{"id":"https://openalex.org/W2108920860","doi":"https://doi.org/10.1109/hldvt.2011.6113989","title":"Verification of composite Galois field multipliers over GF ((2&lt;sup&gt;m&lt;/sup&gt;)&lt;sup&gt;n&lt;/sup&gt;) using computer algebra techniques","display_name":"Verification of composite Galois field multipliers over GF ((2&lt;sup&gt;m&lt;/sup&gt;)&lt;sup&gt;n&lt;/sup&gt;) using computer algebra techniques","publication_year":2011,"publication_date":"2011-11-01","ids":{"openalex":"https://openalex.org/W2108920860","doi":"https://doi.org/10.1109/hldvt.2011.6113989","mag":"2108920860"},"language":"en","primary_location":{"id":"doi:10.1109/hldvt.2011.6113989","is_oa":false,"landing_page_url":"https://doi.org/10.1109/hldvt.2011.6113989","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2011 IEEE International High Level Design Validation and Test Workshop","raw_type":"proceedings-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/A5111646337","display_name":"Jinpeng Lv","orcid":null},"institutions":[{"id":"https://openalex.org/I223532165","display_name":"University of Utah","ror":"https://ror.org/03r0ha626","country_code":"US","type":"education","lineage":["https://openalex.org/I223532165"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jinpeng Lv","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Utah, Salt Lake, UT, USA","Department of Electrical And Computer Engineering, University of Utah, Salt Lake City, UT 84112"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Utah, Salt Lake, UT, USA","institution_ids":["https://openalex.org/I223532165"]},{"raw_affiliation_string":"Department of Electrical And Computer Engineering, University of Utah, Salt Lake City, UT 84112","institution_ids":["https://openalex.org/I223532165"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048751210","display_name":"Priyank Kalla","orcid":"https://orcid.org/0000-0001-7412-5138"},"institutions":[{"id":"https://openalex.org/I223532165","display_name":"University of Utah","ror":"https://ror.org/03r0ha626","country_code":"US","type":"education","lineage":["https://openalex.org/I223532165"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Priyank Kalla","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Utah, Salt Lake, UT, USA","Department of Electrical And Computer Engineering, University of Utah, Salt Lake City, UT 84112"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Utah, Salt Lake, UT, USA","institution_ids":["https://openalex.org/I223532165"]},{"raw_affiliation_string":"Department of Electrical And Computer Engineering, University of Utah, Salt Lake City, UT 84112","institution_ids":["https://openalex.org/I223532165"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041901796","display_name":"Florian Enescu","orcid":"https://orcid.org/0000-0002-9532-2568"},"institutions":[{"id":"https://openalex.org/I181565077","display_name":"Georgia State University","ror":"https://ror.org/03qt6ba18","country_code":"US","type":"education","lineage":["https://openalex.org/I181565077"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Florian Enescu","raw_affiliation_strings":["Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, USA","Department of Mathematics and Statistics Georgia State University, Atlanta, GA 30302-4038#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, USA","institution_ids":["https://openalex.org/I181565077"]},{"raw_affiliation_string":"Department of Mathematics and Statistics Georgia State University, Atlanta, GA 30302-4038#TAB#","institution_ids":["https://openalex.org/I181565077"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5111646337"],"corresponding_institution_ids":["https://openalex.org/I223532165"],"apc_list":null,"apc_paid":null,"fwci":3.6947,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.9390405,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"136","last_page":"143"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9941999912261963,"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"}},"topics":[{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9941999912261963,"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"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9918000102043152,"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/T10054","display_name":"Parallel Computing and Optimization Techniques","score":0.9897000193595886,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/galois-theory","display_name":"Galois theory","score":0.4740428924560547},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4685465097427368},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4347917437553406},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4126669764518738},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.404843807220459},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37728723883628845},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3721769154071808},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3716219961643219},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.12323397397994995}],"concepts":[{"id":"https://openalex.org/C94398972","wikidata":"https://www.wikidata.org/wiki/Q92552","display_name":"Galois theory","level":2,"score":0.4740428924560547},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4685465097427368},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4347917437553406},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4126669764518738},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.404843807220459},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37728723883628845},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3721769154071808},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3716219961643219},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.12323397397994995}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/hldvt.2011.6113989","is_oa":false,"landing_page_url":"https://doi.org/10.1109/hldvt.2011.6113989","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2011 IEEE International High Level Design Validation and Test Workshop","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W92576581","https://openalex.org/W1480909796","https://openalex.org/W1512259823","https://openalex.org/W1518341038","https://openalex.org/W1541795511","https://openalex.org/W1579437898","https://openalex.org/W1595607623","https://openalex.org/W1844774176","https://openalex.org/W1926206757","https://openalex.org/W1967380110","https://openalex.org/W2079983062","https://openalex.org/W2080267935","https://openalex.org/W2104976591","https://openalex.org/W2108304500","https://openalex.org/W2113028625","https://openalex.org/W2144491553","https://openalex.org/W2156058638","https://openalex.org/W2156921282","https://openalex.org/W2164309559","https://openalex.org/W2175098281","https://openalex.org/W4234211459","https://openalex.org/W4237780066","https://openalex.org/W4245015178","https://openalex.org/W4253178675","https://openalex.org/W6634743126","https://openalex.org/W6638843144","https://openalex.org/W6640293377"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2938786841","https://openalex.org/W4249530125","https://openalex.org/W1944601446","https://openalex.org/W2502533382","https://openalex.org/W87529998","https://openalex.org/W4247547172","https://openalex.org/W4398244834","https://openalex.org/W630499370","https://openalex.org/W2498283248"],"abstract_inverted_index":{"Galois":[0,27,59,166],"field":[1,28,60,94,148],"computations":[2],"abound":[3],"in":[4,9],"many":[5,17],"applications,":[6],"such":[7,26,41,120],"as":[8,66,174],"cryptography,":[10],"error":[11],"correction":[12],"codes,":[13],"signal":[14],"processing,":[15],"among":[16],"others.":[18],"Multiplication":[19],"usually":[20],"lies":[21],"at":[22],"the":[23,34,53,58,92,110,115,146,152,158,162,175,188,196],"core":[24],"of":[25,33,40,119,190,198],"computations,":[29],"and":[30,138],"is":[31,172],"one":[32],"most":[35],"complex":[36],"operations.":[37],"Hardware":[38],"implementations":[39],"multipliers":[42,121],"become":[43],"very":[44],"expensive.":[45],"Therefore,":[46],"there":[47],"have":[48],"been":[49],"efforts":[50],"to":[51,99,186],"reduce":[52],"design":[54],"complexity":[55],"by":[56],"decomposing":[57],"GF(2":[61],"<sup":[62,69,73,102,126,130],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[63,70,74,103,127,131],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">k</sup>":[64],")":[65,72,76,105,129],"GF":[67,96,100,124],"((2":[68,125],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">m</sup>":[71,104,128],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</sup>":[75,132],"where":[77],"k":[78],"=":[79],"m":[80],"\u00d7":[81],"n.":[82],"Such":[83],"a":[84,87,135],"decomposition":[85],"introduces":[86],"hierarchical":[88],"abstraction":[89],"-":[90,107],"lifting":[91],"ground":[93],"from":[95],"(2)":[97],"(bit-level)":[98],"(2":[101],"(word-level)":[106],"thus":[108],"simplifying":[109],"design.":[111],"This":[112],"paper":[113],"addresses":[114],"formal":[116],"verification":[117,159],"problem":[118,160],"designed":[122],"over":[123,165],"),":[133],"using":[134,161],"computer":[136],"algebra":[137],"algebraic":[139],"geometry":[140],"based":[141],"approach.":[142,192],"To":[143],"prove":[144],"that":[145],"composite":[147],"multiplier":[149],"implementation":[150],"matches":[151],"original":[153],"specification,":[154],"we":[155],"hierarchically":[156],"formulate":[157],"Hilbert's":[163],"Nullstellensatz":[164],"Fields.":[167],"A":[168],"Gr\u00f6bner":[169],"basis":[170],"engine":[171],"employed":[173],"underlying":[176],"computational":[177],"framework.":[178],"Experiments":[179],"are":[180,205],"performed":[181],"with":[182],"various":[183],"variable/term":[184],"orders":[185],"demonstrate":[187],"efficacy":[189],"our":[191],"We":[193],"can":[194],"verify":[195],"correctness":[197],"upto":[199],"1024bit":[200],"multipliers,":[201],"whereas":[202],"SAT/SMT-based":[203],"approaches":[204],"in-feasible.":[206]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":4}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
