Cryptography builds on computationally hard problems. A cryptographer seeks problems so hard that they can't be solved, on average, in reasonable time, without a hint in the form of a key, i.e. a piece of information which, when processed through a cryptographic algorithm, can encode or decode data.
In public-key cryptography, in particular, the encrypting side uses a publicly known key to encrypt its information, while the decrypting side uses a secret, and different, key. Public-key cryptography is extremely useful in many applications, but it comes with two downsides: it is a slow process and finding hard problems for it is much more difficult than finding hard problems for private-key cryptography.
The kind of computational hardness we need for public-key cryptography is hard to come by, and in 45 years we only have found a handful of candidate hard problems. This is an important problem for society as all forms of communication and electronic commerce crucially rely on public-key encryption. For such applications to be viable in the medium and long term, it is imperative to develop a wide array of alterative encryption methods, to provide fallbacks in case current methods turn out to be insecure, an event that could potentially occur at any given day.
The FGC (Fine-Grained Cryptography) project aims, as one of its central goals, to find new hard problems for public-key cryptography. Traditionally, cryptography has been based on problems for which there is a conjectured exponential complexity gap between the easy and hard directions, where complexity can be considered a measure of the computational power needed to solve a problem.
The project explores a new avenue, where the complexity gap is not exponential, but "only" polynomial. For instance, if the complexity faced by the “good guy” is 128 raised to the power of two, the “bad guy” also faces 128 raised to a power. If the power is sufficiently high, the code is as secure as with an exponential complexity gap: 128 raised to a power of 20, for example, is roughly equal to 2 raised to a power of 140 and thus comparable to the exponential gap case.