Paper 2026/1379

Hierarchical Structure in Attribute-Based Inner-Product Functional Encryption

Abstract

Attribute-based inner-product functional encryption (AB-IPFE), introduced by Abdalla et al. (Asiacrypt'20), is a cryptosystem that combines the access-control capability of attribute-based encryption (ABE) with the linear-computation capability of inner-product functional encryption. By introducing a hierarchical structure between these two functionalities, we can decompose the key generation algorithm into two steps. While such a structure has been extensively studied in contexts such as hierarchical IBE and delegatable ABE, it has received little attention in AB-IPFE despite its naturalness. In this paper, we formalize a framework for two-level hierarchies in AB-IPFE and classify existing schemes according to their hierarchizability. In particular, we focus on adaptively secure schemes that support expressive access control, such as arithmetic programs in the public-index setting and attribute-hiding inner-product predicates in the private-index setting. To the best of our knowledge, the only efficient constructions known to meet these requirements are those obtained from the framework of Abdalla et al. in the private-index setting and from the scheme of Datta and Pal (Asiacrypt'21) in the public-index setting. We propose several new pairing-based constructions that achieve adaptive security. First, we revisit the approach of Abdalla et al. Based on their framework, we propose three types of constructions that trade off hierarchizability, ciphertext size, and secret-key size. These constructions are based on predicate encodings and therefore support arithmetic span programs or attribute-hiding inner-product predicates. Second, we revisit the approach of Datta and Pal. Their scheme supports attribute-weighted sums, which differ slightly from our target functionality. Although this already yields an AB-IPFE scheme, the resulting scheme is limited to the non-zero-type key-policy setting. We efficiently adapt their scheme to obtain AB-IPFE schemes that also support zero-type predicates, and we propose a ciphertext-policy variant. All of the resulting constructions can be hierarchized, but their intermediate secret keys are large. We also propose variant schemes with shorter intermediate secret keys, at the cost of somewhat larger final secret keys. These schemes support arithmetic branching programs.