{"id":"https://openalex.org/W1960797965","doi":"https://doi.org/10.1142/s0218127415501242","title":"Some Polynomial Chaotic Maps Without Equilibria and an Application to Image Encryption with Avalanche Effects","display_name":"Some Polynomial Chaotic Maps Without Equilibria and an Application to Image Encryption with Avalanche Effects","publication_year":2015,"publication_date":"2015-08-01","ids":{"openalex":"https://openalex.org/W1960797965","doi":"https://doi.org/10.1142/s0218127415501242","mag":"1960797965"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127415501242","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127415501242","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018564432","display_name":"Lequan Min","orcid":"https://orcid.org/0000-0002-4657-964X"},"institutions":[{"id":"https://openalex.org/I92403157","display_name":"University of Science and Technology Beijing","ror":"https://ror.org/02egmk993","country_code":"CN","type":"education","lineage":["https://openalex.org/I92403157"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lequan Min","raw_affiliation_strings":["School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","institution_ids":["https://openalex.org/I92403157"]},{"raw_affiliation_string":"School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China","institution_ids":["https://openalex.org/I92403157"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101487316","display_name":"Xiuping Yang","orcid":"https://orcid.org/0000-0003-3790-951X"},"institutions":[{"id":"https://openalex.org/I92403157","display_name":"University of Science and Technology Beijing","ror":"https://ror.org/02egmk993","country_code":"CN","type":"education","lineage":["https://openalex.org/I92403157"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiuping Yang","raw_affiliation_strings":["School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","institution_ids":["https://openalex.org/I92403157"]},{"raw_affiliation_string":"School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China","institution_ids":["https://openalex.org/I92403157"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024633466","display_name":"Guanrong Chen","orcid":"https://orcid.org/0000-0003-1381-7418"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Guanrong Chen","raw_affiliation_strings":["Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China","Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R., China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China","institution_ids":["https://openalex.org/I168719708"]},{"raw_affiliation_string":"Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R., China","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090934036","display_name":"Danling Wang","orcid":"https://orcid.org/0000-0003-0119-441X"},"institutions":[{"id":"https://openalex.org/I92403157","display_name":"University of Science and Technology Beijing","ror":"https://ror.org/02egmk993","country_code":"CN","type":"education","lineage":["https://openalex.org/I92403157"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Danling Wang","raw_affiliation_strings":["School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China","institution_ids":["https://openalex.org/I92403157"]},{"raw_affiliation_string":"School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China","institution_ids":["https://openalex.org/I92403157"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.6664,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.86661956,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"25","issue":"09","first_page":"1550124","last_page":"1550124"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10244","display_name":"Chaos control and synchronization","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9879999756813049,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/encryption","display_name":"Encryption","score":0.6814721822738647},{"id":"https://openalex.org/keywords/rc4","display_name":"RC4","score":0.6535606384277344},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.6530361175537109},{"id":"https://openalex.org/keywords/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.6368269920349121},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.632235586643219},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.622167706489563},{"id":"https://openalex.org/keywords/key-space","display_name":"Key space","score":0.5897435545921326},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5422670841217041},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.4895842969417572},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.480232298374176},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.445125550031662},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.42454782128334045},{"id":"https://openalex.org/keywords/synchronization","display_name":"Synchronization (alternating current)","score":0.42020943760871887},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.34885114431381226},{"id":"https://openalex.org/keywords/stream-cipher","display_name":"Stream cipher","score":0.3065969944000244},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2946377694606781},{"id":"https://openalex.org/keywords/channel","display_name":"Channel (broadcasting)","score":0.17098331451416016},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1454356610774994}],"concepts":[{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.6814721822738647},{"id":"https://openalex.org/C109154458","wikidata":"https://www.wikidata.org/wiki/Q846955","display_name":"RC4","level":4,"score":0.6535606384277344},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.6530361175537109},{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.6368269920349121},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.632235586643219},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.622167706489563},{"id":"https://openalex.org/C105322999","wikidata":"https://www.wikidata.org/wiki/Q1557574","display_name":"Key space","level":3,"score":0.5897435545921326},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5422670841217041},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.4895842969417572},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.480232298374176},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.445125550031662},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.42454782128334045},{"id":"https://openalex.org/C2778562939","wikidata":"https://www.wikidata.org/wiki/Q1298791","display_name":"Synchronization (alternating current)","level":3,"score":0.42020943760871887},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.34885114431381226},{"id":"https://openalex.org/C92950451","wikidata":"https://www.wikidata.org/wiki/Q864718","display_name":"Stream cipher","level":3,"score":0.3065969944000244},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2946377694606781},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.17098331451416016},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1454356610774994},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127415501242","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127415501242","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320309893","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":46,"referenced_works":["https://openalex.org/W95080231","https://openalex.org/W1496745782","https://openalex.org/W1964889805","https://openalex.org/W1970442252","https://openalex.org/W1972021483","https://openalex.org/W1972886462","https://openalex.org/W1975810685","https://openalex.org/W1979208093","https://openalex.org/W1980589305","https://openalex.org/W1983477908","https://openalex.org/W1990970511","https://openalex.org/W1990987436","https://openalex.org/W1994189116","https://openalex.org/W1994859578","https://openalex.org/W2000882094","https://openalex.org/W2002464840","https://openalex.org/W2006464603","https://openalex.org/W2009483907","https://openalex.org/W2013729900","https://openalex.org/W2015304581","https://openalex.org/W2016380472","https://openalex.org/W2017241387","https://openalex.org/W2023573148","https://openalex.org/W2029295251","https://openalex.org/W2032381756","https://openalex.org/W2034338638","https://openalex.org/W2044729063","https://openalex.org/W2046580173","https://openalex.org/W2046673681","https://openalex.org/W2052192147","https://openalex.org/W2054952654","https://openalex.org/W2057835419","https://openalex.org/W2058152830","https://openalex.org/W2058369496","https://openalex.org/W2059172307","https://openalex.org/W2071840664","https://openalex.org/W2074886859","https://openalex.org/W2077765821","https://openalex.org/W2087950050","https://openalex.org/W2141662965","https://openalex.org/W2205046642","https://openalex.org/W2218724876","https://openalex.org/W3100545517","https://openalex.org/W4230463010","https://openalex.org/W4239883234","https://openalex.org/W4388249796"],"related_works":["https://openalex.org/W26169442","https://openalex.org/W2617856299","https://openalex.org/W2183399669","https://openalex.org/W778070508","https://openalex.org/W2184880873","https://openalex.org/W2795569938","https://openalex.org/W2089478084","https://openalex.org/W2953372076","https://openalex.org/W1724491999","https://openalex.org/W2899285859"],"abstract_inverted_index":{"This":[0],"study":[1],"uses":[2],"seven":[3],"four-dimensional":[4],"four-variable":[5],"polynomial":[6],"chaotic":[7,50,64],"maps":[8],"without":[9,29],"equilibria":[10],"in":[11],"combination":[12],"with":[13,36,156],"generalized":[14,24],"chaos":[15,25],"synchronization":[16,26],"(GCS)":[17],"theorem":[18],"to":[19,60,75,160,172,180],"construct":[20],"eight-dimensional":[21],"bidirectional":[22],"discrete":[23],"(8DBDGCS)":[27],"systems":[28],"equilibria.":[30],"By":[31],"combining":[32],"the":[33,37,49,53,77,92,98,102,109,113,126,130,136,142,146,168,174],"8DBDGCS":[34],"system":[35,43],"GCS":[38,42,55],"theorem,":[39],"a":[40,62,94],"12-dimensional":[41,54],"is":[44,58,73,148,154,170],"designed.":[45],"Numerical":[46],"simulation":[47],"verifies":[48],"dynamics":[51],"of":[52,79,84,112,145],"system,":[56],"which":[57,177],"used":[59,74,155],"design":[61],"216-word":[63,137,185],"pseudorandom":[65],"number":[66],"generator":[67],"(CPRNG).":[68],"The":[69,105,152],"SP-8002":[70],"test":[71,76],"suite":[72],"randomness":[78,110],"four":[80],"100-key":[81],"streams":[82,133],"consisting":[83],"1":[85],"000":[86,87],"bits":[88],"generated":[89,134,182],"respectively":[90],"by":[91],"CPRNG,":[93,97],"six-dimensional":[95],"GCS-based":[96],"RC4":[99],"algorithm":[100],"and":[101,129],"ZUC":[103],"algorithm.":[104],"results":[106],"show":[107],"that":[108,119,167],"performances":[111],"two":[114],"CPRNGs":[115],"are":[116,121,178],"promising,":[117],"suggesting":[118],"there":[120],"no":[122],"significant":[123],"correlations":[124],"between":[125],"key":[127,132,143],"stream":[128],"perturbed":[131],"via":[135,183],"CPRNG.":[138],"In":[139],"addition,":[140],"theoretically":[141],"space":[144],"CPRNG":[147,153,169],"larger":[149],"than":[150],"21195.":[151],"an":[157,162],"avalanche-encryption":[158],"scheme":[159],"encrypt":[161],"RGB":[163],"balloon":[164],"image,":[165],"demonstrating":[166],"able":[171],"generate":[173],"avalanche":[175],"effects":[176],"similar":[179],"those":[181],"ideal":[184],"CPRNGs.":[186]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":3}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}