# Practical fully secure unrestricted inner product functional encryption modulo a prime p

Functional encryption (FE) is an advanced cryptographic primitive which allows, for a single encrypted message, to finely control how much information on the encrypted data each receiver can recover. To this end many functional secret keys are derived from a master secret key. Each functional secret key allows, for a ciphertext encrypted under the associated public key, to recover a specific function of the underlying plaintext.

However constructions for general FE are far from practical, or rely on
non-standard and ill-understood cryptographic assumptions.

In this talk I will focus on the construction of efficient FE schemes for linear
functions (i.e. the inner product functionality), and the framework in which
our constructions hold. Such schemes yield many practical applications, and
our constructions are the first FE schemes for inner products modulo a
prime that are both efficient and recover the result whatever its size.
Our framework consist of a cyclic group $G$ where the decision Diffie-Hellman
assumption holds together with a subgroup $F$ of $G$ where the discrete logarithm
problem is easy. This setting can be instantiated with class groups of imaginary
quadratic fields, and allows us to encode information in the exponent of the subgroup
$F$, which can be efficiently recovered whatever its size.