The paper introduces a new methodology called *Dual System Encryption* for proving the security of encryption systems, specifically focusing on Identity-Based Encryption (IBE) and Hierarchical Identity-Based Encryption (HIBE). The authors present a fully secure IBE system and an HIBE system under the decisional Bilinear Diffie-Hellman and decisional Linear assumptions. Key features of the Dual System Encryption include: 1. **Short Parameters**: The ciphertexts, private keys, and public parameters consist of a constant number of group elements, which is a significant improvement over previous systems. 2. **Indistinguishable Forms**: Both ciphertexts and private keys can take on one of two indistinguishable forms: normal or semi-functional. Normal keys and ciphertexts behave as expected in an IBE system, while semi-functional keys and ciphertexts are designed to be indistinguishable from normal ones in certain games. 3. **Security Proof**: The security proof involves a sequence of games where the challenge ciphertext and private keys are gradually changed to semi-functional forms. The proof shows that no adversary can distinguish between successive games, leading to the conclusion that the system is secure. The paper also discusses the implications of this methodology, including the potential for adapting it to other encryption systems such as Anonymous IBE, searchable encryption, and Broadcast Encryption. The authors provide a detailed construction and proof of security for both the IBE and HIBE systems, demonstrating that their approach is both efficient and secure under simple assumptions.The paper introduces a new methodology called *Dual System Encryption* for proving the security of encryption systems, specifically focusing on Identity-Based Encryption (IBE) and Hierarchical Identity-Based Encryption (HIBE). The authors present a fully secure IBE system and an HIBE system under the decisional Bilinear Diffie-Hellman and decisional Linear assumptions. Key features of the Dual System Encryption include: 1. **Short Parameters**: The ciphertexts, private keys, and public parameters consist of a constant number of group elements, which is a significant improvement over previous systems. 2. **Indistinguishable Forms**: Both ciphertexts and private keys can take on one of two indistinguishable forms: normal or semi-functional. Normal keys and ciphertexts behave as expected in an IBE system, while semi-functional keys and ciphertexts are designed to be indistinguishable from normal ones in certain games. 3. **Security Proof**: The security proof involves a sequence of games where the challenge ciphertext and private keys are gradually changed to semi-functional forms. The proof shows that no adversary can distinguish between successive games, leading to the conclusion that the system is secure. The paper also discusses the implications of this methodology, including the potential for adapting it to other encryption systems such as Anonymous IBE, searchable encryption, and Broadcast Encryption. The authors provide a detailed construction and proof of security for both the IBE and HIBE systems, demonstrating that their approach is both efficient and secure under simple assumptions.