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Mathematical Structures for Cryptography
Scientific report: Mathematical Structures for Cryptography
Léo Ducas, Hendrik Lenstra, Alice Silverberg, Marco Streng
Description and aims
The goal of the workshop was to find new ways to use mathematical structures for cryptographic applications.
Very successful existing examples of such mathematical structures are given by RSA and elliptic curves, which your browser uses nowadays to set up a secure connection with online banking and other web-based services. A promising recent example is the use of lattices in fully homomorphic encryption: a form of encryption where untrusted parties can compute properties of encrypted data without learning the content of the original data. This is becoming more and more important with the rise of online ‘cloud’ services.
Algebra, number theory and algebraic geometry have been a fertile source of suitable structures (RSA, lattices, elliptic curves, abelian varieties), and this workshop aims to bring together researchers from the cryptography and mathematics communities to work towards the goal mentioned above.
In addition to talks by cryptographers and mathematicians, the workshop included ample time for informal discussion and interactions and open question sessions. This aspect seemed particularly successful, with many small groups forming to exchange or collaborate in the many offices provided by the Lorentz center, and several participants expressed their enthusiasm for this format.
The expectation for a wide variety of talks was also met, both in terms of topics than of format. Many tutorial talks were given which helped a lot each community (algebraists, number theorists, cryptographers) to understand the motivations and questions of the others. A few advanced lectures on recent research nevertheless showed the depth of the topics of this workshop.
Considering the rather wide range of backgrounds of different participant, no ‘great scientific breakthroughs’ were expected for this workshop. Rather than developing one common goal, the workshop seems to have been successful at cultivating many shared research interests. For example, the open problem sessions were quite interactive, and some of those problems will for sure be “brought back home” by other participants to be solved.
Several deep mathematical talks were given, but always keeping alive the connection with cryptographic matters. In particular the design of algebraic curves well fitted for efficient implementation is still a lively and fertile topic, as well as progress in cryptanalysis on the discrete logarithm problem. But some new mathematical theories made their appearance with interesting cryptographic consequences, such as capacity theory.
The workshop was also successful in portraying the transition of interests of the cryptographic community to other mathematical objects, in particular to lattices, and algebraic number theory; a transition mostly motivated by the goal of designing cryptographic primitive that would resist quantum computing. But the tutorial talk (F. Vercauteren) on the recent SIDH proposal (Singular-Isogeny Diffie-Hellman) has raised a lot of interest. It could be that algebraic geometry remains relevant in cryptography in a post-quantum world. This for sure will help to bring this scheme under the required scrutiny of experts.