Extractable Witness Encryption for the Homogeneous Linear Equations Problem

Witness encryption is a cryptographic primitive which encrypts a message under an instance of an NP language and decrypts the ciphertext using a witness associated with that instance. In the current state of the art, most of the witness encryption constructions are based on multilinear maps. Following the construction of Choi and Vaudenay based on RSA-related problems, we suggest a novel witness key encapsulation mechanism based on the hardness of solving homogeneous linear Diophantine equations (HLE problem). Our arithmetic-based construction aims to solve an issue raised by these authors where the security might be compromised if the adversary is able to find small solutions to a homogeneous linear Diophantine equation, while avoiding the inefficiency of multilinear maps. The security of our scheme is based on a hidden group order and a knowledge assumption.

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Diophantine equation

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations.

Cryptographic primitive

Cryptographic primitives are well-established, low-level cryptographic algorithms that are frequently used to build cryptographic protocols for computer security systems. These routines include, but are not limited to, one-way hash functions and encryption functions. When creating cryptographic systems, designers use cryptographic primitives as their most basic building blocks. Because of this, cryptographic primitives are designed to do one very specific task in a precisely defined and highly reliable fashion.

P versus NP problem

The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term quickly, used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time).

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