Attribute-based encryption (ABE) enables encryption of messages under access policies so that only users with attributes satisfying the policy can decrypt the ciphertext. In standard ABE, an arbitrary number of colluding users, each without an authorized attribute set, cannot decrypt the ciphertext. However, all existing ABE schemes rely on concrete cryptographic assumptions such as the hardness of certain problems over bilinear maps or integer lattices. Furthermore, it is known that ABE cannot be constructed from generic assumptions such as public-key encryption using black-box techniques. In this work, we revisit the problem of constructing ABE that tolerates collusions of arbitrary but a priori bounded size. We present two ABE schemes secure against bounded collusions that require only semantically secure public-key encryption. Our schemes achieve significant improvement in the size of the public parameters, secret keys, and ciphertexts over the previous construction of bounded-collusion ABE from minimal assumptions by Gorbunov et al. (CRYPTO 2012). In fact, in our second scheme, the size of ABE secret keys does not grow at all with the collusion bound. As a building block, we introduce a multidimensional secret-sharing scheme that may be of independent interest. We also obtain bounded-collusion symmetric-key ABE (which requires the secret key for encryption) by replacing the public-key encryption with symmetric-key encryption, which can be built from the minimal assumption of one-way functions.