Quantum computing and Grover's algorithm

Speaker: 

Shahed Sharif

Institution: 

California State University San Marcos

Time: 

Thursday, June 6, 2019 - 9:30am to 10:20am

Host: 

Location: 

RH 510R

Given a database of $N$ entries of which exactly one satisfies some
easily checked condition, classically it takes $O(N)$ trials to find the
satisfying entry. Grover's algorithm is a quantum algorithm which
reduces the work to $O(\sqrt{N})$ trials. One consequence is that in the
post-quantum regime, hash functions and symmetric ciphers only provide
half the security (measured as the log of the number of trials) as
currently provided. In this talk, we will give a brief description of
Grover's algorithm, including all of the necessary background in quantum
computing.

The Charles-Goren-Lauter Hash Function

Speaker: 

Travis Scholl

Institution: 

University of California, Irvine

Time: 

Thursday, May 23, 2019 - 9:30am to 10:20am

Host: 

Location: 

RH 510R

In this talk we will summarize the Charles-Goren-Lauter hash function that is built on isogeny graphs of supersingular curves. We will also look at the quaternion analogue, which is broken. We will attempt to explain why breaking the quaternion model does not immediately break the isogeny model.

The Humbert Invariant and Isogeny Graphs of Abelian Surfaces

Speaker: 

Shahed Sharif

Institution: 

California State University San Marcos

Time: 

Friday, May 10, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

In a series of papers, Kani studies elliptic subcovers of genus 2 curves. This talk will focus on summarizing this work and it's applications to a recent question of Galbraith concerning isogeny graphs of abelian surfaces. Let A be a Jacobian of a curve of genus 2 which is isogeneous, but not isomorphic, to a product of elliptic curves. Galbraith's question is how "far" is A from a product of ellitpic curves in the $\ell$-isogeny graph for a small prime $\ell$.

Multilinear Cryptography using Nilpotent Groups

Speaker: 

Alice Silverberg

Institution: 

University of California, Irvine

Time: 

Friday, April 26, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

In a recent paper, Kahrobaei, Tortora, and Tota proposed a multilinear cryptosystem using nilpotent groups. This talk will be an exposition of that paper which is available at https://arxiv.org/abs/1902.08777.

Smoothing Ideals in Quaternion Algebras

Speaker: 

Travis Scholl

Institution: 

University of California, Irvine

Time: 

Friday, April 19, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

In this talk we will go over one an algorithm of Kohel-Lauter-Petit-Tignol that, given a left ideal for a maximal order in a quaternion algebra, returns an equivalent left ideal with $\ell$-power reduced norm (The preprint can be found here: https://arxiv.org/pdf/1406.0981.pdf). This algorithm was mentioned in the previous talk as a supplementary algorithm to the reductions between certain computational problems. If there is time, we will also discuss a recent signatue scheme proposed by Galbraith-Petit-Silva based on this algorithm.

Computing Supersingular Isogenies and Endomorphism Rings Part 2

Speaker: 

Shahed Sharif

Institution: 

California State University San Marcos

Time: 

Friday, April 12, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

We will continue discussing the paper by Eisentraeger-Hallgren-Lauter-Morrison-Petit (available at https://eprint.iacr.org/2018/371). In this part of the talk, we will focus on the reductions between the major problems. We will also outline the supplementary algorithms used as a black box in these reductions.

Computing Supersingular Isogenies and Endomorphism Rings

Speaker: 

Shahed Sharif

Institution: 

California State University San Marcos

Time: 

Friday, April 5, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

In this talk we will review a recent paper by Eisentraeger-Hallgren-Lauter-Morrison-Petit (available at https://eprint.iacr.org/2018/371). This paper gives reductions between several important computational problems including finding a path in the $\ell$-isogeny graph between two supersingular elliptic curves, and computing the endomorphism ring of a supersingular elliptic curve curve. This work requires some background on quaternion algebras which can be in Voight's book https://math.dartmouth.edu/~jvoight/quat-book.pdf.

A Discussion on Some Open Problems

Speaker: 

Travis Scholl

Institution: 

University of California, Irvine

Time: 

Friday, March 15, 2019 - 11:00am to 11:50am

Host: 

Location: 

PSCB 220

This talk will be more of a discussion on some open problems we have seen so far. It should be independent of the previous talks. We will focus on a few of the problems on class groups and isogenies referred to in the Altug-Chen paper discussed on 2/22/19. The preprint is available here: https://eprint.iacr.org/2018/926 and notes from the previous talk can be found here: https://www.math.uci.edu/~schollt/multilinear_map_seminar/scholl-02-22-1.... The problems we will focus on are finding an elliptic curve with a specified endomorphism ring, and finding l-isogenies over composite rings.

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