In an identity based encryption scheme, each user is identi- fied by a unique identity string. An attribute based encryption scheme (ABE), in contrast, is a scheme in which each user is identified by a set of attributes, and some function of those attributes is used to determine decryption ability for each ciphertext. Sahai and Waters introduced a single authority attribute encryption scheme and left open the question of whether a scheme could be constructed in which multiple authorities were allowed to distribute attributes [SW05]. We answer this question in the affirmative. Our scheme allows any polynomial number of independent authorities to monitor attributes and distribute secret keys. An encryptor can choose, for each authority, a number dk and a set of attributes; he can then encrypt a message such that a user can only decrypt if he has at least dk of the given attributes from each authority k. Our scheme can tolerate an arbitrary number of corrupt authoritites. We also show how to apply our techniques to achieve a multiauthority version of the large universe fine grained access control ABE presented by Gopal et al. [GPSW06].