# Practical Inner Product Encryption with Constant Private Key +
## 1 Introduction
### Related Works  
#### 1.2.1 Pairing-Based IPE Schemes  
The first IPE scheme, introduced by [PERSON] et al. [1], entails the evaluation of predicates over \(\mathbb{Z}_{N}\) using the inner product, where \(N\) is a composite number. After this pioneering work, many studies followed. For example, [PERSON] and [PERSON] [3] proposed the first hierarchical predicate encryption method (or delegable predicate encryption) for inner product predicates; this provides a user with functionality to delegate more restrictive functionality to another user. [PERSON] and [PERSON] [16] constructed an IPE scheme that solves the inefficiency problem of the previous scheme. More precisely, provided that the description of the ciphertext attribute vector is not included in the ciphertext, the ciphertext overhead of the scheme is reduced to \(O(1)\). By combining dual system encryption [21] and dual pairing vector spaces [3] carefully, [PERSON] et al. [22] obtained the first fully secure IPE scheme and hierarchical predicate encryption under the \(n\)-extended decisional Diffie-Hellman assumption. However, the security of all these previous studies was based on nonstandard assumptions. To resolve this issue, [PERSON] [23] developed the first IPE scheme under the standard assumptions (i.e., decisional bilinear Diffie-Hellman and decisional linear (DLIN) assumptions). [PERSON] and [PERSON] [24] then introduced two nonzero inner product encryption schemes that support constant-size ciphertexts and a constant-size secret key, respectively, which are adaptively secure under the DLIN assumption in the standard model. The authors also proposed the first IPE scheme that is fully secure and fully attribute-hiding [25] as well as the first unbounded IPE scheme that is also fully secure and fully attribute-hiding in the standard model under the DLIN assumption [26]. [PERSON] and [PERSON] [27] introduced a new notion, called IPE with ciphertext conversion, which considers the security of predicate-hiding. [PERSON] and [PERSON] [28] then introduced another concept, called multiparty cloud computation IPE with multiplicative homomorphic property, which enables an IPE scheme to support multiparty cloud computation. [PERSON] et al. [29] proposed a new efficient IPE scheme that only requires \(n\) exponentiation and three pairing computations for decryption. [PERSON] et al. [30] proposed the first enabled-disabled IPE, which supports timed-release services and data self-destruction. [PERSON] [15] constructed two IPE schemes using tag-based quasi-adaptivenoninteractive zero knowledge, where the first and second both have the property of constant-size ciphertext but only the second has the property of attribute-hiding. [PERSON] et al. [7] recently proposed a new IPE scheme based on a double encryption system; it has been demonstrated to achieve adaptive security under a weak attribute-hiding model.  
As discussed subsequently, extensive research has focused on the developed and proposed schemes; however, the private key length of most schemes is linearly dependent on the vector length or requires many pairing operations, making these schemes impractical. Thus, determining how to construct a more practical scheme remains a critical area of research.