This paper presents a novel Attribute-Based Encryption (ABE) scheme that allows private keys to represent any access formula over attributes, including non-monotonic ones. Previous ABE schemes were limited to monotonic access structures. The proposed scheme is secure under the Decisional Bilinear Diffie-Hellman (BDH) assumption and performs well compared to existing, less-expressive ABE systems. The scheme addresses the limitation of previous ABE systems in handling negative constraints in access policies. It uses a technique inspired by secret-sharing schemes to enable non-monotonic access structures. The key idea is to use a polynomial to implicitly make a share available to the decryptor only if a given attribute is not present in the ciphertext. This is achieved by adapting a technique from the broadcast revocation scheme of Naor and Pinkas. The scheme is constructed using bilinear maps and linear secret-sharing schemes. It allows for efficient decryption by leveraging the properties of polynomials and interpolation. The construction supports both key-policy and ciphertext-policy ABE, and can be extended to handle any boolean formula involving AND, OR, NOT, and threshold operations. The paper also discusses the efficiency of the scheme, showing that it can be adapted to handle varying numbers of attributes and that it performs well compared to existing ABE systems. The scheme is proven secure in the selective-set model under chosen-plaintext attacks, reducing the security of the scheme to the hardness of the BDH assumption. The construction is flexible and can be applied to various access structures, including those with negative attributes.This paper presents a novel Attribute-Based Encryption (ABE) scheme that allows private keys to represent any access formula over attributes, including non-monotonic ones. Previous ABE schemes were limited to monotonic access structures. The proposed scheme is secure under the Decisional Bilinear Diffie-Hellman (BDH) assumption and performs well compared to existing, less-expressive ABE systems. The scheme addresses the limitation of previous ABE systems in handling negative constraints in access policies. It uses a technique inspired by secret-sharing schemes to enable non-monotonic access structures. The key idea is to use a polynomial to implicitly make a share available to the decryptor only if a given attribute is not present in the ciphertext. This is achieved by adapting a technique from the broadcast revocation scheme of Naor and Pinkas. The scheme is constructed using bilinear maps and linear secret-sharing schemes. It allows for efficient decryption by leveraging the properties of polynomials and interpolation. The construction supports both key-policy and ciphertext-policy ABE, and can be extended to handle any boolean formula involving AND, OR, NOT, and threshold operations. The paper also discusses the efficiency of the scheme, showing that it can be adapted to handle varying numbers of attributes and that it performs well compared to existing ABE systems. The scheme is proven secure in the selective-set model under chosen-plaintext attacks, reducing the security of the scheme to the hardness of the BDH assumption. The construction is flexible and can be applied to various access structures, including those with negative attributes.