[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85859-en":3,"doc-seo-85859-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85859,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","On CC0 Lower Bounds for AND via Torus Polynomials","The paper studies torus polynomial approximation as a route to a long-standing circuit lower-bound question: whether the AND function can be computed by CC0 circuits, i.e., constant-depth polynomial-size circuits with MODm gates. It builds on torus polynomial techniques introduced to target ACC0/CC0 lower bounds. Using degree lower bounds for symmetric torus polynomials approximating AND, it derives size lower bounds for symmetric CC0 circuits at depth h. It further develops constructions linking circuit symmetry to polynomial symmetry and proves degree bounds for related asymmetric classes.","arXiv :2607 . 10236v 1 [ cs .CC] 11 Jul 2026  \nOn CC0 Lower Bounds for AND via Torus Polynomials  \nVaibhav Krishan* Jayalal Sarma†  \nJuly 14, 2026  \nAbstract  \nWe explore the torus polynomial approximation based approach towards a long-standing question: whether AND can be computed by CC0 circuits-the class of constant-depth polynomial size circuits containing MODm gates for some natural number m. Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach for proving lower bounds against ACC0-a class containing CC0 where the circuits are also allowed AND, OR and NOT gates.  \nWe show how lower bounds for torus polynomials approximating AND can be used to make progress on this question. Using lower bounds on the degree of symmetric torus polynomials approximating AND, proved by Krishan and Vishwanathan (ITCS 2026), we prove size lower bounds for symmetric CC0-circuits computing AND. More precisely, we prove that any depth h symmetric CC0 circuit requires 2(n1/O(h) ) size to compute AND.  \nA key ingredient in our proof is an argument that we can construct symmetric torus polynomials to approximate symmetric CC0 circuits. Our construction exhibits an explicit correspondence between the symmetry of the circuit and that of the polynomial. Using this, we also establish lower bounds for weaker notions of circuit symmetry. Lower bounds for symmetric CC0 circuits were also independently established by Pago (ICALP 2026) using different techniques.  \nIn the asymmetric regime, we establish degree upper bounds for depth three circuits of the form MODp ◦ MODm ◦ ANDO(1) where m = pq is a semiprime. This circuit class is a special case of the constant degree hypothesis, introduced by Barrington, Straubing and Thrien (Information and Computation, 1990), where m could be an arbitrary composite number. We argue that improved lower bounds for asymmetric torus polynomials approximating AND imply size lower bounds for semiprime m and hence progress on the constant-degree hypothesis.  \n* The Institute of Mathematical Sciences, Chennai, [India. Email:](India. Email: vaibhavk@imsc.res.in)[ vaibhavk@imsc.res.in](India. Email: vaibhavk@imsc.res.in)[ ](India. Email: vaibhavk@imsc.res.in)†Indian Institute of Technology Madras (IIT Madras), Chennai, India. Email: [jayalal@cse.iitm.ac.in](jayalal@cse.iitm.ac.in)  \nContents  \n1 Introduction 2  \n2 Preliminaries 6  \n3 Size Lower Bounds for Symmetric CC0 Circuits 7  \n3.1 Converting CC0 circuits to Layered Form ......................... 8  \n3.2 Torus Polynomial Approximations for Layered CC0 Circuits .............. 10  \n3.3 Symmetric Circuits Lead to Symmetric Torus Polynomials ............... 12  \n3.4 Nested Block Symmetric Groups .............................. 13  \n4 Towards Constant Degree Hypothesis for Semiprime Moduli 14  \n5 Degree Upper Bounds for Periodic Functions 15  \n1 Introduction  \nThe polynomial method has proven to be a powerful tool for tackling fundamental questions in theoretical computer science. In particular, studying polynomial approximations for Boolean functions has led to advances in cryptography, quantum computing and circuit complexity (see [Aar08, BT21] and references therein) . In circuit complexity, considering the degree of polynomials approximating Boolean functions has led to remarkable progress in proving circuit lower bounds for explicit functions, a notoriously difficult quest in the area.  \nLandmark results in this direction were proved by Razborov [Raz87], and independently by Smolensky [Smo87] . They proved that for any prime p, MODp 1 cannot be computed by constantdepth polynomial size circuits that use AND, OR, NOT and MODq gates for a prime q  p. In addition, they used the same technique to prove that such circuits cannot compute MAJORITY. One main technical step in these arguments is that any function in the circuit class can be approximated by a low-degree polynomial over the finite field Fq. This is complemented by an arg","cbCaiu8mf8h2Cyw6","https://ap.wps.com/l/cbCaiu8mf8h2Cyw6","pdf",406589,2,1,20,"English","en",105,"# Introduction\n# Preliminaries\n# Size Lower Bounds for Symmetric CC0 Circuits\n## Converting CC0 circuits to Layered Form\n## Torus Polynomial Approximations for Layered CC0 Circuits\n## Symmetric Circuits Lead to Symmetric Torus Polynomials\n# Towards Constant Degree Hypothesis for Semiprime Moduli\n# Degree Upper Bounds for Periodic Functions","[{\"question\":\"What is the central question about CC0 circuits and the AND function?\",\"answer\":\"The work asks whether AND can be computed by CC0 circuits, which are constant-depth, polynomial-size circuits containing MODm gates. The paper connects this question to what degree is required for torus polynomial approximations of AND.\"},{\"question\":\"How do lower bounds for torus polynomials lead to circuit size lower bounds?\",\"answer\":\"The paper uses degree lower bounds for symmetric torus polynomials approximating AND. These polynomial degree bounds are then converted into lower bounds on the size of symmetric CC0 circuits computing AND, showing size grows as 2^{n^{1/O(h)}} at depth h.\"},{\"question\":\"What role does symmetry play in the proof and the construction?\",\"answer\":\"A key ingredient is an argument that symmetric torus polynomials can be constructed to approximate symmetric CC0 circuits. The construction explicitly corresponds the symmetry properties of the circuit to the symmetry of the polynomial, and the paper extends this to weaker notions of circuit symmetry.\"}]",1784206748,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-cc0-lower-bounds-for-and-via-torus-polynomials","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-cc0-lower-bounds-for-and-via-torus-polynomials/85859/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the central question about CC0 circuits and the AND function?","Question",{"text":75,"@type":76},"The work asks whether AND can be computed by CC0 circuits, which are constant-depth, polynomial-size circuits containing MODm gates. The paper connects this question to what degree is required for torus polynomial approximations of AND.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do lower bounds for torus polynomials lead to circuit size lower bounds?",{"text":80,"@type":76},"The paper uses degree lower bounds for symmetric torus polynomials approximating AND. These polynomial degree bounds are then converted into lower bounds on the size of symmetric CC0 circuits computing AND, showing size grows as 2^{n^{1/O(h)}} at depth h.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does symmetry play in the proof and the construction?",{"text":84,"@type":76},"A key ingredient is an argument that symmetric torus polynomials can be constructed to approximate symmetric CC0 circuits. The construction explicitly corresponds the symmetry properties of the circuit to the symmetry of the polynomial, and the paper extends this to weaker notions of circuit symmetry.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]