This paper presents a new methodology for realizing Ciphertext-Policy Attribute-Based Encryption (CP-ABE) under concrete and non-interactive cryptographic assumptions in the standard model. The proposed system allows any encryptor to specify access control in terms of any access formula over the attributes in the system. The efficiency of the system is linear in the complexity of the access formula, with ciphertext size, encryption, and decryption times scaling linearly with the formula's complexity. The only previous work achieving these parameters was limited to a proof in the generic group model. Three constructions are presented within the framework. The first system is proven selectively secure under the decisional Parallel Bilinear Diffie-Hellman Exponent (PBDHE) assumption, a generalization of the BDHE assumption. The next two constructions provide performance tradeoffs for provable security under the (weaker) decisional Bilinear-Diffie-Hellman Exponent and decisional Bilinear Diffie-Hellman assumptions. The paper introduces a new construction that realizes expressive functionality and is efficient and provably secure under a concrete, non-interactive assumption. The system uses a Linear Secret Sharing Scheme (LSSS) matrix to express access control. The encryption algorithm distributes a random exponent according to the LSSS matrix, and private keys are randomized to avoid collusion attacks. The decryption algorithm uses the LSSS matrix to verify if the user's attributes satisfy the access structure. The paper also provides two other constructions that trade off performance parameters for provable security under weaker assumptions. The first construction is proven secure under the decisional Bilinear Diffie-Hellman Exponent (d-BDHE) assumption, while the second is proven secure under the decisional Bilinear Diffie-Hellman assumption. The constructions are compared in terms of ciphertext and key sizes, encryption and decryption times, and assumptions in Table 1. The paper also discusses related work, including the roots of ABE in Identity-Based Encryption (IBE), and the challenges of achieving complex access controls for hiding encrypted data. The paper also discusses the use of bilinear maps and the decisional Parallel Bilinear Diffie-Hellman Exponent assumption in the construction of CP-ABE systems. The paper concludes with a proof of the security of the proposed constructions under the decisional Parallel Bilinear Diffie-Hellman Exponent assumption.This paper presents a new methodology for realizing Ciphertext-Policy Attribute-Based Encryption (CP-ABE) under concrete and non-interactive cryptographic assumptions in the standard model. The proposed system allows any encryptor to specify access control in terms of any access formula over the attributes in the system. The efficiency of the system is linear in the complexity of the access formula, with ciphertext size, encryption, and decryption times scaling linearly with the formula's complexity. The only previous work achieving these parameters was limited to a proof in the generic group model. Three constructions are presented within the framework. The first system is proven selectively secure under the decisional Parallel Bilinear Diffie-Hellman Exponent (PBDHE) assumption, a generalization of the BDHE assumption. The next two constructions provide performance tradeoffs for provable security under the (weaker) decisional Bilinear-Diffie-Hellman Exponent and decisional Bilinear Diffie-Hellman assumptions. The paper introduces a new construction that realizes expressive functionality and is efficient and provably secure under a concrete, non-interactive assumption. The system uses a Linear Secret Sharing Scheme (LSSS) matrix to express access control. The encryption algorithm distributes a random exponent according to the LSSS matrix, and private keys are randomized to avoid collusion attacks. The decryption algorithm uses the LSSS matrix to verify if the user's attributes satisfy the access structure. The paper also provides two other constructions that trade off performance parameters for provable security under weaker assumptions. The first construction is proven secure under the decisional Bilinear Diffie-Hellman Exponent (d-BDHE) assumption, while the second is proven secure under the decisional Bilinear Diffie-Hellman assumption. The constructions are compared in terms of ciphertext and key sizes, encryption and decryption times, and assumptions in Table 1. The paper also discusses related work, including the roots of ABE in Identity-Based Encryption (IBE), and the challenges of achieving complex access controls for hiding encrypted data. The paper also discusses the use of bilinear maps and the decisional Parallel Bilinear Diffie-Hellman Exponent assumption in the construction of CP-ABE systems. The paper concludes with a proof of the security of the proposed constructions under the decisional Parallel Bilinear Diffie-Hellman Exponent assumption.