{"id":1624,"date":"2014-10-13T12:16:35","date_gmt":"2014-10-13T12:16:35","guid":{"rendered":"http:\/\/bitcoinromania.ro\/?p=1624"},"modified":"2026-03-01T13:46:42","modified_gmt":"2026-03-01T11:46:42","slug":"matematica-din-spatele-bitcoin","status":"publish","type":"post","link":"https:\/\/bitcoinromania.ro\/blog\/matematica-din-spatele-bitcoin\/","title":{"rendered":"Matematica din spatele Bitcoin"},"content":{"rendered":"

Eric Rykwalder este un inginer de software \u0219i unul din fondatorii Chain.com. El ofer\u0103 o privire de ansamblu asupra fundamentelor matematice ale protocolului Bitcoin.
\n<\/em>
\n\"math-behind-bitcoin-630x376\"<\/a><\/p>\n

\u00a0<\/em><\/p>\n

Un motiv pentru care Bitcoin poate fi confuz pentru \u00eencep\u0103tori este c\u0103 tehnologia din spatele aceastei monede redefine\u0219te conceptul de proprietate.<\/p>\n

Pentru a de\u021bine ceva \u00een sensul tradi\u021bional, fie c\u0103 este vorba de o cas\u0103 sau o sum\u0103 de bani, \u00eenseamn\u0103 s\u0103 ai custodia personal\u0103 de lucru sau acordarea de custodie pentru o entitate de \u00eencredere, cum ar fi o banc\u0103.<\/p>\n

Cu Bitcoin cazul este diferit. Monedele Bitcoin \u00een sine nu sunt stocate la nivel central sau local, \u0219i a\u0219a nu este \u00een custodia unei entit\u0103\u021bi. Ele exist\u0103 ca \u00eenregistr\u0103ri pe un registru distribuit numit lan\u021bul de bloc, copii care sunt \u00eemp\u0103rt\u0103\u0219ite de c\u0103tre o re\u021bea de voluntari de calculatoare conectate. Pentru a „de\u021bine” o moned\u0103 Bitcoin pur \u0219i simplu reprezint\u0103 faptul c\u0103 ai capacitatea de a transfera controlul acesteia altcuiva prin crearea unui record de transfer \u00een lan\u021bul de bloc. Ce ofer\u0103 aceast\u0103 capacitate? Accesul public \u0219i privat la perechea de chei ECDSA. Ce \u00eenseamn\u0103 asta \u0219i cum rezult\u0103 c\u0103 Bitcoin este p moned\u0103 sigur\u0103?<\/p>\n

S\u0103 arunc\u0103m o privire \u00een spatele acestei tehnologii.<\/p>\n

ECDSA este prescurtarea pentru Elliptic Curve Digital Signature Algorithm. Este un proces care folose\u0219te o curb\u0103 eliptic\u0103 \u0219i un domeniu finit pentru a „semna” date \u00een a\u0219a fel \u00eenc\u00e2t ter\u021bii pot verifica autenticitatea semn\u0103turii \u00een timp ce semnatarul \u00ee\u0219i p\u0103streaz\u0103 capacitatea exclusiv\u0103 de a crea semn\u0103tura. Cu Bitcoin, datele care sunt semnat reprezint\u0103 tranzac\u021bia care transfer\u0103 dreptul de proprietate.<\/p>\n

ECDSA are proceduri separate pentru semnare \u0219i verificare. Fiecare procedur\u0103 este un algoritm compus din c\u00e2teva opera\u021bii aritmetice. Algoritmul de semnare face uz de cheia privat\u0103, iar procesul de verificare face uz de cheia public\u0103. Vom ar\u0103ta un exemplu de acest mecanism mai t\u00e2rziu.<\/p>\n

Dar, mai \u00eent\u00e2i, un curs intensiv pe curbe eliptice \u0219i corpuri finite.<\/p>\n

Curbe eliptice<\/p>\n

O curb\u0103 eliptic\u0103 este reprezentat\u0103 algebric ca o ecua\u021bie de forma:<\/p>\n

y2 = x3 + ax + b<\/p>\n

Pentru a = 0 si b = 7 (versiunea utilizat\u0103 de c\u0103tre Bitcoin), arat\u0103 cam a\u0219a:<\/p>\n

\"1\"<\/a><\/p>\n

Curbele eliptice au propriet\u0103\u021bi utile. De exemplu, o linie non-vertical\u0103 intersectat\u0103 la dou\u0103 puncte non-tangente la curb\u0103 va intersecta \u00eentotdeauna un al treilea punct de pe curb\u0103. O proprietate este aceea c\u0103 o tangent\u0103 linie non-vertical\u0103 a curbei la un moment dat va intersecta exact un alt punct de pe curb\u0103.<\/p>\n

Putem folosi aceste propriet\u0103\u021bi pentru a defini dou\u0103 opera\u021biuni: punct de adunare \u0219i punct de dublare.<\/p>\n

Punct plus, P + Q = R, este definit ca reflectarea prin axa x a punctului de intersec\u021bie al treilea R pe o linie care include P \u0219i Q. Este cel mai u\u0219or de \u00een\u021beles acest lucru, folosind o diagram\u0103:<\/p>\n

\"2\"<\/a><\/p>\n

 <\/p>\n

\u00cen mod similar, punctul de dublare, P + P = R este definit prin g\u0103sirea liniei tangent\u0103 de la punctul dublat, P, \u0219i lu\u00e2nd reflec\u021bie prin axa x a punctului de intersec\u021bie R pe curba pentru a ob\u021bine R. Iat\u0103 un exemplu cum ar ar\u0103ta:<\/p>\n

\"3\"<\/a><\/p>\n

\u00cempreun\u0103, aceste dou\u0103 opera\u021bii sunt utilizate pentru multiplicarea scalar\u0103, R = un P, definit prin ad\u0103ugarea punctului P la sine de c\u00e2teva ori. De exemplu:<\/p>\n

R = 7P
\nR = P + (P + (P + (P + (P + (P + P)))))<\/p>\n

Procesul de multiplicare scalar\u0103 este \u00een mod normal simplificat prin utilizarea unei combina\u021bii de ad\u0103ugare de punct \u0219i opera\u021biuni cu virgul\u0103 dublate. De exemplu:<\/p>\n

R = 7P
\nR = P + 6P
\nR = P + 2 (3P)
\nR = P + 2 (P + 2P)<\/p>\n

Aici, 7P a fost defalcat \u00een dou\u0103 etape de dublare de puncte \u0219i dou\u0103 etape de punct de adi\u021bie.<\/p>\n

Corpuri finite<\/strong><\/p>\n

Un domeniu finit, \u00een contextul ECDSA, poate fi considerat ca o serie predefinit\u0103 de numere pozitive \u00een care fiecare calcul trebuie s\u0103 cad\u0103. Orice num\u0103r \u00een afara acestui interval „\u00eenconjoar\u0103”, astfel \u00eenc\u00e2t s\u0103 se \u00eencadreze \u00een interval.<\/p>\n

Cel mai simplu mod de a te g\u00e2ndi la acest lucru este calcularea resturilor, a\u0219a cum este reprezentat de modul operatorul (mod). De exemplu, 9\/7 d\u0103 1 cu un rest de 2:<\/p>\n

9 mod 7 = 2<\/p>\n

Aici domeniul nostru finit este modul 7, \u0219i toate opera\u021biile de mod peste acest domeniu ob\u021bin un rezultat care se \u00eencadreaz\u0103 \u00eentr-un interval de la 0 la 6.<\/p>\n

Punerea \u00eempreun\u0103<\/strong><\/p>\n

ECDSA utilizeaz\u0103 curbe eliptice \u00een contextul unui c\u00e2mp finit, care schimb\u0103 foarte mult aspectul lor, dar nu ecua\u021biile lor subiacente sau propriet\u0103\u021bi speciale. Aceea\u0219i ecua\u021bie reprezentat\u0103 grafic de mai sus, \u00eentr-un domeniu finit de modul 67, arat\u0103 astfel:<\/p>\n

\"4\"<\/a><\/p>\n

Acum este un set de puncte, \u00een care toate valorile x \u0219i y sunt numere \u00eentregi cuprinse \u00eentre 0 \u0219i 66, Re\u021bine\u021bi c\u0103 „curba” p\u0103streaz\u0103 \u00eenc\u0103 o simetrie orizontal\u0103.<\/p>\n

Punctul plus \u0219i dublarea sunt acum u\u0219or diferite de cum se vede. Liniile trase pe acest grafic se vor \u00eencheia \u00een jurul direc\u021biilor orizontale \u0219i verticale, la fel ca \u00eentr-un joc de asteroizi, men\u021bin\u00e2nd acea\u0219i pant\u0103. Deci, ad\u0103ug\u00e2nd punctele (2, 22) \u0219i (6, 25), arat\u0103 astfel:<\/p>\n

\"5\"<\/a><\/p>\n

Al treilea punct de intersec\u021bie este (47, 39) \u0219i punctul s\u0103u de reflexie este (47, 28).<\/p>\n

\u00cenapoi la ECDSA \u0219i Bitcoin<\/strong><\/p>\n

Un protocol, precum Bitcoin, selecteaz\u0103 un set de parametri de curbe eliptice \u0219i a reprezent\u0103rii sale pe domeniul finit, care este fixat pentru to\u021bi utilizatorii de protocol. Parametrii con\u021bin ecua\u021bia folosit\u0103, prim modul de teren, \u0219i un punct de baz\u0103 care cade pe curb\u0103. Ordinea de punctul de baz\u0103, care nu este \u00een mod independent, dar este o func\u021bie de al\u021bi parametri, poate fi g\u00e2ndit\u0103 ca grafic de c\u00e2te ori punctele pot fi ad\u0103ugate la sine p\u00e2n\u0103 panta sa este infinit\u0103 sau o linie vertical\u0103. Punctul de baz\u0103 este selectat astfel \u00eenc\u00e2t ordinea este un num\u0103r mare prim.<\/p>\n

Bitcoin folose\u0219te un num\u0103r foarte mare \u00een punctul s\u0103u de baz\u0103, prim modul \u0219i ordine. De fapt, toate aplica\u021biile practice ale ECDSA utilizeaz\u0103 valori enorme.Securitatea algoritmului se bazeaz\u0103 pe aceste valori fiind mari, \u0219i, prin urmare, este imposibil\u0103 for\u021ba brut\u0103 sau ingineria invers\u0103.<\/p>\n

\u00cen cazul Bitcoin:<\/p>\n

Ecua\u021bie curbe eliptice: y2 = x3 + 7<\/p>\n

Prim modul = 2256 – 232 – 29 – 28 – 27 – 26 – 24 – 1 = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F<\/p>\n

Punct de baz\u0103 = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8<\/p>\n

Ordine = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141<\/p>\n

Cine a ales aceste numere, \u0219i de ce? O mare parte din cercetare \u0219i o valoare just\u0103 de intrigi \u00eenconjoar\u0103 selectarea parametrilor corespunz\u0103toare. La urma urmei, un num\u0103r mare, aparent aleator ar putea ascunde o metod\u0103 backdoor de a reconstrui cheia privat\u0103. Pe scurt, aceast\u0103 realizare special\u0103 sub numele de secp256k1, face parte dintr-o familie de solu\u021bii de curbe eliptice peste corpuri finite propuse pentru a fi utilizate \u00een criptografie.<\/p>\n

Cheile private \u0219i chei publice<\/p>\n

Cu aceste formalit\u0103\u021bi suntem acum \u00een m\u0103sur\u0103 s\u0103 \u00een\u021belegem cheile publice \u0219i private \u0219i modul \u00een care acestea sunt legate. \u00cen ECDSA, cheia privat\u0103 este un num\u0103r imprevizibil ales \u00eentre 1 \u0219i ordinea. Cheia public\u0103 este derivat\u0103 din cheia privat\u0103 de multiplicare scalar\u0103 a punctului de baz\u0103 cu un num\u0103r de ori egal cu valoarea cheii private. Exprimat ca o ecua\u021bie:<\/p>\n

cheie public\u0103 = cheie privat\u0103 * punct de baz\u0103<\/p>\n

Acest lucru arat\u0103 c\u0103 num\u0103rul maxim posibil de chei private (\u0219i, astfel, adrese Bitcoin) este egal cu ordinul.<\/p>\n

\u00centr-un domeniu continuu am putea trasa linia tangent\u0103 \u0219i identifica cheia public\u0103 pe grafic, dar exist\u0103 unele ecua\u021bii care realizeaz\u0103 acela\u0219i lucru, \u00een contextul de corpuri finite. Punctul plus de p + q pentru a g\u0103si r este definit \u00een\u021belept pe componente, dup\u0103 cum urmeaz\u0103:<\/p>\n

c = (QY – PY) \/ (qx – px)
\nrx = C2 – px – QX
\nRy = c (PX – RX) – py<\/p>\n

\u0218i punct dublare p pentru a g\u0103si r este urm\u0103torul:<\/p>\n

c = (3px2 + a) \/ 2PY
\nrx = c2 – 2px
\nRy = c (PX – RX) – py<\/p>\n

\u00cen practic\u0103, calcului cheii publice este defalcat \u00eentr-un num\u0103r de opera\u021biuni cu punct de dublare \u0219i punct de adi\u021bie pornind de la punctul de baz\u0103.<\/p>\n

Uit\u00e2ndu-ne \u00een urm\u0103 la numerele mici, pentru a ob\u021bine o intui\u021bie cu privire la modul \u00een care sunt construite \u0219i utilizate \u00een semnarea \u0219i verificarea cheilor. Parametrii pe care \u00eei vom folosi sunt:<\/p>\n

Ecua\u021bia: y2 = x3 + 7 (cum s-ar spune, a = 0 si b = 7)
\nPrim Modul: 67
\nPunctul de baz\u0103: (2, 22)
\nComanda: 79
\nCheie privat\u0103: 2<\/p>\n

\u00cen primul r\u00e2nd, haide\u021bi s\u0103 g\u0103sim cheia public\u0103. Din moment ce ne-am ales cel mai simplu posibil cheia privat\u0103 cu valoare = 2, va fi nevoie de doar un singur punct de dublare de func\u021bionare din punctul de baz\u0103. Calculul arat\u0103 astfel:<\/p>\n

c = (3 * 22 + 0) \/ (2 * 22) mod 67
\nc = (3 * 4) \/ (44) mod 67
\nc = 12\/44 mod 67<\/p>\n

Aici trebuie s\u0103 ne oprim pentru un pic de prestidigita\u021bie de m\u00e2n\u0103: cum vom efectua divizia \u00een cadrul unui c\u00e2mp finit, \u00een cazul \u00een care rezultatul trebuie s\u0103 fie \u00eentotdeauna un num\u0103r \u00eentreg? Trebuie s\u0103 se \u00eenmul\u021beac\u0103 cu inversul. \u00cen cazul de fa\u021b\u0103, va trebui s\u0103 ai \u00eencredere \u00een noi pentru momentul \u00een care:<\/p>\n

44-1 = 32<\/p>\n

Mutarea dreapt\u0103 de-a lungul:<\/p>\n

c = 12 * 32 mod 67
\nc = 384 mod 67
\nc = 49<\/p>\n

rx = (492-2 * 2) mod 67
\nrx = (2401-4) mod 67
\nrx = 2,397 mod 67
\nrx = 52<\/p>\n

riu = (49 * (2-52) – 22) mod 67
\nriu = (49 * (-50) – 22) mod 67
\nriu = (-2450 – 22) mod 67
\nriu = -2472 mod 67
\nry = 7<\/p>\n

\"math-behind-bitcoin-630x376\"<\/a><\/p>\n

Semn\u0103tura noastr\u0103 este perechea (r, s) = (62, 47).<\/p>\n

Ca \u0219i \u00een cazul cheilor private \u0219i publice, aceast\u0103 semn\u0103tur\u0103 este \u00een mod normal reprezentat\u0103 de un \u0219ir hexazecimal.<\/p>\n

Verificarea semn\u0103turii cu cheia public\u0103<\/strong><\/p>\n

Avem acum c\u00e2teva date \u0219i o semn\u0103tur\u0103 pentru date. O a treia parte care are cheia noastr\u0103 public\u0103 poate primi date \u0219i semn\u0103tura noastr\u0103 ca s\u0103 verifice c\u0103 suntem expeditorii. S\u0103 vedem cum func\u021bioneaz\u0103 acest lucru.<\/p>\n

Cu Q fiind cheia public\u0103 \u0219i celelalte variabile definite ca mai \u00eenainte, etapele de verificare a semn\u0103turii sunt urm\u0103toarele:<\/p>\n

Asigura\u021bi-v\u0103 c\u0103 r \u0219i s sunt \u00eentre 1 \u0219i n – 1.
\nCalcula\u021bi w = s-1 mod n
\nCalcula\u021bi u = z * w n mod
\nCalcula\u021bi v = r * w mod n
\nCalcula\u021bi punctul (x, y) = S + VQ
\nVerifica\u021bi c\u0103 r = x mod n. Semn\u0103tura nu este valabil\u0103 \u00een cazul \u00een care acesta nu este.
\nVariabile noastre, \u00eenc\u0103 o dat\u0103:
\nz = 17 (date)
\n(r, s) = (62, 47) (semn\u0103tur\u0103)
\nn = 79 (pentru)
\nG = (2, 22), (punct de baz\u0103)
\nQ = (52, 7) (cheia public\u0103)<\/p>\n

Asigura\u021bi-v\u0103 c\u0103 r \u0219i s sunt \u00eentre 1 – 1
\nr: 1 <= 62 <79
\ns: 1 <= 47 <79<\/p>\n

Calcula\u021bi w:
\nw = s-1 mod n
\nw = 47-1 mod 79
\nw = 37<\/p>\n

Calcula\u021bi u:
\nu = ZW mod n
\nu = 17 * 37 mod 79
\nu = 629 mod 79
\nu = 76<\/p>\n

Calcula\u021bi v:
\nv = RW n mod
\nv = 62 * 37 mod 79
\nv = 2,294 mod 79
\nv = 3<\/p>\n

Calcula\u021bi punctul (x, y):
\n(x, y) = ug + VQ<\/p>\n

Descompunem dublarea punctului \u0219i ad\u0103ugarea \u00een UG \u0219i VQ separat.<\/p>\n

S = 76g
\nS = 2 (38G)
\nS = 2 (2 (19G))
\nS = 2 (2 (G + 18G))
\nS = 2 (2 (P + 2 (9G)))
\nS = 2 (2 (P + 2 (G + 8G)))
\nS = 2 (2 (P + 2 (P + 2 (4G))))
\nS = 2 (2 (P + 2 (P + 2 (2 (2G)))))<\/p>\n

Prin utilizarea trucului de grupare vom reduce 75 de opera\u021bii de adunare succesive la doar \u0219ase opera\u021biuni de la punctul de dublare \u0219i dou\u0103 opera\u021biuni de punct plus.
\nLucr\u00e2nd de la interior spre exterior:<\/p>\n

S = 2 (2 (P + 2 (P + 2 (2 (2 (2, 22))))))
\nS = 2 (2 (P + 2 (P + 2 (2 (52, 7)))))
\nS = 2 (2 (P + 2 (P + 2 (25, 17))))
\nS = 2 (2 (P + 2 ((2, 22) + (21, 42))))
\nS = 2 (2 (P + 2 (13, 44)))
\npG = 2 (2 ((2, 22) + (66, 26)))
\nS = 2 (2 (38, 26))
\npG = 2 (27, 40)
\npG = (62, 4)<\/p>\n

\u0218i acum pentru VQ:<\/p>\n

VQ = T3
\nVQ = Q + T2
\nVQ = Q + 2 (52, 7)
\nVQ = (52, 7) + (25, 17)
\nVQ = (11, 20)<\/p>\n

Pun\u00e2ndu-le \u00eempreun\u0103:<\/p>\n

(x, y) = ug + VQ
\n(x, y) = (62, 4) + (11, 20)
\n(x, y) = (62, 63)<\/p>\n

Pentru etapa final\u0103.<\/p>\n

Verifica\u021bi c\u0103 r = x mod n
\nr = x mod n
\n62 = 62 mod 79
\n62 = 62<\/p>\n

Semn\u0103tura este valid\u0103!<\/p>\n

Concluzie<\/strong><\/p>\n

Pentru cei dintre voi care au v\u0103zut toate ecua\u021biile \u0219i au s\u0103rit la partea de jos, ce am \u00eenv\u0103\u021bat?<\/p>\n

Am dezvoltat intui\u021bie \u00een rela\u021bia matematic\u0103 profund\u0103 care exist\u0103 \u00eentre cheile publice \u0219i private. Am v\u0103zut cum, chiar \u0219i \u00een cele mai simple exemple de matematic\u0103 din spatele semn\u0103turii \u0219i verific\u0103rii rapide se complic\u0103 \u0219i putem aprecia complexitatea enorm\u0103 pe care trebuie s\u0103 o implic\u0103m atunci c\u00e2nd parametrii implica\u021bi sunt numere de 256 de bi\u021bi. Am v\u0103zut modul \u00een care aplicarea inteligent\u0103 a celor mai simple proceduri matematice pot crea \u00eentr-o direc\u021bie func\u021biile „trap\u0103” necesare pentru a p\u0103stra asimetria informa\u021biilor care define\u0219c dreptul de proprietate asupra Bitcoin. Avem \u00eencrederea dob\u00e2ndit\u0103 \u00een soliditatea sistemului, cu condi\u021bia ca s\u0103 protej\u0103m cu aten\u021bie cuno\u0219tin\u021bele de chei private.<\/p>\n

Cu alte cuvinte, acesta este motivul pentru care se spune de obicei c\u0103 Bitcoin este „sus\u021binut\u0103 de matematic\u0103<\/em>„.<\/p>\n

Sursa: coindesk<\/p>\n","protected":false},"excerpt":{"rendered":"

Eric Rykwalder este un inginer de software \u0219i unul din fondatorii Chain.com. El ofer\u0103 o privire de ansamblu asupra fundamentelor matematice ale protocolului Bitcoin. \u00a0 Un motiv pentru care Bitcoin poate fi confuz pentru \u00eencep\u0103tori este c\u0103 tehnologia din spatele aceastei monede redefine\u0219te conceptul de proprietate. Pentru a de\u021bine ceva \u00een sensul tradi\u021bional, fie c\u0103 este vorba de o cas\u0103 sau o sum\u0103 de bani, \u00eenseamn\u0103 s\u0103 ai custodia personal\u0103 de lucru sau acordarea de custodie pentru o entitate deRead More<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[362,361],"class_list":["post-1624","post","type-post","status-publish","format-standard","hentry","category-bitcoin","tag-calcule-bitcoin","tag-matematica-bitcoin"],"yoast_head":"\nMatematica din spatele Bitcoin<\/title>\n<meta name=\"description\" content=\"Matematica din spatele Bitcoin\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/bitcoinromania.ro\/blog\/matematica-din-spatele-bitcoin\/\" \/>\n<meta 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