{"id":"https://openalex.org/W7135227146","doi":"https://doi.org/10.48550/arxiv.2603.11196","title":"Primitive-Root Ratio over Prime Fields: A Shifted-Prime Distribution of Hausdorff Dimension Zero and Implications for PRIM-LWE","display_name":"Primitive-Root Ratio over Prime Fields: A Shifted-Prime Distribution of Hausdorff Dimension Zero and Implications for PRIM-LWE","publication_year":2026,"publication_date":"2026-03-11","ids":{"openalex":"https://openalex.org/W7135227146","doi":"https://doi.org/10.48550/arxiv.2603.11196"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.11196","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.11196","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2603.11196","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006963894","display_name":"Vipin Singh Sehrawat","orcid":"https://orcid.org/0000-0002-1603-3929"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Sehrawat, Vipin Singh","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5006963894"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9132999777793884,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9132999777793884,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.032499998807907104,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.014100000262260437,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/pointwise","display_name":"Pointwise","score":0.6474000215530396},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.48429998755455017},{"id":"https://openalex.org/keywords/prime-number","display_name":"Prime number","score":0.4675999879837036},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.4577000141143799},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.45089998841285706},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.44209998846054077},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.44179999828338623},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.44110000133514404}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8208000063896179},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.6474000215530396},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.48429998755455017},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.47699999809265137},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.4675999879837036},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.4577000141143799},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.45089998841285706},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.44209998846054077},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.44179999828338623},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.44110000133514404},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.42179998755455017},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3808000087738037},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3702000081539154},{"id":"https://openalex.org/C121089165","wikidata":"https://www.wikidata.org/wiki/Q6889796","display_name":"Moduli","level":2,"score":0.34700000286102295},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3467000126838684},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.314300000667572},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.2883000075817108},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28700000047683716},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C2779960059","wikidata":"https://www.wikidata.org/wiki/Q7113681","display_name":"Overhead (engineering)","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.2578999996185303}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.11196","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.11196","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.11196","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.11196","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"For":[0],"a":[1,19,28,92,125,141],"prime":[2],"$p$,":[3],"let":[4],"$c(p)$":[5,26],"denote":[6],"the":[7,33,43,49,54,60,68,76,102,107,119,146,158,173,186,192],"limiting":[8,50,108,120],"fraction":[9],"of":[10,32,48,98,128,167],"$n\\times":[11],"n$":[12],"matrices":[13],"over":[14,42],"$\\mathbb{F}_p$":[15],"whose":[16],"determinant":[17],"is":[18,27,71,110,122],"primitive":[20],"root":[21],"modulo":[22],"$p$.":[23],"The":[24,132,149,182,204],"quantity":[25,183],"natural":[29],"multiplicative":[30],"deformation":[31],"totient":[34],"ratio":[35],"$\u03c6(p-1)/(p-1)$":[36],"and":[37,46,67,85,113,139,157,220],"inherits":[38],"its":[39],"distributional":[40],"behaviour":[41],"primes.":[44,148],"Existence":[45],"continuity":[47],"law":[51],"follow":[52],"from":[53,194],"shifted-prime":[55],"Erd\u0151s--Wintner--Hildebrand":[56],"framework.":[57],"We":[58,90],"prove":[59],"following":[61],"new":[62],"results:":[63],"unconditionally,":[64],"$\\inf_p":[65],"c(p)=0$":[66],"sharp":[69],"order":[70],"$\\min_{p\\le":[72],"x}c(p)\\asymp":[73],"1/\\log\\log":[74],"x$;":[75],"reciprocal":[77],"satisfies":[78],"$\\limsup_{p\\to\\infty,":[79],"\\,":[80],"p\\text{":[81],"prime}}":[82],"1/(c(p)\\log\\log":[83],"p)=e^\u03b3$,":[84],"no":[86],"smaller":[87],"constant":[88,180],"suffices.":[89],"give":[91],"complete":[93],"proof,":[94],"combining":[95],"an":[96,164,178],"adaptation":[97],"Erd\u0151s's":[99],"argument":[100],"with":[101,177,196],"Jessen--Wintner":[103],"pure-type":[104],"dichotomy,":[105],"that":[106],"distribution":[109,133],"purely":[111],"singular,":[112],"strengthen":[114],"this":[115],"to":[116,163,199],"$\\dim_H(\u03bc_G)=0$,":[117],"i.e.":[118],"measure":[121],"carried":[123],"by":[124,145,153],"Borel":[126],"set":[127],"Hausdorff":[129],"dimension":[130],"zero.":[131],"has":[134],"full":[135],"topological":[136],"support":[137],"$[0,\\tfrac12]$":[138],"admits":[140],"Bernoulli-product":[142],"representation":[143],"indexed":[144],"odd":[147],"moments":[150],"are":[151],"given":[152],"convergent":[154],"Euler":[155],"products,":[156],"Mellin":[159],"transform":[160],"$\\mathbb{E}[X^s]$":[161],"extends":[162],"entire":[165],"function":[166],"$s$,":[168],"non-vanishing":[169],"on":[170],"$\\operatorname{Re}(s)&gt;0$.":[171],"Near":[172],"right":[174],"endpoint,":[175],"$1-G(\\tfrac12-\\varepsilon)\\sim\u03ba/\\log(1/\\varepsilon)$":[176],"explicit":[179,205],"$\u03ba$.":[181],"$1/c(p)$":[184],"equals":[185],"dimension-uniform":[187],"expected":[188],"rejection-sampling":[189],"overhead":[190,209],"in":[191,201,215],"reduction":[193],"learning":[195],"errors":[197],"(LWE)":[198],"PRIM-LWE":[200],"lattice-based":[202],"cryptography.":[203],"bounds":[206],"yield":[207],"concrete":[208],"estimates":[210],"for":[211],"all":[212],"primes":[213],"appearing":[214],"current":[216],"NIST":[217],"post-quantum":[218],"standards":[219],"representative":[221],"NTT-friendly":[222],"moduli.":[223]},"counts_by_year":[],"updated_date":"2026-05-09T06:09:20.037420","created_date":"2026-03-14T00:00:00"}
