From Quadratic Functions to Polynomials: Generic Functional Encryption from Standard Assumptions

Linru Zhang, Yuechen Chen, Jun Zhang, Meiqi He, Siu-Ming Yiu

Abstract: The “all-or-nothing” notion of traditional public-key encryptions is found to be insufficient for many emerging applications in which users are only allowed to obtain a functional value of the ciphertext without any other information about the ciphertext. Functional encryption was proposed to address this issue. However, existing functional encryption schemes for generic circuits either have bounded collusions or rely on not well studied assumptions. Recently, Abdalla et al. started a new line of work that focuses on specific functions and well-known standard assumptions. Several efficient schemes were proposed for inner-product and quadratic functions. There are still a lot of unsolved problems in this direction, in particular, whether a generic FE scheme can be constructed for quadratic functions and even higher degree polynomials. In this paper, we provide affirmative answers to these questions. First, we show an IND-secure generic functional encryption scheme against adaptive adversary for quadratic functions from standard assumptions. Second, we show how to build a functional encryption scheme for cubic functions (the first in the literature in public-key setting) from a functional encryption scheme for quadratic functions. Finally, we give a generalized method that transforms an IND-secure functional encryption scheme for degree-m polynomials to an IND-secure functional encryption scheme for degree-\((m+1)\) polynomials.