All is fair in war, love, and mathematics. Eric Temple Bell
Alice and Bob are two millionaires. They want to determine who is richer, but do not want to disclose their actual wealth. A possible solution is to provide a trusted third party with their accounts’ balances and let this party announce who is richer. The use this crutch is unsatisfactory and leaves every cryptographer with an uncomfortable feeling. Through secure multi-party computations, we provide Alice and Bob with a protocol to jointly compute the answer to their question using cryptographic primitives instead of a trusted third party.
Cryptography’s model of users is quite simplistic: every user is either good or bad. Game theory offers a more nuanced version: every participant is trying to achieve his or her individual goals and behaves “rationally” in order to do so. We revisit multi-party computations with such rational players and arrive at surprising results.
By the way, the example above is the classic Yao’s Millionaires’ Problem. A recent application is the Bitcoin cryptocurrency, where the users jointly compute the “history” of transactions.
All models are wrong, but some are useful. George Box
The majority of attacks on cryptosystems is statistical, ranging from frequency analysis (of human languages) to correlations (of bit patterns). Differential and linear cryptanalysis are prime examples of the latter.
The classic field of statistics has gotten a fresh impulse from the emergence of artificial intelligence/machine learning in the last century. Both being based on huge amounts of data, they offer different perspectives. We take this as inspiration to attack cryptosystems using machine-learning methods.
Polynomials over Finite Fields
Polynomials appear all over mathematics and computer science. They inherit many basic notions from the integers, e.g. primality and squarefreeness. But the absence of carries for sums and products of polynomials makes them more accessible.
In the business of understanding polynomials, we measure our success by our ability to count the ones with certain properties – approximately and exactly. From this point of view, I approached:
- special multivariate polynomials
- decomposable univariate polynomials.
The decomposition of polynomials has many applications.
- In the field of signal processing, its application to the Fourier transform improves the transmission rate.
- It is closely related to the factorization of projective polynomials. These play a prominent role in the recent advances on the discrete logarithm problem.
- It is closely related to the factorization of skew polynomials. This problem is at the heart of some cryptographic key-exchanges.
- It is the underlying structure of iterated block ciphers. Then decomposition is equivalent to breaking.
- Finally, you find it in Beethoven’s compositions.