{"id":9837,"date":"2023-10-22T17:00:57","date_gmt":"2023-10-22T10:00:57","guid":{"rendered":"https:\/\/bitcoinvn.io\/news\/?p=9837"},"modified":"2024-05-04T09:03:27","modified_gmt":"2024-05-04T02:03:27","slug":"phan-tich-toan-hoc-an-giau-dang-sau-bitcoin","status":"publish","type":"post","link":"https:\/\/bitcoinvn.io\/insights\/vi\/phan-tich-toan-hoc-an-giau-dang-sau-bitcoin\/","title":{"rendered":"Ph\u00e2n t\u00edch To\u00e1n h\u1ecdc \u201c\u1ea9n gi\u1ea5u” \u0111\u1eb1ng sau Bitcoin"},"content":{"rendered":"

\u0110\u1ed1i v\u1edbi nh\u1eefng ng\u01b0\u1eddi m\u1edbi b\u1eaft \u0111\u1ea7u, m\u1ed9t trong nh\u1eefng l\u00fd do khi\u1ebfn Bitcoin.<\/span>tr\u1edf n\u00ean kh\u00f3 hi\u1ec3u ch\u00ednh l\u00e0: c\u00f4ng ngh\u1ec7 ph\u00eda sau n\u00f3 \u0111\u00e3 cho ch\u00fang ta m\u1ed9t \u0111\u1ecbnh ngh\u0129a ho\u00e0n to\u00e0n kh\u00e1c.<\/span>v\u1ec1 kh\u00e1i ni\u1ec7m \u201c<\/i><\/em>quy\u1ec1n s\u1edf h\u1eefu<\/i><\/em>\u201d<\/i><\/em>.<\/i><\/em> V\u1eady quy\u1ec1n s\u1edf h\u1eefu trong Bitcoin l\u00e0 g\u00ec? L\u00e0m th\u1ebf n\u00e0o \u0111\u1ec3 t\u1ea1o ra kh\u00f3a ri\u00eang t\u01b0 (private key), kh\u00f3a c\u00f4ng khai (public key), ch\u1eef k\u00fd\u2026 \u0111\u1ec3 gi\u00fap b\u1ea1n to\u00e0n quy\u1ec1n s\u1edf h\u1eefu Bitcoin? <\/i><\/em><\/p>\n

T\u1ea5t c\u1ea3 \u0111\u1ec1u \u0111\u01b0\u1ee3c th\u1ef1c hi\u1ec7n b\u1edfi c\u00e1c thu\u1eadt to\u00e1n khoa h\u1ecdc v\u00e0 ch\u00ednh x\u00e1c. Ngay b\u00e2y gi\u1edd, m\u1eddi b\u1ea1n c\u00f9ng ch\u00fang t\u00f4i.<\/span>\u0111i s\u00e2u ph\u00e2n t\u00edch c\u00e1c ph\u00e9p t\u00ednh To\u00e1n h\u1ecdc t\u1ea1o n\u00ean th\u00e0nh c\u00f4ng c\u1ee7a Bitcoin. Ri\u00eang \u0111\u1ed1i v\u1edbi nh\u1eefng ng\u01b0\u1eddi kh\u00f4ng c\u00f3 ni\u1ec1m \u0111am m\u00ea v\u1edbi nh\u1eefng con s\u1ed1,.<\/span>b\u1ea1n h\u00e3y l\u01b0\u1edbt \u0111\u1ebfn ph\u1ea7n k\u1ebft lu\u1eadn \u1edf cu\u1ed1i b\u00e0i<\/strong> \u0111\u1ec3 t\u00ecm ra b\u00e0i h\u1ecdc cho ri\u00eang m\u00ecnh nh\u00e9!<\/p>\n

\"\"
D\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u00e2n t\u00edch to\u00e1n h\u1ecdc \u0111\u1eb1ng sau Bitcoin<\/figcaption><\/figure>\n

Bitcoin \u0111\u1ecbnh ngh\u0129a l\u1ea1i kh\u00e1i ni\u1ec7m v\u1ec1 \u201cQuy\u1ec1n S\u1edf h\u1eefu\u201d<\/b><\/strong><\/h2>\n

Theo ngh\u0129a truy\u1ec1n th\u1ed1ng, \u0111\u1ec3 s\u1edf h\u1eefu m\u1ed9t c\u00e1i g\u00ec \u0111\u00f3. V\u00ed d\u1ee5 nh\u01b0 s\u01a1 h\u1eefu m\u1ed9t c\u0103n nh\u00e0, m\u1ed9t chi\u1ebfc xe hay m\u1ed9t kho\u1ea3n ti\u1ec1n\u2026 c\u00f3 ngh\u0129a l\u00e0 nh\u1eefng th\u1ee9 n\u00e0y thu\u1ed9c quy\u1ec1n n\u1eafm gi\u1eef c\u1ee7a ch\u00fang ta (ho\u1eb7c m\u1ed9t \u0111\u01a1n v\u1ecb \u0111\u00e1ng tin c\u1eady nh\u01b0 ng\u00e2n h\u00e0ng).<\/p>\n

Tuy nhi\u00ean, v\u1edbi Bitcoin, quy\u1ec1n s\u1edf h\u1eefu l\u1ea1i \u0111\u01b0\u1ee3c th\u1ec3 hi\u1ec7n m\u1ed9t c\u00e1ch ho\u00e0n to\u00e0n kh\u00e1c. Bitcoin l\u00e0 phi t\u1eadp trung v\u00e0 kh\u00f4ng m\u1ed9t ai hay m\u1ed9t t\u1ed5 ch\u1ee9c n\u00e0o c\u00f3 kh\u1ea3 n\u0103ng n\u1eafm gi\u1eef n\u00f3 trong tay<\/b><\/strong>. Nh\u1eefng \u0111\u1ed3ng ti\u1ec1n Bitcoin (BTC)<\/a> n\u00e0y t\u1ed3n t\u1ea1i d\u01b0\u1edbi d\u1ea1ng c\u00e1c b\u1ed9 h\u1ed3 s\u01a1 \u0111\u01b0\u1ee3c ph\u00e2n ph\u1ed1i tr\u00ean m\u1ed9t cu\u1ed1n s\u1ed5 k\u1ebf to\u00e1n g\u1ecdi l\u00e0\u00a0chu\u1ed7i kh\u1ed1i (blockchain). B\u1ea3n sao c\u1ee7a d\u1eef li\u1ec7u s\u1ebd \u0111\u01b0\u1ee3c chia s\u1ebb cho c\u00e1c m\u1ea1ng l\u01b0\u1edbi m\u00e1y t\u00ednh ch\u1ee7 \u0111\u1ed9ng k\u1ebft n\u1ed1i v\u1edbi n\u00f3.<\/p>\n

Hi\u1ec3u m\u1ed9t c\u00e1ch \u0111\u01a1n gi\u1ea3n, khi \u201cs\u1edf h\u1eefu\u201d 1 Bitcoin c\u00f3 ngh\u0129a l\u00e0 b\u1ea1n c\u00f3 m\u1ed9t h\u1ed3 s\u01a1 l\u01b0u tr\u1eef 1 BTC.<\/span> tr\u00ean blockchain v\u00e0 b\u1ea1n to\u00e0n quy\u1ec1n chuy\u1ec3n nh\u01b0\u1ee3ng quy\u1ec1n s\u1edf h\u1eefu h\u1ed3 s\u01a1 n\u00e0y.<\/span>cho m\u1ed9t ng\u01b0\u1eddi kh\u00e1c n\u1ebfu b\u1ea1n n\u1eafm gi\u1eef m\u00e3 kh\u00f3a c\u00f4ng khai (public) v\u00e0 kh\u00f3a private ECDSA<\/u><\/a>. V\u1eady c\u1eb7p m\u00e3 kh\u00f3a n\u00e0y c\u00f3 ngh\u0129a l\u00e0 g\u00ec? V\u00e0 t\u1ea1i sao c\u1eb7p m\u00e3 kh\u00f3a n\u00e0y gi\u00fap Bitcoin an to\u00e0n?<\/p>\n

Ch\u00fang ta s\u1ebd c\u00f9ng \u0111i s\u00e2u ph\u00e2n t\u00edch nh\u00e9!<\/p>\n

ECDSA<\/b><\/strong> l\u00e0 vi\u1ebft t\u1eaft c\u1ee7a Elliptic Curve Digital Signature Algorithm –.<\/span>m\u1ed9t thu\u1eadt to\u00e1n ch\u1eef k\u00fd \u0111i\u1ec7n t\u1eed \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean t\u1eeb \u0111\u01b0\u1eddng cong Elliptic. Theo \u0111\u00f3, ECDSA k\u1ebft h\u1ee3p 1\u00a0\u0111\u01b0\u1eddng cong elliptic<\/u><\/a>\u00a0v\u00e0 1\u00a0tr\u01b0\u1eddng h\u1eefu h\u1ea1n<\/u><\/a> \u0111\u1ec3 t\u1ea1o ra ch\u1eef k\u00fd (sign) nh\u1eb1m x\u00e1c nh\u1eadn ph\u00ea duy\u1ec7t chuy\u1ec3n nh\u01b0\u1ee3ng quy\u1ec1n s\u1edf h\u1eefu BTC. Ch\u1eef k\u00fd n\u00e0y bu\u1ed9c ph\u1ea3i \u0111\u1ea3m b\u1ea3o 2 y\u1ebfu t\u1ed1: ch\u1eef k\u00fd \u0111\u1ed9c quy\u1ec1n v\u00e0 m\u1ecdi ng\u01b0\u1eddi c\u00f3 th\u1ec3 x\u00e1c minh t\u00ednh x\u00e1c th\u1ef1c c\u1ee7a ch\u1eef k\u00fd \u0111\u00f3.<\/p>\n

Nh\u00ecn chung, thu\u1eadt to\u00e1n ECDSA c\u00f3 nh\u1eefng c\u00f4ng th\u1ee9c ri\u00eang bi\u1ec7t.<\/span>trong vi\u1ec7c t\u1ea1o ra ch\u1eef k\u00fd v\u00e0 x\u00e1c minh giao d\u1ecbch. M\u1ed7i c\u00f4ng th\u1ee9c l\u00e0 m\u1ed9t thu\u1eadt to\u00e1n ri\u00eang. C\u00e1c thu\u1eadt to\u00e1n t\u1ea1o ra ch\u1eef k\u00fd n\u00e0y s\u1eed d\u1ee5ng c\u00e1c kh\u00f3a ri\u00eang (private key), v\u00e0 qu\u00e1 tr\u00ecnh x\u00e1c minh d\u00f9ng kh\u00f3a c\u00f4ng khai (public key). \u0110\u1ec3 hi\u1ec3u h\u01a1n v\u1ec1 v\u1ea5n \u0111\u1ec1 n\u00e0y, ch\u00fang ta s\u1ebd c\u00f9ng nhau ph\u00e2n t\u00edch c\u00e1c v\u00ed d\u1ee5 c\u1ee5 th\u1ec3.<\/p>\n

Tuy nhi\u00ean, tr\u01b0\u1edbc ti\u00ean, h\u00e3y c\u00f9ng nhau t\u00ecm hi\u1ec3u nhanh v\u1ec1 2 y\u1ebfu t\u1ed1 t\u1ea1o ra ch\u1eef k\u00fd, bao g\u1ed3m: \u0111\u01b0\u1eddng cong Elliptic<\/strong> v\u00e0 c\u00e1c tr\u01b0\u1eddng h\u1eefu h\u1ea1n<\/strong>.<\/p>\n

T\u1eeb A-Z v\u1ec1 \u0111\u01b0\u1eddng cong Elliptic<\/b><\/strong><\/h2>\n

\u0110\u01b0\u1eddng cong Elliptic l\u00e0 m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 d\u1ea1ng:<\/p>\n

\n

y\u00a02<\/sup>\u00a0= x\u00a03<\/sup>\u00a0+ ax + b<\/strong><\/p>\n<\/blockquote>\n

V\u1edbi phi\u00ean b\u1ea3n c\u1ee7a Bitcoin, c\u00e1c gi\u00e1 tr\u1ecb a,b \u0111\u01b0\u1ee3c t\u00ednh l\u00e0: a\u00a0= 0<\/i><\/em>\u00a0v\u00e0\u00a0b = 7<\/i><\/em>. Khi bi\u1ec3u di\u1ec5n tr\u00ean bi\u1ec3u \u0111\u1ed3, ch\u00fang \u0111\u01b0\u1ee3c th\u1ec3 hi\u1ec7n nh\u01b0 sau:<\/p>\n

\"Bi\u1ec3u
Bi\u1ec3u \u0111\u1ed3 elliptic c\u1ee7a Bitcoin<\/figcaption><\/figure>\n

M\u1ed7i \u0111\u01b0\u1eddng cong Elliptic gi\u1ea3i th\u00edch r\u1ea5t nhi\u1ec1u \u0111i\u1ec1u \u00fd ngh\u0129a. V\u00ed d\u1ee5, m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng n\u1eb1m x\u00e9o giao v\u1edbi \u0111\u01b0\u1eddng cong t\u1ea1i 2 \u0111i\u1ec3m.<\/span>th\u00ec ch\u00fang s\u1ebd lu\u00f4n giao v\u1edbi \u0111\u01b0\u1eddng cong t\u1ea1i m\u1ed9t \u0111i\u1ec3m th\u1ee9 3. M\u1ed9t t\u00ednh ch\u1ea5t kh\u00e1c l\u00e0: m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng n\u1eb1m x\u00e9o so v\u1edbi \u0111\u01b0\u1eddng cong.<\/span>v\u00e0 giao v\u1edbi \u0111\u01b0\u1eddng cong t\u1ea1i 1 \u0111i\u1ec3m th\u00ec.<\/span>ch\u00fang s\u1ebd ti\u1ebfp t\u1ee5c giao v\u1edbi \u0111\u01b0\u1eddng cong \u1ea5y t\u1ea1i m\u1ed9t \u0111i\u1ec3m th\u1ee9 2.<\/p>\n

D\u1ef1a v\u00e0o c\u00e1c \u0111\u1eb7c t\u00ednh n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh 2 ph\u00e9p to\u00e1n:.<\/span>C\u1ed9ng \u0110i\u1ec3m v\u00e0 Nh\u00e2n \u0111\u00f4i \u0110i\u1ec3m.<\/p>\n

Tr\u01b0\u1edbc ti\u00ean l\u00e0 ph\u00e9p\u00a0C\u1ed9ng \u0110i\u1ec3m<\/b><\/strong>\u00a0\u0111\u01b0\u1ee3c t\u00ednh b\u1eb1ng c\u00f4ng th\u1ee9c: P + Q = R<\/i><\/b><\/em><\/strong>. <\/i><\/b><\/em><\/strong>C\u00f4ng th\u1ee9c n\u00e0y \u0111\u01b0\u1ee3c t\u00ednh\u00a0b\u1eb1ng\u00a0d\u1ef1a tr\u00ean \u0111i\u1ec3m giao nhau th\u1ee9 3 l\u00e0 R\u2019.<\/i><\/em> \u0110\u1ec3 hi\u1ec3u h\u01a1n v\u1ec1 c\u00f4ng th\u1ee9c n\u00e0y, ch\u00fang ta s\u1ebd c\u00f9ng nhau.<\/span>bi\u1ec3u di\u1ec5n ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean tr\u1ee5c s\u1ed1 nh\u01b0 sau:<\/p>\n

\"Bi\u1ec3u
Bi\u1ec3u di\u1ec5n ph\u00e9p C\u1ed9ng \u0110i\u1ec3m tr\u00ean tr\u1ee5c s\u1ed1<\/figcaption><\/figure>\n

T\u01b0\u01a1ng t\u1ef1, ph\u00e9p Nh\u00e2n \u0111\u00f4i \u0110i\u1ec3m<\/b><\/strong>\u00a0\u0111\u01b0\u1ee3c t\u00ednh nh\u01b0 sau: P + P = R<\/b><\/strong>. <\/i><\/em>Theo \u0111\u00f3, R\u2019 \u0111\u01b0\u1ee3c t\u00ednh b\u1eb1ng c\u00e1ch nh\u00e2n \u0111\u00f4i \u0111i\u1ec3m P v\u00e0 R.<\/span>s\u1ebd \u0111\u01b0\u1ee3c t\u00ecm ra b\u1eb1ng c\u00e1ch \u0111\u1ed1i x\u1ee9ng R\u2019 th\u00f4ng qua tr\u1ee5c X. \u0110\u1ec3 hi\u1ec3u r\u00f5 h\u01a1n v\u1ec1 c\u00f4ng th\u1ee9c n\u00e0y, m\u1eddi b\u1ea1n tham kh\u1ea3o:<\/p>\n

\"Ph\u01b0\u01a1ng
Ph\u01b0\u01a1ng tr\u00ecnh Nh\u00e2n \u0111\u00f4i \u0110i\u1ec3m tr\u00ean tr\u1ee5c s\u1ed1<\/figcaption><\/figure>\n

2 ph\u00e9p t\u00ednh C\u1ed9ng \u0110i\u1ec3m v\u00e0 Nh\u00e2n \u0111\u00f4i \u0110i\u1ec3m \u0111\u00e3 \u0111\u01b0\u1ee3c d\u00f9ng \u0111\u1ec3 t\u1ea1o n\u00ean ph\u00e9p Nh\u00e2n V\u00f4 H\u01b0\u1edbng v\u1edbi c\u00f4ng th\u1ee9c: R=<\/b><\/strong>a<\/b><\/strong>*<\/b><\/strong>P<\/b><\/strong>. <\/i><\/em>Theo \u0111\u00f3, R \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh b\u1eb1ng c\u00e1ch th\u00eam nhi\u1ec1u l\u1ea7n \u0111i\u1ec3m P v\u1edbi ch\u00ednh n\u00f3. V\u00ed d\u1ee5:<\/p>\n

\n

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

Qu\u00e1 tr\u00ecnh nh\u00e2n v\u00f4 h\u01b0\u1edbng c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c \u0111\u01a1n gi\u1ea3n h\u00f3a.<\/span>b\u1eb1ng c\u00e1ch k\u1ebft h\u1ee3p ph\u00e9p C\u1ed9ng \u0110i\u1ec3m v\u00e0 Nh\u00e2n \u0111\u00f4i \u0110i\u1ec3m. V\u00ed d\u1ee5:<\/p>\n

\n

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

Theo c\u00f4ng th\u1ee9c R = P + 2 (P + 2P)<\/strong>\u00a0th\u00ec 7P \u0111\u00e3 \u0111\u01b0\u1ee3c chia th\u00e0nh\u00a02 b\u01b0\u1edbc: nh\u00e2n \u0111\u00f4i \u0111i\u1ec3m 2 b\u01b0\u1edbc c\u1ed9ng \u0111i\u1ec3m.<\/p>\n

Nh\u1eefng \u0111i\u1ec1u c\u1ea7n bi\u1ebft v\u1ec1 Tr\u01b0\u1eddng H\u1eefu h\u1ea1n<\/b><\/strong><\/h2>\n

\u0110\u1ed1i v\u1edbi ECDSA, m\u1ed9t tr\u01b0\u1eddng h\u1eefu h\u1ea1n \u0111\u01b0\u1ee3c hi\u1ec3u nh\u01b0 l\u00e0 m\u1ed9t t\u1ed5 h\u1ee3p (c\u00f3 gi\u1edbi h\u1ea1n) c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh tr\u01b0\u1edbc v\u00e0 to\u00e0n b\u1ed9 c\u00e1c ph\u00e9p t\u00ednh s\u1ebd n\u1eb1m trong ph\u1ea1m vi nh\u1eefng con s\u1ed1 ch\u00fang ta \u0111\u00e3 c\u00f3. B\u1ea5t k\u1ef3 con s\u1ed1 n\u00e0o ngo\u00e0i t\u1ed5 h\u1ee3p n\u00e0y s\u1ebd \u0111\u01b0\u1ee3c quy ho\u1ea1ch v\u1ec1 trong t\u1ed5 h\u1ee3p<\/b><\/strong>.<\/p>\n

C\u00e1ch \u0111\u01a1n gi\u1ea3n nh\u1ea5t \u0111\u1ec3 th\u1ec3 hi\u1ec7n qu\u00e1 tr\u00ecnh \u0111\u01b0a c\u00e1c con s\u1ed1 ngo\u00e0i t\u1ed5 h\u1ee3p v\u1ec1 trong ph\u1ea1m vi t\u1ed5 h\u1ee3p<\/b><\/strong> ch\u00ednh l\u00e0 ph\u00e9p t\u00ecm s\u1ed1 d\u01b0 (hay c\u00f2n g\u1ecdi l\u00e0 ph\u00e9p modulus (mod)). V\u00ed d\u1ee5 ch\u00fang ta ph\u00e2n t\u00edch ph\u00e9p chia 9\/7 s\u1ebd ra con s\u1ed1 d\u01b0 l\u00e0 2.<\/p>\n

\n

9 mod <\/i><\/em>7 = 2<\/i><\/em><\/p>\n<\/blockquote>\n

Theo v\u00ed d\u1ee5 tr\u00ean, tr\u01b0\u1eddng h\u1eefu h\u1ea1n c\u1ee7a ta l\u00e0 modulo 7. T\u1ea5t c\u1ea3 c\u00e1c ph\u00e9p to\u00e1n mod 7 \u0111\u1ec1u s\u1ebd cho m\u1ed9t con s\u1ed1 d\u01b0 n\u1eb1m trong ph\u1ea1m vi t\u1ed5 h\u1ee3p t\u1eeb 0-6.<\/p>\n

S\u1ef1 k\u1ebft h\u1ee3p c\u1ee7a \u0111\u01b0\u1eddng cong Elliptic v\u00e0 Tr\u01b0\u1eddng H\u1eefu h\u1ea1n<\/b><\/strong><\/h2>\n

Nh\u01b0 \u0111\u00e3 n\u00f3i \u1edf tr\u00ean, ECDSA k\u1ebft h\u1ee3p c\u00e1c \u0111\u01b0\u1eddng cong Elliptic v\u1edbi c\u00e1c th\u00f4ng s\u1ed1 n\u1eb1m trong m\u1ed9t tr\u01b0\u1eddng h\u1eefu h\u1ea1n. Ch\u00ednh \u0111i\u1ec1u n\u00e0y \u0111\u00e3 l\u00e0m thay \u0111\u1ed5i \u0111\u00e1ng k\u1ec3 c\u00e1c h\u00ecnh d\u1ea1ng c\u1ee7a n\u00f3 nh\u01b0ng v\u1eabn b\u1ea3o to\u00e0n ph\u01b0\u01a1ng tr\u00ecnh c\u01a1 b\u1ea3n v\u00e0 t\u00ednh ch\u1ea5t \u0111\u1eb7c bi\u1ec7t c\u1ee7a n\u00f3. M\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng t\u1ef1 n\u1eb1m trong tr\u01b0\u1eddng h\u1eefu h\u1ea1n c\u1ee7a modulo 67\u00a0s\u1ebd c\u00f3 h\u00ecnh d\u1ea1ng nh\u01b0 th\u1ebf n\u00e0y:<\/p>\n

\"S\u1ef1
S\u1ef1 k\u1ebft h\u1ee3p c\u1ee7a \u0111\u01b0\u1eddng cong elliptic v\u00e0 tr\u01b0\u1eddng h\u1eefu h\u1ea1n tr\u1edf th\u00e0nh m\u1ed9t t\u1eadp h\u1ee3p \u0111i\u1ec3m<\/figcaption><\/figure>\n

L\u00fac n\u00e0y, t\u1ea5t c\u1ea3 tr\u1edf th\u00e0nh\u00a0m\u1ed9t t\u1eadp h\u1ee3p \u0111i\u1ec3m. Trong \u0111\u00f3 m\u1ecdi gi\u00e1 tr\u1ecb\u00a0x<\/i><\/em>\u00a0v\u00e0\u00a0y<\/i><\/em>\u00a0\u0111\u1ec1u l\u00e0 s\u1ed1 nguy\u00ean, n\u1eb1m trong ph\u1ea1m vi t\u1ed5 h\u1ee3p s\u1ed1 t\u1eeb 0-66.<\/p>\n

X\u00e9t v\u1ec1 m\u1eb7t tr\u1ef1c quan, ph\u00e9p c\u1ed9ng \u0111i\u1ec3m v\u00e0 nh\u00e2n \u0111\u00f4i \u0111i\u00eam theo c\u00e1ch bi\u1ec3u di\u1ec5n n\u00e0y s\u1ebd c\u00f3 h\u01a1i kh\u00e1c nhau m\u1ed9t ch\u00fat. \u0110\u01b0\u1eddng v\u1ebd tr\u00ean \u0111\u1ed3 th\u1ecb n\u00e0y s\u1ebd qu\u1ea5n quanh h\u01b0\u1edbng ngang v\u00e0 d\u1ecdc, gi\u1ed1ng nh\u01b0 trong m\u1ed9t tr\u00f2 ch\u01a1i c\u00e1c ti\u1ec3u h\u00e0nh tinh, v\u00e0 v\u1eabn duy tr\u00ec \u0111\u1ed9 d\u1ed1c. V\u00ec v\u1eady, n\u1ebfu c\u1ed9ng \u0111i\u1ec3m (2, 22) v\u00e0 (6, 25) th\u00ec \u0111\u1ed3 th\u1ecb s\u1ebd tr\u00f4ng nh\u01b0 th\u1ebf n\u00e0y:<\/p>\n

\"\"
\u0110i\u1ec3m giao nhau th\u1ee9 ba l\u00e0 (47, 39) v\u00e0 \u0111i\u1ec3m \u00e1nh x\u1ea1 c\u1ee7a n\u00f3 l\u00e0 (47, 28)<\/figcaption><\/figure>\n

Quay t<\/b><\/strong>r\u1edf l\u1ea1i <\/b><\/strong>ph\u00e2n t\u00edch<\/b><\/strong>\u00a0ECDSA v\u00e0 Bitcoin<\/b><\/strong><\/h2>\n

Giao th\u1ee9c Bitcoin<\/a> ch\u1ecdn m\u1ed9t t\u1eadp c\u00e1c tham s\u1ed1 cho c\u00e1c \u0111\u01b0\u1eddng cong elliptic v\u00e0 tr\u01b0\u1eddng h\u1eefu h\u1ea1n c\u1ee7a n\u00f3\u00a0\u0111\u1ec3 \u00e1p d\u1ee5ng c\u1ed1 \u0111\u1ecbnh cho t\u1ea5t c\u1ea3 ng\u01b0\u1eddi d\u00f9ng giao th\u1ee9c. C\u00e1c th\u00f4ng s\u1ed1 n\u00e0y bao g\u1ed3m: c\u00e1c\u00a0ph\u01b0\u01a1ng tr\u00ecnh<\/b><\/strong>,\u00a0modulo s\u1ed1 nguy\u00ean t\u1ed1<\/b><\/strong>\u00a0c\u1ee7a tr\u01b0\u1eddng n\u00e0y, v\u00e0 m\u1ed9t\u00a0\u0111i\u1ec3m c\u01a1 s\u1edf<\/b><\/strong>\u00a0n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng cong.\u00a0B\u1eadc<\/b><\/strong>\u00a0c\u1ee7a \u0111i\u1ec3m c\u01a1 s\u1edf kh\u00f4ng \u0111\u01b0\u1ee3c l\u1ef1a ch\u1ecdn \u0111\u1ed9c l\u1eadp m\u00e0 l\u00e0 m\u1ed9t h\u00e0m c\u1ee7a\u00a0nhi\u1ec1u th\u00f4ng s\u1ed1 kh\u00e1c. Ch\u00fang\u00a0c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c xem nh\u01b0 s\u1ed1 l\u1ea7n \u0111i\u1ec3m \u0111\u00f3 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c c\u1ed9ng v\u00e0o ch\u00ednh n\u00f3 cho t\u1edbi khi \u0111\u1ed9 d\u1ed1c c\u1ee7a n\u00f3 l\u00e0 v\u00f4 h\u1ea1n (t\u1ee9c l\u00e0 m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u1ee9ng). C\u00e1c \u0111i\u1ec3m c\u01a1 s\u1edf \u0111\u01b0\u1ee3c l\u1ef1a ch\u1ecdn sao cho b\u1eadc l\u00e0 m\u1ed9t s\u1ed1 nguy\u00ean t\u1ed1 l\u1edbn.<\/p>\n

\u0110\u1ed1i v\u1edbi \u0111i\u1ec3m c\u01a1 s\u1edf, modulo nguy\u00ean t\u1ed1, v\u00e0 b\u1eadc, Bitcoin th\u01b0\u1eddng s\u1eed d\u1ee5ng c\u00e1c con s\u1ed1 r\u1ea5t l\u1edbn.\u00a0T\u1ea5t c\u1ea3 c\u00e1c \u1ee9ng d\u1ee5ng th\u1ef1c t\u1ebf c\u1ee7a ECDSA\u00a0\u0111\u1ec1u\u00a0s\u1eed d\u1ee5ng c\u00e1c gi\u00e1 tr\u1ecb r\u1ea5t l\u1edbn.\u00a0An ninh c\u1ee7a c\u00e1c thu\u1eadt to\u00e1n b\u1ea3o m\u1eadt d\u1ef1a tr\u00ean c\u00e1c gi\u00e1 tr\u1ecb l\u1edbn. Do \u0111\u00f3\u00a0r\u1ea5t kh\u00f3 c\u00f3 ai \u0111\u1ea3o ng\u01b0\u1ee3c ho\u1eb7c t\u1ea5n c\u00f4ng\u00a0h\u00e0ng r\u00e0o b\u1ea3o m\u1eadt c\u1ee7a Bitcoin\u00a0b\u1eb1ng c\u00e1c ph\u00e9p th\u1eed t\u1ea5n c\u00f4ng l\u1ea7n l\u01b0\u1ee3t.<\/p>\n

\u0110\u1ed1i v\u1edbi Bitcoin:<\/p>\n

\n

Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng congElliptic l\u00e0:\u00a0y\u00a02<\/sup>\u00a0= x\u00a03<\/sup>\u00a0+ 7<\/p>\n

Modulo nguy\u00ean t\u1ed1 =\u00a02256<\/sup>\u00a0\u2013 232<\/sup>\u00a0\u2013 29<\/sup>\u00a0\u2013 28<\/sup>\u00a0\u2013 27<\/sup>\u00a0\u2013 26<\/sup>\u00a0\u2013 24<\/sup>\u00a0\u2013 1 = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F<\/p>\n

\u0110i\u1ec3m c\u01a1 s\u1edf = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8<\/p>\n

B\u1eadc c\u1ee7a \u0111i\u1ec3m = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141<\/p>\n<\/blockquote>\n

C\u00e2u h\u1ecfi \u0111\u1eb7t ra l\u00e0: ai \u0111\u00e3 ch\u1ecdn nh\u1eefng con s\u1ed1 n\u00e0y v\u00e0 t\u1ea1i sao h\u1ecd l\u1ea1i l\u1ef1a ch\u1ecdn? \u0110\u00e3 c\u00f3 r\u1ea5t nhi\u1ec1u nghi\u00ean c\u1ee9u<\/u><\/a>, v\u00e0 m\u1ed9t l\u01b0\u1ee3ng kha kh\u00e1\u00a0nh\u1eefng thuy\u1ebft \u00e2m m\u01b0u<\/u><\/a>\u00a0xung quanh vi\u1ec7c l\u1ef1a ch\u1ecdn c\u00e1c th\u00f4ng s\u1ed1 n\u00eau tr\u00ean. Tuy nhi\u00ean, sau t\u1ea5t c\u1ea3, s\u1ed1 \u0111\u00f4ng c\u00e1c nghi\u00ean c\u1ee9u \u0111\u1ec1u \u0111\u1ec1u ch\u1ec9 ra l\u00e0 c\u00f3 m\u1ed9t ph\u01b0\u01a1ng ph\u00e1p backdoor \u0111\u1ec3 t\u00ecm ra kh\u00f3a private. Ph\u01b0\u01a1ng ph\u00e1p \u0111\u1eb7c bi\u1ec7t n\u00e0y c\u00f3 t\u00ean l\u00e0 secp256k1. Ch\u00fang l\u00e0\u00a0m\u1ed9t ph\u1ea7n \u0111\u01b0\u1eddng cong elliptic tr\u00ean tr\u01b0\u1eddng v\u00f4 h\u1ea1n\u00a0\u0111\u00e3 \u0111\u01b0\u1ee3c \u0111\u1ec1 xu\u1ea5t \u0111\u1ec3 s\u1eed d\u1ee5ng trong ng\u00e0nh m\u1eadt m\u00e3.<\/p>\n

K<\/b><\/strong>h\u00f3a <\/b><\/strong>Private v\u00e0 <\/b><\/strong>kh\u00f3a <\/b><\/strong>Public<\/b><\/strong>\u00a0l\u00e0 g\u00ec?<\/b><\/strong><\/h3>\n

Ngay b\u00e2y gi\u1edd, ch\u00fang ta s\u1ebd c\u00f9ng t\u00ecm hi\u1ec3u nhanh v\u1ec1 kh\u00f3a c\u00f4ng khai (Public key) v\u00e0 kh\u00f3a ri\u00eang t\u01b0 (private key). Sau \u0111\u00f3 ch\u00fang ta s\u1ebd ph\u00e2n t\u00edch s\u1ef1 li\u00ean quan m\u1eadt thi\u1ebft c\u1ee7a ch\u00fang.<\/p>\n

Hi\u1ec3u m\u1ed9t c\u00e1ch \u0111\u01a1n gi\u1ea3n, trong ECDSA, kh\u00f3a ri\u00eang t\u01b0\u00a0l\u00e0 m\u1ed9t d\u00e3y s\u1ed1 ng\u1eabu nhi\u00ean, kh\u00f4ng ai c\u00f3 th\u1ec3 \u0111o\u00e1n \u0111\u01b0\u1ee3c. D\u00e3y s\u1ed1 n\u00e0y n\u1eb1m gi\u1eefa s\u1ed1 1 v\u00e0 b\u1eadc c\u1ee7a \u0111i\u1ec3m.\u00a0Trong khi \u0111\u00f3, kh\u00f3a c\u00f4ng khai \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean t\u1eeb kh\u00f3a ri\u00eang b\u1eb1ng c\u00e1ch nh\u00e2n v\u00f4 h\u01b0\u1edbng kh\u00f3a ri\u00eang t\u01b0 v\u1edbi \u0111i\u1ec3m c\u01a1 s\u1edf. Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0y \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n nh\u01b0 sau:<\/p>\n

\n

Kh\u00f3a public = kh\u00f3a private * \u0111i\u1ec3m c\u01a1 s\u1edf<\/i><\/em><\/p>\n<\/blockquote>\n

Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0y cho th\u1ea5y: s\u1ed1 l\u01b0\u1ee3ng t\u1ed1i \u0111a kh\u00f3a private\u00a0(c\u00e1c \u0111\u1ecba ch\u1ec9 Bitcoin) l\u00e0 b\u1eb1ng b\u1eadc c\u1ee7a \u0111i\u1ec3m.<\/p>\n

Trong m\u1ed9t tr\u01b0\u1eddng li\u00ean t\u1ee5c,\u00a0ch\u00fang ta c\u00f3 th\u1ec3 v\u1ebd \u0111\u01b0\u1eddng ti\u1ebfp tuy\u1ebfn v\u00e0 x\u00e1c \u0111\u1ecbnh kh\u00f3a public tr\u00ean \u0111\u1ed3 th\u1ecb. Tuy nhi\u00ean, c\u00f3 m\u1ed9t s\u1ed1\u00a0ph\u01b0\u01a1ng tr\u00ecnh<\/u><\/a>\u00a0c\u00f3\u00a0th\u1ec3\u00a0th\u1ef1c hi\u1ec7n \u0111i\u1ec1u t\u01b0\u01a1ng t\u1ef1 trong b\u1ed1i c\u1ea3nh tr\u01b0\u1eddng h\u1eefu h\u1ea1n. T\u1eeb ph\u00e9p c\u1ed9ng \u0111i\u1ec3m\u00a0p + q<\/strong>\u00a0\u0111\u1ec3 t\u00ecm ra\u00a0r<\/i><\/em>\u00a0\u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh nh\u01b0 sau:<\/p>\n

\n

c = (qy<\/sub>\u00a0\u2013 py<\/sub>) \/ (qx<\/sub>\u00a0\u2013 px<\/sub>)
\nrx<\/sub>\u00a0= c2<\/sup>\u00a0\u2013 px<\/sub>\u00a0\u2013 qx<\/sub>
\nry<\/sub>\u00a0= c (px<\/sub>\u00a0\u2013 rx<\/sub>) \u2013 py<\/sub><\/p>\n<\/blockquote>\n

\u0110\u1ed3ng th\u1eddi, t\u1eeb ph\u00e9p nh\u00e2n \u0111\u00f4i \u0111i\u1ec3m\u00a0p<\/strong>, ch\u00fang ta c\u00f3 th\u1ec3 t\u00ecm ra r<\/strong> nh\u01b0 sau:<\/p>\n

\n

c = (3px<\/sub>2<\/sup>\u00a0+ a) \/ 2py<\/sub>
\nrx<\/sub>\u00a0= c2<\/sup>\u00a0\u2013 2px<\/sub>
\nry<\/sub>\u00a0= c (px<\/sub>\u00a0\u2013 rx<\/sub>) \u2013 py<\/sub><\/p>\n<\/blockquote>\n

Tr\u00ean th\u1ef1c t\u1ebf, qu\u00e1 tr\u00ecnh t\u00ednh to\u00e1n kh\u00f3a public \u0111\u01b0\u1ee3c chia th\u00e0nh c\u00e1c\u00a0ph\u00e9p nh\u00e2n \u0111\u00f4i \u0111i\u1ec3m v\u00e0 ph\u00e9p c\u1ed9ng \u0111i\u1ec3m \u00a0b\u1eaft \u0111\u1ea7u t\u1eeb \u0111i\u1ec3m c\u01a1 s\u1edf.<\/p>\n

\u0110\u1ec3 c\u00f3 c\u00e1i nh\u00ecn tr\u1ef1c quan v\u1ec1 v\u1ec1 \u0111\u1ec1 n\u00e0y, ch\u00fang ta h\u00e3y tham kh\u1ea3o m\u1ed9t s\u1ed1 v\u00ed d\u1ee5 v\u1edbi c\u00e1c con s\u1ed1 nh\u1ecf d\u01b0\u1edbi \u0111\u00e2y:<\/p>\n

\n

Ph\u01b0\u01a1ng tr\u00ecnh: y\u00a02<\/sup>\u00a0= x\u00a03<\/sup> + 7 (a = 0 v\u00e0 b = 7)
\nModulo nguy\u00ean t\u1ed1: 67
\n\u0110i\u1ec3m c\u01a1 b\u1ea3n: (2, 22)
\nB\u1eadc c\u1ee7a \u0111i\u1ec3m: 79
\nKh\u00f3a ri\u00eang: 2<\/p>\n<\/blockquote>\n

\u0110\u1ea7u ti\u00ean, h\u00e3y c\u00f9ng nhau t\u00ecm kh\u00f3a public\u00a0v\u1edbi kh\u00f3a private c\u00f3 gi\u00e1 tr\u1ecb =\u00a02<\/strong>. Ngay b\u00e2y gi\u1edd, ch\u00fang ta ch\u1ec9 c\u1ea7n th\u1ef1c hi\u1ec7n ph\u00e9p to\u00e1n nh\u00e2n \u0111\u00f4i \u0111i\u1ec3m m\u1ed9t l\u1ea7n t\u1eeb \u0111i\u1ec3m c\u01a1 s\u1edf:<\/p>\n

\n

c = (3 * 22<\/sup>\u00a0+ 0) \/ (2 * 22) mod 67
\nc = (3 * 4) \/ (44) mod 67
\nc = 12 \/ 44 mod 67<\/p>\n<\/blockquote>\n

H\u00e3y d\u1eebng l\u1ea1i \u0111\u1ec3 xem x\u00e9t m\u1ed9t v\u1ea5n \u0111\u1ec1 nh\u1ecf:<\/b><\/strong>\u00a0L\u00e0m sao \u0111\u1ec3 th\u1ef1c hi\u1ec7n m\u1ed9t ph\u00e9p chia v\u1edbi s\u1ed1 d\u01b0 thu\u1ed9c tr\u01b0\u1eddng h\u1eefu h\u1ea1n – n\u01a1i k\u1ebft qu\u1ea3 ph\u00e9p chia ph\u1ea3i lu\u00f4n l\u00e0 s\u1ed1 nguy\u00ean? V\u1edbi tr\u01b0\u1eddng h\u1ee3p n\u00e0y, ch\u00fang ta ph\u1ea3i l\u00e0m th\u00eam m\u1ed9t ph\u00e9p nh\u00e2n v\u1edbi ngh\u1ecbch \u0111\u1ea3o (n\u1ebfu b\u1ea1n quan t\u00e2m, c\u00f3 th\u1ec3 \u1edf\u00a0\u0111\u00e2y<\/u><\/a>\u00a0v\u00e0\u00a0\u1edf \u0111\u00e2y<\/u><\/a>). Trong tr\u01b0\u1eddng h\u1ee3p hi\u1ec7n t\u1ea1i, b\u1ea1n s\u1ebd ph\u1ea3i tin t\u01b0\u1edfng ch\u00fang t\u00f4i khi vi\u1ebft r\u1eb1ng:<\/p>\n

\n

44-1<\/sup>\u00a0= 32<\/p>\n<\/blockquote>\n

B\u00e2y gi\u1edd, ch\u00fang ta s\u1ebd ti\u1ebfp t\u1ee5c v\u1edbi c\u00e1c ph\u00e9p t\u00ednh:<\/p>\n

\n

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

rx<\/sub>\u00a0= (492<\/sup>\u00a0\u2013 2 * 2) mod 67
\nrx<\/sub>\u00a0= (2401 \u2013 4) mod 67
\nrx<\/sub>\u00a0= 2397 mod 67
\nrx<\/sub>\u00a0= 52<\/p>\n

ry<\/sub>\u00a0= (49 * (2 \u2013 52) \u2013 22) mod 67
\nry<\/sub>\u00a0= (49 * (-50) \u2013 22) mod 67
\nry<\/sub>\u00a0= (-2450 \u2013 22) mod 67
\nry<\/sub>\u00a0= -2472 mod 67
\nry<\/sub>\u00a0= 7<\/p>\n<\/blockquote>\n

Nh\u01b0 v\u1eady, t\u1eeb k\u1ebft qu\u1ea3 tr\u00ean, kh\u00f3a public c\u1ee7a ch\u00fang ta t\u01b0\u01a1ng \u1ee9ng v\u1edbi \u0111i\u1ec3m\u00a0(52, 7). T\u1ea5t c\u1ea3 ch\u1ec9 \u0111\u1ec3 t\u00ednh cho kh\u00f3a ri\u00eang b\u1eb1ng 2!<\/p>\n

Nh\u00ecn chung, ph\u00e9p to\u00e1n \u0111i t\u1eeb kh\u00f3a <\/b><\/strong>private t\u00ednh ra kh\u00f3a public<\/b><\/strong>\u00a0<\/b><\/strong>l\u00e0 m\u1ed9t ph\u00e9p to\u00e1n kh\u00e1 \u0111\u01a1n gi\u1ea3n v\u00e0 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c t\u00ednh tr\u00ean m\u00e1y t\u00ednh. \u00a0Tuy nhi\u00ean, n\u1ebfu mu\u1ed1n l\u00e0m ng\u01b0\u1ee3c l\u1ea1i – \u0111i t\u1eeb kh\u00f3a public sang kh\u00f3a private l\u1ea1i l\u00e0 m\u1ed9t ph\u01b0\u01a1ng \u00e1n kh\u00f4ng kh\u1ea3 thi. \u0110\u01a1n gi\u1ea3n v\u00ec c\u00e1c th\u00f4ng s\u1ed1 trong m\u1eadt m\u00e3 elliptic l\u00e0 r\u1ea5t l\u1edbn.<\/p>\n

N\u00f3i c\u00e1ch kh\u00e1c, vi\u1ec7c \u0111i t\u1eeb kh\u00f3a private \u0111\u1ec3 t\u00ednh ra kh\u00f3a public l\u00e0 chuy\u1ebfn \u0111i m\u1ed9t chi\u1ec1u<\/b><\/strong>\u00a0v\u00e0 kh\u00f4ng th\u1ec3 \u0111\u1ea3o ng\u01b0\u1ee3c.<\/p>\n

Th\u00f4ng th\u01b0\u1eddng, kh\u00f3a public th\u01b0\u1eddng \u0111\u01b0\u1ee3c \u0111\u1ea1i di\u1ec7n b\u1edfi m\u1ed9t chu\u1ed7i th\u1eadp l\u1ee5c ph\u00e2n. \u0110\u1ee3i \u0111\u00e3! L\u00e0m th\u1ebf n\u00e0o \u0111\u1ec3 ch\u00fang ta nh\u1eadn \u0111\u01b0\u1ee3c t\u1eeb m\u1ed9t \u0111i\u1ec3m tr\u00ean m\u1eb7t ph\u1eb3ng, \u0111\u01b0\u1ee3c m\u00f4 t\u1ea3 b\u1edfi hai con s\u1ed1, \u0111\u1ec3 ra m\u1ed9t s\u1ed1 duy nh\u1ea5t? Trong m\u1ed9t kh\u00f3a public kh\u00f4ng \u0111\u01b0\u1ee3c n\u00e9n, hai s\u1ed1 256-bit \u0111\u1ea1i di\u1ec7n cho c\u00e1c t\u1ecda \u0111\u1ed9\u00a0x<\/i><\/em>\u00a0v\u00e0\u00a0y<\/i><\/em>\u00a0\u0111\u01b0\u1ee3c \u0111\u00ednh v\u1edbi nhau trong m\u1ed9t chu\u1ed7i k\u00fd t\u1ef1 d\u00e0i. Ch\u00fang ta c\u0169ng c\u00f3 th\u1ec3 t\u1eadn d\u1ee5ng l\u1ee3i th\u1ebf c\u1ee7a t\u00ednh \u0111\u1ed1i x\u1ee9ng c\u1ee7a c\u00e1c \u0111\u01b0\u1eddng cong elliptic \u0111\u1ec3 t\u1ea1o ra m\u1ed9t kh\u00f3a public n\u00e9n, ch\u1ec9 b\u1eb1ng c\u00e1ch gi\u1eef gi\u00e1 tr\u1ecb\u00a0x<\/i><\/em>\u00a0v\u00e0 ghi nh\u1eadn n\u1eeda n\u00e0o c\u1ee7a \u0111\u01b0\u1eddng cong m\u00e0 \u0111i\u1ec3m \u0111\u00f3 n\u1eb1m tr\u00ean. T\u1eeb ph\u1ea7n th\u00f4ng tin n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 ph\u1ee5c h\u1ed3i c\u1ea3 hai t\u1ecda \u0111\u1ed9.<\/p>\n

C\u00e1ch t\u1ea1o ra ch\u1eef k\u00fd ph\u00ea duy\u1ec7t m\u1ed9t s\u1ed1 d\u1eef li\u1ec7u v\u1edbi kh\u00f3a Private<\/b><\/strong><\/h2>\n

Gi\u1ea3 s\u1eed ch\u00fang ta \u0111ang c\u00f3 m\u1ed9t c\u1eb7p kh\u00f3a private v\u00e0 kh\u00f3a public, qu\u00e1 tr\u00ecnh t\u1ea1o ra ch\u1eef k\u00fd ph\u00ea duy\u1ec7t d\u1eef li\u1ec7u s\u1ebd \u0111\u01b0\u1ee3c di\u1ec5n ra nh\u01b0 sau:<\/p>\n

B\u1ea1n bi\u1ebft \u0111\u1ea5y, c\u00e1c d\u1eef li\u1ec7u c\u00f3 th\u1ec3 c\u00f3 \u0111\u1ed9 d\u00e0i b\u1ea5t k\u1ef3. Th\u00f4ng th\u01b0\u1eddng, qu\u00e1 tr\u00ecnh t\u1ea1o ra ch\u1eef k\u00fd s\u1ebd b\u1eaft \u0111\u1ea7u b\u1eb1ng b\u01b0\u1edbc b\u0103m (hash) d\u1eef li\u1ec7u \u0111\u1ec3 t\u1ea1o ra m\u1ed9t s\u1ed1 c\u00f3 ch\u1ee9a c\u00f9ng m\u1ed9t s\u1ed1 bit (256) nh\u01b0 b\u1eadc c\u1ee7a \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng cong. Tuy nhi\u00ean, \u0111\u1ec3 gi\u00fap b\u1ea1n d\u1ec5 hi\u1ec3u h\u01a1n, ch\u00fang ta s\u1ebd b\u1ecf qua b\u01b0\u1edbc b\u0103m v\u00e0 ch\u1ec9 k\u00fd\u00a0d\u1eef li\u1ec7u th\u00f4\u00a0z<\/i><\/b><\/em><\/strong>. N\u1ebfu g\u1ecdi G<\/b><\/strong>\u00a0l\u00e0 \u0111i\u1ec3m c\u01a1 s\u1edf,\u00a0n<\/i><\/b><\/em><\/strong>\u00a0l\u00e0 b\u1eadc c\u1ee7a \u0111i\u1ec3m\u00a0v\u00e0\u00a0<\/b><\/strong>d<\/i><\/b><\/em><\/strong>\u00a0l\u00e0 kh\u00f3a private. Qu\u00e1 tr\u00ecnh t\u1ea1o ra ch\u1eef k\u00fd s\u1ebd tr\u1ea3i qua 5 b\u01b0\u1edbc:<\/p>\n