About

Jeffrey Hoffstein is a professor in the Department of Mathematics at Brown University, with research interests spanning number theory, automorphic forms, and cryptography. His extensive work includes contributions to the development of cryptographic schemes such as NTRUEncrypt and NTRUSign, which are based on lattice theory, as well as research on automorphic L-series, multiple Dirichlet series, and theta functions. Hoffstein has authored numerous publications in these fields, advancing the understanding of automorphic forms, L-functions, and their applications to number theory and cryptography. His work is characterized by a focus on the intersection of pure mathematics and practical cryptographic applications, contributing significantly to both theoretical and applied aspects of modern mathematics.

Research topics

• Computer Science
• Mathematics
• Geometry
• Pure mathematics
• Mathematical analysis
• Algorithm
• Physics
• History

Selected publications

• NTRU

  2025-01-01

  book-chapterSenior author
  PublisherDOI

• Privately Generated Key Pairs for Post Quantum Cryptography in a Distributed Network

  Applied Sciences · 2024-10-02 · 5 citations

  articleOpen access

  In the proposed protocol, a trusted entity interacts with the terminal device of each user to verify the legitimacy of the public keys without having access to the private keys that are generated and kept totally secret by the user. The protocol introduces challenge–response–pair mechanisms enabling the generation, distribution, and verification of cryptographic public–private key pairs in a distributed network with multi-factor authentication, tokens, and template-less biometry. While protocols using generic digital signature algorithms are proposed, the focus of the experimental work was to implement a solution based on Crystals-Dilithium, a post-quantum cryptographic algorithm under standardization. Crystals-Dilithium generates public keys consisting of two interrelated parts, a matrix generating seed, and a vector computed from the matrix and two randomly picked vectors forming the secret key. We show how such a split of the public keys lends itself to a two-way authentication of both the trusted entity and the users.

  PublisherOA PDFDOI

• The shifted convolution L-function for Maass forms

  Research in Number Theory · 2024-10-16 · 1 citations

  articleSenior author
  PublisherDOI

• Methods to Encrypt and Authenticate Digital Files in Distributed Networks and Zero-Trust Environments

  Axioms · 2023-05-29 · 8 citations

  articleOpen access

  The methods proposed in this paper are leveraging Challenge–Response–Pair (CRP) mechanisms that are directly using each digital file as a source of randomness. Two use cases are considered: the protection and verification of authenticity of the information distributed in storage nodes and the protection of the files kept in terminal devices operating in contested zero-trust environments comprised of weak signals in the presence of obfuscating electromagnetic noise. With the use of nonces, the message digests of hashed digital files can be unique and unclonable; they can act as Physical Unclonable Functions (PUF)s in challenge–response mechanisms. During enrollment, randomly selected “challenges” result in unique output data known as the “responses” which enable the generation and distribution of cryptographic keys. During verification cycles, the CRP mechanisms are repeated for proof of authenticity and deciphering. One of the main contributions of the paper is the development of mechanisms accommodating the injection of obfuscating noises to mitigate several vectors of attacks, disturbing the side channel analysis of the terminal devices. The method can distribute error-free cryptographic keys in noisy networks with light computing elements without relying on heavy Error Correcting Codes (ECC), fuzzy extractors, or data helpers.

  PublisherOA PDFDOI

• Lynne Heather Walling (1958–2021)

  Notices of the American Mathematical Society · 2023

  • History
  DOI

• The shifted convolution L-function for Maass forms

  arXiv (Cornell University) · 2023-11-11

  preprintOpen accessSenior author

  Let $Φ_1,Φ_2$ be Maass forms for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $C_1(n),C_2(n)$. For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution L-function $$L_h(s,{Φ_1,Φ_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For ${\rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{1}{2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}

  PublisherOA PDFDOI

• Non‐vanishing of symmetric cube L$L$‐functions

  Journal of the London Mathematical Society · 2022-10-03

  article1st authorCorresponding

  We prove that there are infinitely many Maass-Hecke cuspforms over the field Q[ √ -3] such that the corresponding symmetric cube L-series does not vanish at the center of the critical strip.This is done by using a result of Ginzburg, Jiang and Rallis which shows that the symmetric cube non-vanishing happens if and only if a certain triple product integral involving the cusp form and the cubic theta function on Q[ √ -3] does not vanish.We use spectral theory and the properties of the cubic theta function to show that the non-vanishing of this triple product occurs for infinitely many cusp forms.We also formulate a conjecture about the meaning of the absolute value squared of the triple product which is reminiscent of Watson's identity.J. H. would like to thank S.

  PublisherDOI

• Homomorphic Encryption Standard

  2021-01-01 · 37 citations

  preprint
  PublisherDOI

• Non-vanishing of symmetric cube $L$-functions

  Bristol Research (University of Bristol) · 2021-09-21 · 1 citations

  preprintOpen access1st authorCorresponding

  We prove that there are infinitely many Maass--Hecke cuspforms over the field $\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series does not vanish at the center of the critical strip. This is done by using a result of Ginzburg, Jiang and Rallis which shows that the symmetric cube non-vanishing happens if and only if a certain triple product integral involving the cusp form and the cubic theta function on $\mathbb{Q}[\sqrt{-3}]$ does not vanish. We use spectral theory and the properties of the cubic theta function to show that the non-vanishing of this triple product occurs for infinitely many cusp forms. We also formulate a conjecture about the meaning of the absolute value squared of the triple product which is reminiscent of Watson's identity.

  PublisherDOI

• First moments of Rankin–Selberg convolutions of automorphic forms on $${{\,\mathrm{GL}\,}}(2)$$

  Research in Number Theory · 2021 · 2 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  PublisherOA PDFDOI

Recent grants

• FRG: Collaborative Research: Combinatorial representation theory, multiple Dirichlet series and moments of L-functions.

  NSF · $225k · 2007–2012

• Collaborative Research: FRG: Applications of Multiple Dirichlet Series to Analytic Number Theory

  NSF · $319k · 2004–2008

• TWC: Medium: Collaborative: Development and Evaluation of Next Generation Homomorphic Encryption Schemes

  NSF · $358k · 2016–2019

Frequent coauthors

• Joseph H. Silverman

  71 shared

• Jill Pipher

  69 shared

• William Whyte

  Qualcomm (United States)

  25 shared

• Daniel Bump

  23 shared

• Solomon Friedberg

  Boston College

  21 shared

• Min Lee

  20 shared

• Samuel Dickman

  Planned Parenthood

  16 shared

• Michelle Manes

  American Institute of Mathematics

  16 shared

Labs

• AlgebraPI