This paper introduces a new methodology for proving the security of encryption systems called Dual System Encryption. It enables the construction of fully secure Identity-Based Encryption (IBE) and Hierarchical Identity-Based Encryption (HIBE) systems under the simple and well-established decisional Bilinear Diffie-Hellman (BDH) and decisional Linear assumptions. The IBE system features ciphertexts, private keys, and public parameters that each consist of a constant number of group elements, making them highly efficient. This is the first HIBE system and the first IBE system with short parameters under simple assumptions. Dual System Encryption allows both ciphertexts and private keys to take on two indistinguishable forms: normal and semi-functional. A semi-functional private key can decrypt normally generated ciphertexts, but decryption fails if a semi-functional ciphertext is decrypted with a semi-functional private key. Conversely, semi-functional ciphertexts can only be decrypted by normal private keys. The security proof involves a sequence of games where the challenge ciphertext and private keys are gradually transformed into semi-functional forms. This approach enables the proof of security under the decisional Linear and BDH assumptions. The method avoids the limitations of previous partitioning strategies, which were insufficient for more complex systems like HIBE and Attribute-Based Encryption. The paper also presents a fully secure HIBE system based on the IBE construction, demonstrating that the added complexity is minimal. The system leverages the structure of the Boneh-Boyen selective-ID HIBE. The methodology is expected to be applicable to other encryption systems, including Anonymous IBE, searchable encryption, broadcast encryption, and Attribute-Based Encryption, under simple assumptions. The paper provides a detailed construction of the IBE and HIBE systems, including the algorithms for setup, encryption, key generation, and decryption. It also describes semi-functional algorithms for ciphertexts and keys, which are crucial for the security proof. The proof of security is structured as a sequence of games, demonstrating that an adversary cannot distinguish between successive games under the given assumptions. The techniques used in the proof are based on a reduction algorithm that can generate keys for any identity and use any identity as a challenge identity, eliminating the need for an abort condition.This paper introduces a new methodology for proving the security of encryption systems called Dual System Encryption. It enables the construction of fully secure Identity-Based Encryption (IBE) and Hierarchical Identity-Based Encryption (HIBE) systems under the simple and well-established decisional Bilinear Diffie-Hellman (BDH) and decisional Linear assumptions. The IBE system features ciphertexts, private keys, and public parameters that each consist of a constant number of group elements, making them highly efficient. This is the first HIBE system and the first IBE system with short parameters under simple assumptions. Dual System Encryption allows both ciphertexts and private keys to take on two indistinguishable forms: normal and semi-functional. A semi-functional private key can decrypt normally generated ciphertexts, but decryption fails if a semi-functional ciphertext is decrypted with a semi-functional private key. Conversely, semi-functional ciphertexts can only be decrypted by normal private keys. The security proof involves a sequence of games where the challenge ciphertext and private keys are gradually transformed into semi-functional forms. This approach enables the proof of security under the decisional Linear and BDH assumptions. The method avoids the limitations of previous partitioning strategies, which were insufficient for more complex systems like HIBE and Attribute-Based Encryption. The paper also presents a fully secure HIBE system based on the IBE construction, demonstrating that the added complexity is minimal. The system leverages the structure of the Boneh-Boyen selective-ID HIBE. The methodology is expected to be applicable to other encryption systems, including Anonymous IBE, searchable encryption, broadcast encryption, and Attribute-Based Encryption, under simple assumptions. The paper provides a detailed construction of the IBE and HIBE systems, including the algorithms for setup, encryption, key generation, and decryption. It also describes semi-functional algorithms for ciphertexts and keys, which are crucial for the security proof. The proof of security is structured as a sequence of games, demonstrating that an adversary cannot distinguish between successive games under the given assumptions. The techniques used in the proof are based on a reduction algorithm that can generate keys for any identity and use any identity as a challenge identity, eliminating the need for an abort condition.