{"id":"https://openalex.org/W7225751488","doi":"https://doi.org/10.21203/rs.3.rs-11213217/v1","title":"Bi-Skew Generalized Triangular Matrix Rings and Their Applications in Post-Quantum Cryptography","display_name":"Bi-Skew Generalized Triangular Matrix Rings and Their Applications in Post-Quantum Cryptography","publication_year":2026,"publication_date":"2026-11-10","ids":{"openalex":"https://openalex.org/W7225751488","doi":"https://doi.org/10.21203/rs.3.rs-11213217/v1"},"language":"en","primary_location":{"id":"doi:10.21203/rs.3.rs-11213217/v1","doi":"https://doi.org/10.21203/rs.3.rs-11213217/v1","is_oa":true,"landing_page_url":"https://doi.org/10.21203/rs.3.rs-11213217/v1","pdf_url":null,"source":{"id":"https://openalex.org/S4306525896","display_name":"Research Square","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Research Square","raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.21203/rs.3.rs-11213217/v1","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009120094","display_name":"Mohamed A. Farahat","orcid":"https://orcid.org/0000-0003-4731-7139","observed_orcids":["https://orcid.org/0000-0003-4731-7139"]},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Mohamed A. FARAHAT","raw_affiliation_strings":["Al-Azhar University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Al-Azhar University","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008850056","display_name":"REFAAT M. SALEM","orcid":"https://orcid.org/0000-0001-6213-2795","observed_orcids":["https://orcid.org/0000-0001-6213-2795"]},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Refaat M. SALEM","raw_affiliation_strings":["Al-Azhar University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Al-Azhar University","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5114134090","display_name":"H. A. A. Mohamed","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Hanan A. MOHAMED","raw_affiliation_strings":["Ain Shams University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ain Shams University","institution_ids":["https://openalex.org/I107720978"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.4731000065803528,"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"}},"topics":[{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.4731000065803528,"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/T10237","display_name":"Cryptography and Data Security","score":0.38449999690055847,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.05380000174045563,"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":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.722599983215332},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.4641000032424927},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4399999976158142},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.42340001463890076},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.4041999876499176},{"id":"https://openalex.org/C68797384","wikidata":"https://www.wikidata.org/wiki/Q2746792","display_name":"Noncommutative geometry","level":2,"score":0.39959999918937683},{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.39730000495910645},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39730000495910645},{"id":"https://openalex.org/C2779765290","wikidata":"https://www.wikidata.org/wiki/Q1674414","display_name":"Isogeny","level":3,"score":0.3921999931335449},{"id":"https://openalex.org/C2778153370","wikidata":"https://www.wikidata.org/wiki/Q2896709","display_name":"Endomorphism ring","level":3,"score":0.3874000012874603},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.3840999901294708},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.3774999976158142},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.3686000108718872},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.3617999851703644},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.33169999718666077},{"id":"https://openalex.org/C172252984","wikidata":"https://www.wikidata.org/wiki/Q368988","display_name":"Idempotence","level":2,"score":0.3172999918460846},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.3118000030517578},{"id":"https://openalex.org/C27563320","wikidata":"https://www.wikidata.org/wiki/Q650670","display_name":"Nilpotent matrix","level":5,"score":0.3100000023841858},{"id":"https://openalex.org/C116858840","wikidata":"https://www.wikidata.org/wiki/Q1340800","display_name":"Endomorphism","level":2,"score":0.3077000081539154},{"id":"https://openalex.org/C99674996","wikidata":"https://www.wikidata.org/wiki/Q1414155","display_name":"Key exchange","level":4,"score":0.3046000003814697},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.2976999878883362},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2784000039100647},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.2773999869823456},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.27559998631477356},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.2630999982357025},{"id":"https://openalex.org/C113313756","wikidata":"https://www.wikidata.org/wiki/Q332791","display_name":"Diagonal matrix","level":3,"score":0.26089999079704285},{"id":"https://openalex.org/C15927051","wikidata":"https://www.wikidata.org/wiki/Q246593","display_name":"Cryptographic primitive","level":4,"score":0.2540999948978424},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.2540000081062317},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.25270000100135803},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.21203/rs.3.rs-11213217/v1","doi":"https://doi.org/10.21203/rs.3.rs-11213217/v1","is_oa":true,"landing_page_url":"https://doi.org/10.21203/rs.3.rs-11213217/v1","pdf_url":null,"source":{"id":"https://openalex.org/S4306525896","display_name":"Research Square","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Research Square","raw_type":"posted-content"}],"best_oa_location":{"id":"doi:10.21203/rs.3.rs-11213217/v1","doi":"https://doi.org/10.21203/rs.3.rs-11213217/v1","is_oa":true,"landing_page_url":"https://doi.org/10.21203/rs.3.rs-11213217/v1","pdf_url":null,"source":{"id":"https://openalex.org/S4306525896","display_name":"Research Square","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Research Square","raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2,75,103,137],"paper,":[3],"we":[4,101,210],"introduce":[5],"a":[6,38,79,108,122,129,203],"new":[7],"algebraic":[8],"structure":[9,73],"called":[10],"the":[11,71,89,115,134,187,225,230],"bi-skew":[12],"generalized":[13],"triangular":[14],"matrix":[15,166],"ring,":[16],"denoted":[17],"by":[18,77,88,114,145,176],"(S,":[19,40],"R,":[20],"M,":[21],"0)":[22],"\u03c6,\u03c8":[23],",":[24,54,100,168],"where":[25],"S":[26,34,47,53],"and":[27,55,151,169,181,233],"R":[28,36,61],"are":[29,173,250],"associative":[30],"rings":[31],"with":[32,48,62],"unity,":[33],"M":[35],"is":[37,43,57,126,139,189],"unitary":[39],"R)-bimodule,":[41],"\u03c6":[42],"an":[44,58,212],"endomor-phism":[45],"on":[46,60],"\u03c6(1":[49],"S)":[50],"=":[51,65],"1":[52,66],"\u03c8":[56],"endomorphism":[59],"\u03c8(1":[63],"R)":[64],"R.":[67],"We":[68,118,158],"rigorously":[69],"determine":[70],"ideal":[72],"of":[74,82,92,186],"ring":[76],"establishing":[78],"complete":[80],"characterization":[81],"its":[83],"Jacobson":[84],"radical":[85],"J(A).":[86,157],"Driven":[87],"emerging":[90],"vulner-abilities":[91],"classical":[93],"public-key":[94],"cryptosystems":[95],"to":[96,106,128,239,252],"quantum":[97],"polynomial-time":[98],"algorithms":[99],"deploy":[102],"non-commutative":[104],"framework":[105],"construct":[107,211],"post-quantum":[109],"key":[110],"exchange":[111],"protocol":[112,125,144],"inspired":[113],"Stickel":[116],"architecture.":[117],"demonstrate":[119],"that":[120,160],"while":[121],"naive":[123],"pure-exponentiation":[124],"vulnerable":[127],"canonical":[130,208],"projection":[131,161,231],"homomorphism":[132],"onto":[133],"diagonal":[135],"subrings,":[136],"threat":[138],"completely":[140],"neutralized":[141],"within":[142,229],"our":[143],"integrating":[146],"orthogonal":[147],"idempotents":[148],"(zero-divisor":[149],"confinement)":[150],"ephemeral":[152],"nilpotent":[153,182],"perturbations":[154],"drawn":[155],"from":[156,237],"prove":[159],"decoupling,":[162],"centralizer":[163],"decompositions,":[164],"Shpilrain\u2013Ushakov":[165],"inversions":[167],"minimal":[170],"polynomial":[171],"linearizations":[172],"simultaneously":[174],"obstructed":[175],"Frobenius":[177],"non-linearity,":[178],"idempotent":[179],"singularity,":[180],"masking.":[183],"The":[184],"security":[185],"scheme":[188],"formulated":[190],"under":[191],"explicit":[192],"multivariate":[193],"quadratic":[194],"(MQ)":[195],"hardness":[196],"conjectures":[197],"against":[198,207],"passive":[199],"adversaries.":[200],"Furthermore,":[201],"as":[202],"second":[204],"structural":[205],"countermeasure":[206],"projection,":[209],"asymmetric":[213],"data":[214],"encapsulation":[215],"mechanism":[216],"(DEM)":[217],"deploying":[218],"Radi\u0107\u2019s":[219],"right-invertibility":[220],"over":[221],"rectangular":[222],"bimodules,":[223],"confining":[224],"secret":[226],"carrier":[227],"entirely":[228],"kernel":[232],"leveraging":[234],"dimensional":[235],"expansion":[236],"2-vectors":[238],"3-vectors":[240],"for":[241],"entropy":[242],"hiding.":[243],"Finally,":[244],"two":[245],"open":[246],"cryptographic":[247],"challenge":[248],"problems":[249],"provided":[251],"facilitate":[253],"community":[254],"cryptanalysis.":[255],"2020":[256],"Mathematics":[257],"Subject":[258],"Classification:":[259],"Primary":[260],"16D25,":[261],"16D90,":[262],"16S50;":[263],"Secondary":[264],"11T71,":[265],"16W20,":[266],"16N20,":[267],"94A60.":[268]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-11T00:00:00"}
