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We describe the popular BB84 protocol and critically examine its securityproof as presented by Shor and Preskill. The proof requires the use of quantumerror correcting codes called the Calderbank-Shor-Steanne (CSS) quantum codes.These quantum codes are constructed in the quantum domain from two suitableclassical linear codes, one used to correct for bit-flip errors and the otherfor phase-flip errors. Consequently, as a prelude to the security proof, thereport reviews the essential properties of linear codes, especially the conceptof cosets, before building the quantum codes that are utilized in the proof.The proof considers a security entanglement-based protocol, which issubsequently reduced to a "Prepare and Measure" protocol similar in structureto the BB84 protocol, thus establishing the security of the BB84 protocol. Theproof, however, is not without assumptions, which are also enumerated. Thetreatment throughout is pedagogical, and this report, therefore, serves auseful tutorial for researchers, practitioners, and students, new to the fieldof quantum information science, in particular, quantum cryptography, as itdevelops the proof in a systematic manner, starting from the properties oflinear codes, and then advancing to the quantum error correcting codes, whichare critical to the understanding of the security proof.

Quantum Error Correcting Codes and the Security Proof of the BB84 Protocol