In 1984, Shamir [1] proposed a public-key encryption scheme such that the public key could be an arbitrary string, in particular, some form of unique identity of the user. This kind of scheme is known as Identity-Based Encryption (IBE). Shamir’s original motivation for constructing IBE was to simplify key management in email systems. There are two basic approaches to the construction of IBE system. The first one, upon which Boneh-Franklin [2] scheme in based, builds IBE systems using bilinear maps [3,4,5]. The resulting systems are efficient both in performance and ciphertext length. The second approach, due to Cocks [6], builds an elegant IBE system based on the quadratic residuosity problem modulo an RSA composite N. The ciphertext in this system contains two elements of Z/NZ to each bit of the plaintext. Hence, the encryption of an l-bit message yields a ciphertext of size 2l · log2N bits. For example, encrypting a 128-bit message using a 1024 bits modulo, the resulting ciphertext is of size 32678 bytes. For comparison, pairing based methods produce a 36 byte ciphertext. An open problem since Cocks scheme was the construction of a space efficient IBE scheme without pairings, namely a system with short ciphertexts. In 2007, Boneh, Gentry and Hamburg [7] proposed such a system. In their scheme, the ciphertext size is about l + log2N. Encrypting a 128-bit message produces a ciphertext of size 145 bytes. The security of the system is based on the quadratic residuosity problem. Encryption time in this system is quartic on the security parameter, while in the most part of practical public-key system the encryption is cubic on the security parameter. The objective of this work is to study and spell out the Boneh-Gentry-Hamburg scheme.