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1. Arithmetic Sketching

   Boneh, Dan; Boyle, Elette; Corrigan-Gibbs, Henry; Gilboa, Niv; Ishai, Yuval (January 2023, IACR International Cryptology Conference (CRYPTO))

   This paper introduces arithmetic sketching, an abstraction of a primitive that several previous works use to achieve lightweight, low-communication zero-knowledge verification of secret-shared vectors. An arithmetic sketching scheme for a language L ⊆ F^n consists of (1) a randomized linear function compressing a long input x to a short “sketch,” and (2) a small arithmetic circuit that accepts the sketch if and only if x ∈ L, up to some small error. If the language L has an arithmetic sketching scheme with short sketches, then it is possible to test membership in L using an arithmetic circuit with few multiplication gates. Since multiplications are the dominant cost in protocols for computation on secret-shared, encrypted, and committed data, arithmetic sketching schemes give rise to lightweight protocols in each of these settings. Beyond the formalization of arithmetic sketching, our contributions are: – A general framework for constructing arithmetic sketching schemes from algebraic varieties. This framework unifies schemes from prior work and gives rise to schemes for useful new languages and with improved soundness error. – The first arithmetic sketching schemes for languages of sparse vectors: vectors with bounded Hamming weight, bounded L1 norm, and vectors whose few non-zero values satisfy a given predicate. – A method for “compiling” any arithmetic sketching scheme for a language L into a low-communication malicious-secure multi-server protocol for securely testing that a client-provided secret-shared vector is in L. We also prove the first nontrivial lower bounds showing limits on the sketch size for certain languages (e.g., vectors of Hamming-weight one) and proving the non-existence of arithmetic sketching schemes for others (e.g., the language of all vectors that contain a specific value). 

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2. Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers

   https://doi.org/10.1093/imrn/rnx271  

   Chung, Hee-Joong; Kim, Dohyeong; Kim, Minhyong; Pappas, Georgios; Park, Jeehoon; Yoo, Hwajong (November 2017, International Mathematics Research Notices)

   null (Ed.)

   Abstract Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of $$n$$-th power residue symbols. This formalism leads to a precise arithmetic analogue of a “path-integral formula” for linking numbers. 

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3. Problems in Arithmetic Topology

   https://doi.org/10.1007/s40687-021-00264-5  

   Gómez-Gonzáles, C.; Wolfson, J. (April 2021, Research in the mathematical sciences)

   We present a list of problems in arithmetic topology posed at the June 2019 PIMS/NSF workshop on “Arithmetic Topology.” Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Participants were explicitly asked to provide problems of various levels of difficulty, with the goal of capturing a cross section of exciting challenges in the field that could help guide future activity. The problems, together with references and brief discussions when appropriate, are collected below into three categories: (1) topological analogues of arithmetic phenomena, (2) point counts, stability phenomena and the Grothendieck ring, and (3) tools, methods and examples. 

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4. Arithmetic of Borcherds products

   https://doi.org/10.24033/ast.1128  

   Howard, Benjamin; Madapusi Pera, Keerthi (January 2020, Asterisque)
5. The arithmetic of Coxeter permutahedra

   https://doi.org/10.18257/raccefyn.1189  

   Ardila, Federico; Beck, Matthias; McWhirter, Jodi (December 2020, Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales)

   null (Ed.)

   Ehrhart theory mesures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, ..... We compute the Ehrhart theory of four families of polytopes of great importance in several areas of mathematics: the standard Coxeter permutahedra for the classical Coxeter groups An, Bn, Cn, Dn. A central tool, of independent interest, is a description of the Ehrhart theory of a rational translate of an integer projection of a cube. 

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