
parimita kshetraalapai bahupadaala kaarakatvam

ganitam mariyu computer beejaganitamlo bahupadi yokka kaarakatvam anedi daanini irreducible kaarakaala utpattigaa vichchinnam cheyadaanni kaligi untundi. eee kullipovadam siddhaantaparamgaa saadhyame mariyu edainaa rangamlo gunakaalato kuudina bahupadaalaku pratyekamainadi, conny oka algorithm dwara karakatva ganananu anumatinchadaaniki gunakaala kshetrampai balamaina parimitulu avasaram. aacharanalo, algorithamlu parimita kshetramlo, hetubaddhata rangamlo ledhaa vaatilo okadaani yokka parimitamgaa utpatti cheyabadina kshetra vistaranalo gunakaalato kuudina bahupadaala kosam matrame roopondinchabaddaayi.
hetubaddha sankhyalapai bahula variable bahupadula kesutho sahaa anni kaaraka algorithamlu, eee kesuku samasyanu taggistaayi; bahupadi karakanni chudandi. idhi coding siddhaantam (chakriya redendency sankethaalu mariyu bch sankethaalu), guudha lipi shaastram (deerghavruttaakaara vakrarekhala dwara public kii guudha lipi shaastram) mariyu ganana sankhya siddhaantam vanti parimita kshetraala yokka vividha anuvartanaalaku kooda upayoginchabadutundi.
bahula variable bahupadaalanu yeka variable bahupadaalaku karakamga tagginchadam valla parimita kshetramloni gunakaala vishayamlo etuvanti nirdishtata undadu kabatti, eee vyaasamlo oka variable unna bahupadaalu matrame pariganinchabadataayi.
nepathyam.
parimita kshetram.
parimita kshetraala siddhaantam, dheeni muulaalanu gaas mariyu galois rachanala nundi gurtinchavacchu, idhi ganita saastramlooni vividha saakhalalo oka paatra pooshinchindi. ganitam mariyu computer sains vanti itara amsaalalo eee bhavana vartimpajeyadam valla parimita rangaalalo aasakti punarujjeevanam chendindi mariyu idhi coding siddhaantam mariyu guudha lipi saastramlo mukhyamaina anuvartanaala kaaranamgaa paakshikamgaa undhi. parimita kshetraala anuvartanaalu guudha lipi shaastram, computer beejaganitam mariyu coding siddhaantamlo eee parinamalalo konnintini parichayam chestaayi.
parimita kshetram ledhaa galois kshetram anedi oka kramam (muulakaala sankhya) thoo kuudina kshetram. parimita kshetram yokka kramam ellappuduu oka pradhana ledhaa pradhana shakti. prati pradhana saktiki, isomorphism varaku "q" muulakaalatoe sarigga oka parimita kshetram untundi. eee kshetraanni "gf" ("q") ledhaa f "q" anie suuchistaaru. p" anedi prime ayithe, "gf" ("p") anedi arder "p" yokka prime feeled; idhi avashesha taragatula maadyulo "p" feeled, mariyu daani "p" muulakaalu 0,1, anie suuchimchabadataayi...p" −1. kabatti "gf" ("p") loe adhe ardham.
irreducible polynamials.
f" anedi parimita kshetramgaa undanivvandi. saadhaarana kshetraala vishayaanikosthe, sthiramkaani bahupadi "f" anedi "f" ["x"] loe, adhi saanukuula degree yokka rendu bahupadaala gunakam kakapothe, daanini "f" paie irreducible anie antaaru. f" paie irreducible kaani saanukuula degree yokka bahupadini "f" paie reducible "anie pilustaaru.
irreducible bahupadulu prime kaani kramamlo parimita kshetraalanu nirminchadaaniki anumatistaayi. nijaaniki, oka pradhana shakti "q" kosam, f "q" anedi "q" muulakaalatoe parimita kshetram, isomorphism varaku pratyekamainadi. oka bahupadi "f" yokka degree "n" okati kante ekkuva, idhi f "q" paie irreducible, idhi "q" n "muulakaalatoe feeled kii isomorphic ayina degree" n "yokka feeled extension nirvachistundihi eee extension yokka muulakaalu" n "kante takkuva degree yokka bahupadaalu; f" q "yokka moolakam dwara adanamgaa, vyavakalanam mariyu gunakaaram anevi bahupadaalu; rendu muulakaala gunakam anedi vaati utpatti yokka" f "dwara vibhajana yokka migilina bhaagam bahupadaalugaa untundi; oka moolakam yokka vilomaanni podiginchina gcd algorithm dwara lekkinchavacchu (beejaganitam yokka ankaganita podigimpulanu chudandi).
idhi, pradhanam kaani parimita rangamlo ganinchadaaniki, irreducible polynomialnu utpatti cheyalsina avasaram undani anusaristundi. dheeni kosam, saadhaarana paddhati yaadrucchikamgaa bahupadini teesukoni irreducibility kosam pareekshinchadam. kshetramlo gunakaaram yokka saamarthyam kosam, "x" "n" + "" x "" + "" b "aakaaramlo unna bahupadaalanu vetakadam saadhaaranam.
parimita kshetraalapai irreducible bahupadaalu feedback shift registerlu mariyu f2 "n" paie vivikta lagarithamnu upayoginchi sudorandam number generatorlaku kooda upayogapadataayi.
f "q" paie degree n yokka irreducible monic bahupadaala sankhya anedi morau yokka neckless-counting function "m" q "(" n ") dwara ivvabadina aperiodic necklacela sankhya. daggari sambandham unna neckless function "n" q "(" n ") anedi praadhamika (irreducible yokka shakti) degree" n "yokka monic bahupadaalanu lekkistundi; ledhaa pratyamnaayamgaa anni degreela yokka irreducible bahupadaalu nnii vibhajistaayi.
udaaharana.
bahupadi q paie irreducible conny yea parimita kshetrampai kaadu.
sanklishtata.
bahupadi kaaraka algorithamlu utpattulu, vibhagalu, zii. sii. dii, oka bahupadi maadyulo marokati vanti praathamika bahupadi karyakalapalanu upayogistaayi. n" degree yokka rendu bahupadula gunakaaram "o" ("n" 2) loe "q" loe "shaastreeya" ankaganitam upayoginchi ledhaa "o" ("n" logg ("n") logg ("n")) loe "q" loe "vegavantamaina" ankaganitam upayoginchi operationlalo cheyavachu. euclidean vibhajana (migilina vaatitho vibhajana) adhe samaya parimitullo nirvahinchavacchu. garishtamgaa "n" degree unna rendu bahupadaala madhya bahupadi yokka athi pedda saadhaarana bhagakam yokka kharchunu shaastreeya paddhatulanu upayoginchi f "q" loe "o" ("n" 2) karyakalapaluga ledhaa vegavantamaina paddhatulanu upayoginchi f "q" loe "o" ("n" log2 ("n") log ("log (" n ")) karyakalapalaga teesukovachhu. bahupadaalu "h", "g" degree garishtamgaa "n", ghaataankam "hq" mode "g" nuu "o" (logg ("q")) bahupadi utpattulato cheyavachu, squaring paddhati dwara ghaataankam upayoginchi, antey "o" ("n" 2log ("q")) shaastreeya paddhatulanu upayoginchi f "q" loe karyakalapalu, ledhaa "o" ("n" logg ("q") logg ("n") logg ("n")) vegavantamaina paddhatulanu upayoginchi f "q" loe karyakalapalu.
anusarinche algorithamlalo, sanklishtatalu f "q" loe ankaganita karyakalapala sankhya paramgaa vyakteekarinchabadataayi, bahupadaala ankaganitam kosam shaastreeya algorithamlanu upayogistaayi.
kaaraka algorithamlu.
parimita kshetraalapai bahupadaalanu karakam cheyadaaniki aneka algorithamlu eee krindi moodu dasalanu kaligi untayih
oka mukhyamaina minahaayimpu berlecomp yokka algorithm, idhi 2 mariyu 3 dasalanu militam chestundi.
berlecomp yokka algorithm.
berlecomp yokka algorithm modati kaaraka algorithamgaa chaaritraatmakamgaa mukhyamainadi, idhi aacharanalo baga panichestundi. ayithe, idhi ground feeled yokka muulakaalapai oka loopnu kaligi untundi, idhi chinna parimita kshetraalapai matrame aacharaneeyamani suuchistumdi. sthiramaina ground feeled kosam, daani samaya sanklishtata bahupadi, conny, saadhaarana ground feeldla kosam, ground feeled parimaanamlo sanklishtata ghatankamga untundi.
chadarapu-rahita kaarakatvam.
algorithm bahupadaalaku chadarapu-rahita karakanni nirnayistundi, dheeni gunakaalu "p" oka pradhaanamtho kramamlo parimita kshetram f "q" nundi vastaayi. eee algorithm modata utpannaanni nirnayistundi, aapai bahupadi yokka gcd mariyu daani utpannaanni lekkistundi. okavela adhi okati kakapothe, gcd mallee asalu bahupadigaa vibhajinchabadutundi, ayithe utpannam sunna kaadu (parimita kshetraalapai nirvachinchabadina sthiraamkam kaani bahupadaalaku unna sandarbham).
eee algorithm, oka bahupadi yokka utpannam sunna ayithe, adhi "x" p "loe bahupadi, antey, gunakaalu f" p "kii chendinavi ayithe," x "nuu" x "1/" p "dwara pratyaamnaayam cheyadam dwara pondina bahupadi yokka" p "th shakti. gunakaalu f "p" kii chendinavi kakapothe, sunna utpannamtho kuudina bahupadi yokka "p" va muulam "x" paie adhe pratyaamnaayam dwara pondabadutundi, idhi gunakaalaku frobenius automorphism yokka vilomaanni vartimpajeyadam dwara puurti cheyabadutundi. 
eee algorithm lakshanam sunna yokka kshetrampai kooda panichestundi, "p" th muulaalu lekkinchabadina suuchanala blacklalo idhi eppuduu pravesinchadu. ayithe, eee sandarbhamlo, yun yokka algorithm chaalaa samarthavantamgaa untundi endukante idhi takkuva degreela bahupadaala yokka goppa saadhaarana bhaagakaalanu lekkistundi. paryavasanamga, puurnaamkaalapai bahupadaanni karakamga chesetappudu, eee krindi algorithm upayoginchabadaduh okati modata puurnaamkaalapai chadarapu-rahita karakanni lekkistundi, mariyu phalitamgaa vachhe bahupadaalanu karakamga cheyadaaniki, oka "p" nuu enchukuntundi, tadwara avi chadarapu-rahita maadyulo "p" gaa untaayi. 
 algorithm sff (chadarapu-rahita kaarakatvam)
 input f "q" ["x"] loe oka monic bahupadi "f", ikkada "q" = "p" m "
 avtput "f" yokka chadarapu-rahita kaarakatvam"
 "r" 1
 #"f" yokka anni kaarakaala yokka utpatti (gunakaaram lekunda) gaa "w" nii cheyandi 
 #"p" thoo bhaginchaleni gunakaaram"
 "c" "Âgcd (" "f", "f" ")
 "doubleu" "Â" "eff/sii" 
 #dasha 1: "doubleu" looni anni kaarakaalanu gurtinchandi"
 "neenu" 1 
 ayithe "w" inc 1 cheyandi
 "y" "Âgcd (" "w", "c" ")
 "mukham" "Â" "/" "y" ""
 "r" "r" "·" "mukham" "i"
 "" "c" "" "c" "/" "y" ";" "i" "" "i" "+ 1 
 mugimpu samayamlo
 #"sii" anedi ippudu "eff" yokka migilina kaarakaala yokka utpatti (bahulatvamtho)"
 dasha 2: punaraavrutaanni upayoginchi migilina anni kaarakaalanu gurtinchandi
 ivi "p" thoo bhaaginchagala gunakaranni kaligi unna "f" karakalu anie gamaninchandi."
 c" ≤1 ayithe
 "c" "Â" "c" "1/p"
 "r" "Â" "r" "· sff (" "c" ")" "p" ""
 mugiste 
 avtput ("r")
f" yokka anni irreducible kaarakaala utpattini oke gunakaaramtho gurtinchadame aalochana. idhi rendu dasallo jarugutundi. modati dasha "p" thoo vibhajinchaleni gunakaaramtho kuudina anni kaarakaalanu kanugonadaaniki "f" yokka adhikaarika utpannaanni upayogistundi. rendava dasha migilina kaarakaalanu gurtistundi. migilina anni karakalu "p" thoo bhaaginchadagina gunakaranni kaligi unnanduna, avi "p" yokka saktulu anie ardham, okaru kevalam "p" va vargamuulaanni teesukoni punaraavrutaanni vartimpajeyavachu.
chadarapu-rahita kaarakatvam yokka udaaharana.
veelu.
moodu amsaalatoe kshetrampai karakamga undataaniki.
algorithm modata ganistundi
derivative sunna kaanidi kabatti mana vadda undhi mariyu manam vail looploki pravesistaamu. oka loop tarvaata manaku, mariyu naveekaranalato, mariyu. loop dwara rendava saari, naveekaranalato, mariyu istundi. loop dwara moodava saari kooda maaradu. naalugosaari loop dwara manaku, naveekaranalato, mariyu. w" = 1 kabatti, manam vail loop nundi nishkramistaamu. endukante, idhi khachitamgaa khachitamaina cube ayi undaali. cube ruut, dheenini bharti cheyadam dwara pondabadutundi, mariyu chadarapu-rahita vidhananni punaraavrutamgaa pilavadam anedi adhi chadarapu-rahitamgaa undani nirnayistundi. anduvalla, daanini cubing chesi, aaa binduvu viluvatho kalapadam valla chadarapu-rahita kullipovadam jarugutundi.
vilakshanhamaina-sthaayi kaarakatvam.
eee algorithm oka chadarapu-rahita bahupadaanni bahupadaala yokka utpattigaa vibhajistundi, dheeni irreducible karakalu annee oke degreeni kaligi untaayi. degree anedi karakamga undaalsina bahupadigaa undanivvandi.
 algorithm vilakshana-sthaayi kaarakatvam (ddf)
 input oka monic square-free bahupadi 
 avtput anni jatala samiti, antey 
 degree yokka irreducible phyaaktar kaligi undhi mariyu
 idhi degree yokka anni monok irreducible kaarakaala yokka utpatti.
 praarambhinchandi
 suutram _ 10
 suutram _ 11 chestunnappudu 
 suutram _ 12
 appudu, 
 suutram _ 13;
 suutram _ 14
 mugiste
 mugimpu samayam;
 suutram _ 15 ayithe, suutram _ 16;
 suutram _ 17 ayithe, tirigi vellandi,
 vaerae tirigi 
 mugimpu
algorithm yokka khachitatvam eee krindi vaatipai aadharapadi untundih
lemma. i" ≤1 bahupadi kosam
idhi f "q" ["x"] looni anni monic irreducible bahupadaala yokka utpatti, dheeni degree "i" nuu vibhajistundi.
modati chuupuloo, idhi samarthavantamgaa undadu, endukante idhi input bahupadi yokka degreelo ghaataankamaina degree yokka bahupadaala yokka gcd nii ganistundi. ayithe,
dwara bharti cheyavachu
anduvalla, manam lekkinchaali
rendu paddhatulu unnaih
paddhati I. viluva nundi praarambhinchandi
munupati dasalo ganinchabadindi mariyu squaring paddhati dwara ghaataankaanni upayoginchi daani "q" th shakti maadyulo kotta "f *" nuu ganinchadaaniki. eee avasaralu
prati dasalo f "q" loe ankaganita karyakalapalu, mariyu eee vidhamgaa
mottam algorithm kosam ankaganita karyakalapalu.
paddhati II. q" th shakti anedi f "q" paie oka sarala patam anae vaastavaanni upayoginchi manam daani maatrikanu dheenitho lekkinchavacchu.
operations. appudu loop yokka prati punaraavrutamloo, oka vector dwara oka maatrika yokka utpattini lekkinchandi ("o" ("f") 2) aapareshanlatho). idhi f "q" loe mottam karyakalapala sankhyanu prerepistundi, idhi
anduvalla eee rendava paddhati marinta samarthavantamgaa untundi mariyu saadhaaranamgaa praadhaanyata ivvabadutundi. antekakunda, eee paddhatiloo ganinchabadina maatrikanu chaalaa algorithamlu, samaana-sthaayi kaarakatvam kosam upayogistaayi (krinda chudandi); tadwara vibhina-sthaayi kaarakatvam kosam dheenini upayoginchadam valana marinta computing samayanni aadaa chestundi.
samaana-sthaayi kaarakatvam.
contore-jassenhaus algorithm.
eee vibhaagamlo, manamu oka parimita kshetram f "q" paie, oka monok square free ekakaalika bahupadi "f", degree "n" yokka karakanni pariganistaamu, idhi jatha vaareegaa pratyaekamaina irreducible kaaraka suutram _ 27 degree "d" nuu kaligi untundi.
modata contore mariyu jassenhaus (1981) dwara oka algorithamnu, aapai konchem merugaina sanklishtata kaligina oka vaividhyaanni vivaristaamu. rendoo sambhaavya algorithamlu, veeti nadustunna samayam yaadruchchika empikalapai (laas vegas algorithamlu) aadharapadi untundi mariyu manchi sagatu nadustunna samayanni kaligi untundi. tadupari vibhaagamlo manam shoup (1990) dwara oka algorithamnu vivaristaamu, idhi kooda samaana-sthaayi kaaraka algorithm, conny nirnayaatmakamainadi. eee algorithamlannintiki gunakaala kshetraaniki besi kramam "q" avasaram. marinta kaaraka algorithamla kosam e chudandi.zii. naath yokka pustakam dhi art off computer prograaming valume 2.
 algorithm contore-jassenhaus algorithm.
 input besi kramamlo f "q" anae parimita kshetram.
 oka monic square free polynamial "f" loe f "q" ["x"] degree "n" = "rd", 
 idhi "r" ≤2 irreducible factors prati degree "d" kaligi untundi"
 avtput "f" yokka monic irreducible kaarakaala samiti.
 karakalu = {"f"};
 parimaanam (karakalu) <"r" chesthe,
 yaadrucchikamgaa digg ("hetch") <"enn" thoo f "q" ["x"] loe "h" enchukondi;
 suutram _ 28
 prati "yuu" kii digg ("yuu")> "dii" thoo karakalu
 okavela gcd ("g", "u") ≤1 mariyu gcd ("g", "u") ≤ "u" ayithe, appudu
 karakalu = karakalu _ 29;
 mugimpu
 mugimpu samayam
 tirigi vachhe karakalu
eee algorithm yokka khachitatvam ring f "q" ["x"] "f" anedi f "q" ["x"] "fi" kshetraala yokka pratyaksha utpatti, ikkada "f" anedi "f" yokka irreducible karakalapai nadustundi. eee kshetraalannintilo "qd" muulakaalu unnanduna, eee kshetraalalo deeninainaa "g" yokka bhaagam sambhaavyatatoo sunnaga untundi.
idhi bahupadi gcd ("g", "u") anedi "g" yokka kaarakaala yokka utpatti anie suuchistumdi, dheeniki "g" yokka bhaagam sunna.
algorithm yokka vail loop yokka itareshanla sagatu sankhya formula _ 31 kante takkuvagaa undani choopabadindi, idhi f "q" loe sagatu sankhya ankaganita karyakalapalanu istundi, idhi formula _ 32.
d" logg ("q")> "n" anae vilakshanhamaina sandarbhamlo, eee sanklishtata tagginchavachhu
leanier mapp yokka kernallo "h" nuu enchukovadam dwara
mariyu bodhananu bharti cheyandi
dwara
chellubaatu rujuvu paina perkonnatle untundi, f "q" ["x"] "fi" kshetraala pratyaksha utpattini vaati upakshetraala pratyaksha utpattini "q" muulakaalatoe bharti chestundi. sanklishtata algorithm kosam suutram 37 loe kullipotundi, leanier mapp yokka maatrika yokka ganana kosam suutram 38 (idhi ippatike chadarapu-rahita kaarakamlo lekkinchabadavachu) mariyu daani kernalnu ganinchadaaniki "o" ("n" 3). karakalu oke sthaayini kaligi undakapothe kooda eee algorithm panichestundani gamaninchavacchu (eee sandarbhamlo vail loopnu aapadaniki avasaramaina kaarakaala sankhya "r", kernell yokka parimaanamgaa kanugonabadutundi). ayinappatikee, eee algorithamnu upayoginche mundu chadarapu-rahita kaarakatvam cheyabadithe sanklishtata konchem merugga untundi (chadarapu-rahita kaarakatvamto "n" taggavacchu, idhi klishtamaina dasala sanklishtatanu taggistundi).
victor shoup yokka algorithm.
munupati vibhagam yokka algorithamla maadirigaane, victor shoup yokka algorithm oka samaana-sthaayi kaaraka algorithm. vatiki bhinnamgaa, idhi oka nirnayaatmaka algorithm. ayithe, aacharanalo, munupati vibhagam yokka algorithamla kante idhi takkuva samarthavantamgaa untundi. shoup yokka algorithm kosam, input anedi pradhana kshetraala f "p" paie bahupadaalaku parimitam cheyabadindi.
shoup yokka algorithm yokka chettha sandarbham samaya sanklishtata kaaraka suutraanni kaligi untundi _ 39 ayithe ghaataanka, eee sanklishtata oka karakamga unna munupati nirnayaatmaka algorithamla (berlecomp yokka algorithm) kante chaalaa merugga untundi. ayithe, computing samayam ghatankamga unde chaalaa takkuva bahupadaalu unnaayi, mariyu algorithm yokka sagatu samaya sanklishtata suutram _ 40 loe bahupadi, ikkada bahupadi yokka degree, mariyu ground feeled yokka muulakaala sankhya.
g" = "g" 1 anukundam... "gk" "kaavalasina karakamga untundi, ikkada" "g" "anedi degree" "d" "yokka vibhinnamaina yeka-irreducible bahupadaalu". n" = deeg ("g") = "kd". manam ring "r" = f "q" ["x"] "g" nuu pariganistaamu mariyu "r" loe "x" yokka chitram "x" dwara kooda suuchistaamu. ring "r" anedi "r" = f "q" ["x"] "gi" kshetraala pratyaksha utpatti, mariyu manam "r" nundi "r" varaku sahaja homomorphism "paie" dwara suuchistaamu. f "q" paie "r" yokka galois samuuham anedi "d" kramamlo chakreeyamgaa untundi, idhi kshetra automorphism "u" "" up "dwara utpatti avutundi. idhi "rii" loe "zii" yokka muulaalu anie anusaristundi
munupati algorithm maadirigaane, eee algorithm berlecomp yokka algorithm vale "r" yokka adhe upa beejaganitam "b" nuu upayogistundi, konnisaarlu dheenini "berlecomp subzebra" anie pilustaaru mariyu ilaa nirvachinchabadindih
b" yokka upasamiti "s" anedi, prati 1 ≤ "i" <"j" ≤ "k" kii suutram _ 43 unna "s" ±s "unte, vaeru chese samiti anie cheppabadutundi. munupati algorithamlo," s "yokka yaadruchchika muulakaalanu enchukovadam dwara vaeru chese samitini nirmistaaru. shoup yokka algorithamlo, vibhajana samitini eee krindi vidhamgaa nirminchaaru. s" loe "r" ["y"] ilaa undanivvandi
appudu suutram _ 45 anedi oka vibhajana samiti endukante suutram _ 46 "i" = 1,...k" (rendu monic bahupadaalu oke muulaalanu kaligi untaayi). zii" anedi jatha vaareegaa vibhinnamgaa unnanduna, prati jatha vibhina suuchikalaku ("ii", "zhe"), kaneesam oka gunakam "shaa" suutraanni _ 47 thoo santrupti parustundi.
oka vaeru sett kaligi, shoup yokka algorithm munupati vibhagam yokka chivari algorithamgaa konasaagutundi, sarala patam suutram yokka kernalloni "yaadruchchika" h vadda enchukondi "suuchananu bharti cheyadam dwara sarala patam suutram _ 34" dwara "" h "+" "i" "thoo" "h" "loe" "s" "mariyu" "i" "loe" "{1" "thoo enchukondi"..., k" −1} ".
samaya sanklishtata.
munupati vibhagalalo vivarinchinatlugaa, parimita kshetraalapai kaarakatvam kosam, bahupadi samaya sanklishtata yokka yaadruchchika algorithamlu unnaayi (udaaharanaku contore-jassenhaus algorithm). bahupadi sagatu sanklishtatatoe nirnayaatmaka algorithamlu kooda unnaayi (udaaharanaku shoup yokka algorithm).
bahupadi chettha-sandarbha sanklishtatatoe nirnayaatmaka algorithm uniki ippatikee oka bahiranga samasya.
irreducibility yokka rabin pareeksha.
vilakshanhamaina-sthaayi kaaraka algorithm vale, rabin yokka algorithm paina perkonna lemma aadhaaramgaa untundi. pratyeka-sthaayi kaaraka algorithm prati "d" nii input bahupadi yokka sagam degree kante ekkuva kaadu. takkuva "d" nuu parigananaloki teesukoovadaaniki karakalu avasaram ledani rabin yokka algorithm prayojanaanni pondutundi. lekapothe, idhi vilakshanhamaina-sthaayi kaaraka algorithm maadirigaane untundi. idhi eee krindi vaastavam aadhaaramgaa undhi.
p" 1, anie anukundam...pk", "n" yokka anni pradhana bhagaluga undandi, mariyu suutram _ 49nii suuchistumdi, f "q" loe 1 ≤ "i" ≤ "k" bahupadi "f" kosam, degree "n" yokka degree "q" ["x"] anedi f "q" ["x"] loe irreducible ayithe mariyu suutram _ 50 ayithe, 1 ≤ "i" ≤ "k" kosam, mariyu "f" suutram _ 51nii vibhajiste matrame. vaastavaaniki, "f" kii "n" nuu vibhajinchani degree karakanni kaligi unte, "f" suutram _ 51nii vibhajinchadu; "f" kii degree "n" karakanni kaligi unte, appudu eee amsam suutram _ 53loe kaneesam okadaanilo vibhajistundi.
 algorithm rabin irreducibility test
 input degree "n" yokka f "q" ["x"] loe oka monic bahupadi "f", 
 "p", 1...pk", "n" yokka anni vibhina pradhana bhagalu.
 avtput "f" anedi irreducible ledhaa "f" anedi reducible.
 zhe" = 1 nundi "keey" kosam cheyandi 
 suutram _ 54;
 i" = 1 nundi "k" kosam cheyandi 
 suutram _ 55;
 "g": = gcd ("" f "," h ");
 g" ≤1 ayithe, tirigi "f" tagginchadaginadi mariyu aapandi;
 mugimpu;
 suutram _ 56;
 g" = 0 ayithe, tirigi "f" anedi irreducible avutundi, 
 lekapothe "f" tagginchadaginadi "
eee algorithm yokka praathamika aalochana emitante, formula _ 57 nuu atichinna formula _ 58 nundi punaraavruta squaring ledhaa frobenius automorphismnu upayoginchi praarambhinchi, aapai correspondent gcd nii teesukovadam. praathamika bahupadi ankaganitam upayoginchi, frobenius automorphism yokka maatrika yokka gananaku f "q" loe suutram _ 59 operations avasaram, dheeni ganana
o" ("n" 3) tadupari karyakalapalu avasaram, mariyu algorithanku "o" ("kn" 2) karyakalapalu avasaram, idhi f "q" loe mottam suutram _ 59 karyakalapalanu istundi. vegavantamaina ankaganitaanni upayoginchi (gunakaaram mariyu vibhajana kosam sanklishtata suutram _ 62, mariyu gcd ganana kosam suutram _ 63), suutram _ 64 yokka ganana suutram _ 65, mariyu algorithm suutram _ 66, idhi f "q" loe mottam suutram _ 65 karyakalapalanu istundi.

