associated design criteria are examined. The binary ASSCs based on natural sequence generators are subsequently overviewed. The Legendre cipher, the two-prime cipher of order 2 and the stream ciphers with keystream sequences with three-level autocorrelation are mainly discussed. The ASSCs based on other generators are also introduced and briefly analysed. Next, a simple ASSC with proven properties is proposed and examined. Both theoretical and experimental results show that this stream cipher is secure with respect to a number of possible attacks and has reasonable performance. In order to make the stream cipher applicable, we also describe some methods of detecting the loss of synchronization and present the resynchronization process. The advantage of the stream cipher over existing ones is that it has more proven security properties.