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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 webbased 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. Format 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. Scientific developments 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 (SingularIsogeny DiffieHellman)
has raised a lot of interest. It could be that algebraic geometry remains
relevant in cryptography in a postquantum world. This for sure will help to
bring this scheme under the required scrutiny of experts. [Back] 