It has been an appealing but challenging goal in research on attribute-based encryption (ABE) and attribute-based signatures (ABS) to design a secure scheme with short ciphertexts and signatures, respectively. While recent results show that some promising progress has been made in this direction, they do not always offer a satisfactory level of security, i.e. achieving selective rather than full security. In this paper, we aim to achieve both full security and short ciphertexts/signatures for threshold access structures in the ABE/ABS setting. Towards achieving this goal, we propose generic property-preserving conversions from inner-product systems to attribute-based systems. We first give concrete constructions of fully secure IPE/IPS with constant-size ciphertexts/signatures in the composite order groups. By making use of our IPE/IPS schemes as building blocks, we then present concrete constructions of fully secure key-policy ABE (KP-ABE) and ciphertext-policy ABE (CP-ABE) with constant-size ciphertexts, and a fully secure ABS with constant-size signatures with perfect privacy for threshold access structures. These results give rise to the first constructions satisfying the aforementioned requirements. Our schemes reduce the number of pairing evaluations to a constant, a very attractive property for practical attribute-based systems. Furthermore, we show that our schemes can be extended to support large attribute universes and more expressive access structures.