{"id":"https://openalex.org/W2053266393","doi":"https://doi.org/10.1109/tit.2011.2169129","title":"Differential Properties of ${x\\mapsto x^{2^{t}-1}}$","display_name":"Differential Properties of ${x\\mapsto x^{2^{t}-1}}$","publication_year":2011,"publication_date":"2011-12-01","ids":{"openalex":"https://openalex.org/W2053266393","doi":"https://doi.org/10.1109/tit.2011.2169129","mag":"2053266393"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2011.2169129","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2169129","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","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/A5049943235","display_name":"C\u00e9line Blondeau","orcid":"https://orcid.org/0000-0002-7777-8545"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"C\u00e9line Blondeau","raw_affiliation_strings":["SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050062480","display_name":"Anne Canteaut","orcid":"https://orcid.org/0000-0002-6292-8336"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Anne Canteaut","raw_affiliation_strings":["SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048217797","display_name":"Pascale Charpin","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Pascale Charpin","raw_affiliation_strings":["SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"SECRET project-team, INRIA-Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"SECRET Project-Team, INRIA Paris-Rocquencourt Domaine de Voluceau, Le Chesnay, France","institution_ids":["https://openalex.org/I1326498283"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.8361,"has_fulltext":false,"cited_by_count":69,"citation_normalized_percentile":{"value":0.94973677,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"57","issue":"12","first_page":"8127","last_page":"8137"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"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/T11130","display_name":"Coding theory and cryptography","score":1.0,"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":0.9883000254631042,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9731000065803528,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/computer-science","display_name":"Computer science","score":0.449375182390213},{"id":"https://openalex.org/keywords/differential","display_name":"Differential (mechanical device)","score":0.435077428817749},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.36937618255615234},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34894102811813354},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3344794809818268},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2721058130264282}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.449375182390213},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.435077428817749},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.36937618255615234},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34894102811813354},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3344794809818268},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2721058130264282},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2011.2169129","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2169129","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-00610099v1","is_oa":false,"landing_page_url":"https://hal.science/hal-00610099","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Information Theory, 2011, 57 (12), pp.8127 - 8137. &#x27E8;10.1109/TIT.2011.2169129&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1528065150","https://openalex.org/W1575149347","https://openalex.org/W1585566083","https://openalex.org/W1587687098","https://openalex.org/W1629301835","https://openalex.org/W1977206724","https://openalex.org/W1980099675","https://openalex.org/W1984812887","https://openalex.org/W1990605160","https://openalex.org/W2015894712","https://openalex.org/W2016744051","https://openalex.org/W2041328554","https://openalex.org/W2082143349","https://openalex.org/W2087896741","https://openalex.org/W2097345111","https://openalex.org/W2105985720","https://openalex.org/W2131820257","https://openalex.org/W2138238405","https://openalex.org/W2245380141","https://openalex.org/W2286695063","https://openalex.org/W2963192837","https://openalex.org/W4248135912"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694"],"abstract_inverted_index":{"We":[0,46,105,179],"provide":[1],"an":[2],"extensive":[3],"study":[4],"of":[5,9,53,61,63,114,170,175,186,199,202,207],"the":[6,10,50,59,64,111,167,171,176,182,200],"differential":[7,51,112,168,184],"properties":[8,169],"functions":[11,55],"<i":[12,16,22,31,38,42,67,73,77,84,88,92,101,115,119,125,130,134,140,145,149,153,187,191,208,212,218,223,228,233,237],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[13,17,20,23,29,32,39,43,68,71,74,78,81,85,89,93,99,102,116,120,123,126,131,135,138,141,146,150,154,188,192,195,209,213,216,219,224,229,234,238],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x</i>":[14,18,69,90,117,121,132,136,189,193,210,214],"\u2192":[15,118,133,190,211],"<sup":[19,70,80,122,137,194,215],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sup>":[21,72,82,124,139,217],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">t</i>":[24,40,75,127,155,220,225],"-1":[25,128,143,221],"over":[26],"\\BBF":[27,97],"<sub":[28,98],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[30,100],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>":[33,44,103,151,230,235,239],",":[34],"for":[35,144,222],"1":[36],"<;":[37,41],".":[45,104],"notably":[47],"show":[48],"that":[49],"spectra":[52,113,185],"these":[54],"are":[56],"determined":[57],"by":[58,197],"number":[60],"roots":[62],"linear":[65],"polynomials":[66],"+":[76],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">bx</i>":[79],"+(":[83],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">b</i>":[86,94],"+1)":[87],"where":[91],"varies":[95],"in":[96],"prove":[106],"a":[107,158,164],"strong":[108],"relationship":[109],"between":[110,166],"and":[129,174,206],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">s</i>":[142,147],"=":[148],"-":[152],"+1.":[156],"As":[157],"direct":[159],"consequence,":[160],"this":[161],"result":[162],"enlightens":[163],"connection":[165],"cube":[172],"function":[173],"inverse":[177],"function.":[178],"also":[180],"determine":[181],"complete":[183],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">7</sup>":[196],"means":[198],"value":[201],"some":[203],"Kloosterman":[204],"sums,":[205],"\u2208":[226],"{[":[227],"/2],":[231],"[":[232],"/2]+1,":[236],"-2}.":[240]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":12},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}