AMS Sectional Meeting Special Session on The Mathematics of Cryptography

The American Mathematical Society Spring Central and Western Joint Sectional
Meeting was held at the University of Hawaii at Manoa, in Honolulu during March 22-24, 2019.

There was a Special Session on The Mathematics of Cryptography organised by Shahed Sharif and Alice Silverberg. Slides of (some of) the talks are available here.

Among the talks I attended, I mention these:

  • Isogeny cryptography: strengths, weaknesses and challenges, by Steven Galbraith.

    I talked about CSIDH and SeaSign, and then said a little bit about some work of my PhD student Yan Bo Ti on hash functions from dimension 2 supersingular abelian varieties. The slides online also cover some other topics that I did not mention (Kuperberg’s algorithm and quaternion algebras).

  • Identity-Based Encryption from the Diffie-Hellman Assumption by Sanjam Garg.

    This was a really nice talk that described a “hash encryption” scheme based on the (decisional) Diffie-Hellman problem and explained how this enables identity-based encryption from the Diffie-Hellman problem (no pairings needed). This is joint work with Nico Döttling. The schemes are not practical.

  • Ramanujan Graphs In Cryptography by Kristin Lauter.

    Kristin reported joint work with Anamaria Costache, Brooke Feigon, Maike Massierer and Anna Puskas about some computational problems in isogeny graphs. A paper on this work is eprint 2018/593.

  • Numerical Method for Comparison on Homomorphically Encrypted Numbers by Jung Hee Cheon.

    Jung Hee talked about some basic mathematical functions (such as max and min) that are useful for practical computations on encrypted data. He explained some iterated processes (in his words “nowadays I am working in numerical analysis”) that give low-depth circuits to compute approximations to these functions.

  • Multiparty Non-Interactive Key Exchange From Isogenies on Elliptic Curves by Shahed Sharif.

    Shahed talked (on the blackboard — there are no slides) about his paper eprint 2018/665 with Dan Boneh, Darren Glass, Daniel Krashen, Kristin Lauter, Alice Silverberg, Mehdi Tibouchi and Mark Zhandry. The scheme is still incomplete as no suitable efficiently computable isomorphism invariant of abelian varieties has been found. Shahed discussed attempts to find such invariant, and I learned some interesting facts about polarizations on abelian surfaces.

  • The Hidden Quadratic Form Problem by Joseph Silverman.

    Joe presented joint work with Jeff Hoffstein and others on a new candidate number-theoretical problem that might be interesting for new signature schemes. This is a work-in-progress and is not published yet.

  • Isolated Curves and Cryptography by Travis Scholl.

    Travis presented his papers eprint 2017/383, eprint 2018/307 and some newer work on “isolated curves”.

  • Fun with the hidden number problem by Nadia Heninger.

    Nadia surveyed her joint work with Breitner, published as “Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies”. Her talk also included an overview of lattice algorithms for the hidden number problem, and a very clear sketch of Bleichenbacher’s approach using Fourier analysis to the hidden number problem.

  • Short digital signatures via isomorphisms between modular lattices based on finite field isomorphisms by Jeffrey Hoffstein.

    Jeff presented very new joint work with Joe Silverman on yet another number-theoretical problem that might be interesting for new signature schemes. This is related to their previous work on isomorphisms of finite fields, but with new ideas and applications. I was not able to follow the details of the talk. There is no preprint yet on this work.

  • Computing isogenies and endomorphism rings of supersingular elliptic curves by Travis Morrison.

    Travis gave an overview of his EUROCRYPT 2018 paper with Eisentraeger, Hallgren, Lauter and Petit.

  • Lower bounds for Hilbert class polynomials by Reinier Broker.

    Much work on algorithms to compute Hilbert class polynomials requires proving good upper bounds on the size (e.g., bitlength) of these polynomials. Reinier spoke about his current work-in-progress trying to prove lower bounds on the size of these polynomials.

There was also a Special Session on Emerging Connections with Number Theory organised by Kate Stange and Renate Scheidler, plus a lot of other sessions, that included talks of some interest to readers of this blog. However, I stayed in the Mathematics of Cryptography room.

— Steven Galbraith

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