{"id":36779,"date":"2025-05-12T16:02:34","date_gmt":"2025-05-12T08:02:34","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/36779.html"},"modified":"2025-05-12T16:02:34","modified_gmt":"2025-05-12T08:02:34","slug":"%e3%80%90%e5%8c%ba%e5%9d%97%e9%93%be%e3%80%91%e6%a4%ad%e5%9c%86%e6%9b%b2%e7%ba%bf%e6%95%b0%e5%ad%97%e7%ad%be%e5%90%8d%e7%ae%97%e6%b3%95%ef%bc%88ecdsa%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/36779.html","title":{"rendered":"\u3010\u533a\u5757\u94fe\u3011\u692d\u5706\u66f2\u7ebf\u6570\u5b57\u7b7e\u540d\u7b97\u6cd5\uff08ECDSA\uff09"},"content":{"rendered":"

\u672c\u6587\u4e3b\u8981\u53c2\u8003：<\/p>\n

\u4e00\u6587\u8bfb\u61c2ECDSA\u7b97\u6cd5\u5982\u4f55\u4fdd\u62a4\u6570\u636e \u692d\u5706\u66f2\u7ebf\u6570\u5b57\u7b7e\u540d\u7b97\u6cd5<\/p>\n

1. ECDSA\u7b97\u6cd5\u7b80\u4ecb<\/h4>\n

\u2003\u2003ECDSA \u662f Elliptic Curve Digital Signature Algorithm \u7684\u7b80\u79f0，\u4e3b\u8981\u7528\u4e8e\u5bf9\u6570\u636e（\u6bd4\u5982\u4e00\u4e2a\u6587\u4ef6）\u521b\u5efa\u6570\u5b57\u7b7e\u540d，\u4ee5\u4fbf\u4e8e\u4f60\u5728\u4e0d\u7834\u574f\u5b83\u7684\u5b89\u5168\u6027\u7684\u524d\u63d0\u4e0b\u5bf9\u5b83\u7684\u771f\u5b9e\u6027\u8fdb\u884c\u9a8c\u8bc1\u3002 \u2003\u2003\u4f60\u4e0d\u5e94\u8be5\u5c06 ECDSA \u4e0e\u7528\u6765\u5bf9\u6570\u636e\u8fdb\u884c\u52a0\u5bc6\u7684 AES（\u9ad8\u7ea7\u52a0\u5bc6\u6807\u51c6）\u76f8\u6df7\u6dc6\u3002ECDSA \u4e0d\u4f1a\u5bf9\u6570\u636e\u8fdb\u884c\u52a0\u5bc6\u3001\u6216\u963b\u6b62\u522b\u4eba\u770b\u5230\u6216\u8bbf\u95ee\u4f60\u7684\u6570\u636e，\u5b83\u53ef\u4ee5\u9632\u6b62\u7684\u662f\u786e\u4fdd\u6570\u636e\u6ca1\u6709\u88ab\u7be1\u6539\u3002 \u2003\u2003ECDSA \u539f\u7406\u975e\u5e38\u7b80\u5355，\u6709\u4e00\u4e2a\u6570\u5b66\u65b9\u7a0b，\u5728\u56fe\u4e0a\u753b\u4e86\u4e00\u6761\u66f2\u7ebf，\u7136\u540e\u4f60\u5728\u8fd9\u6761\u66f2\u7ebf\u4e0a\u9762\u968f\u673a\u9009\u53d6\u4e86\u4e00\u4e2a\u70b9\u4f5c\u4e3a\u4f60\u7684 \u539f\u70b9 G\u3002\u63a5\u7740\u4f60\u4ea7\u751f\u4e86\u4e00\u4e2a \u968f\u673a\u6570 k，\u4f5c\u4e3a\u4f60\u7684 \u79c1\u94a5，\u6700\u540e\u4f60\u7528\u4e0a\u9762\u7684 \u968f\u673a\u6570 k \u548c \u539f\u70b9 G \u901a\u8fc7\u4e00\u4e9b\u590d\u6742\u7684\u9b54\u6cd5\u6570\u5b66\u65b9\u7a0b\u5f97\u5230\u8be5\u6761\u66f2\u7ebf\u4e0a\u9762\u7684\u7b2c\u4e8c\u4e2a\u70b9，\u8fd9\u662f\u4f60\u7684 \u516c\u94a5 P\u3002<\/p>\n

\u2003\u2003\u5f53\u4f60\u60f3\u8981\u5bf9\u4e00\u4e2a\u6587\u4ef6\u8fdb\u884c\u7b7e\u540d\u7684\u65f6\u5019，\u7b7e\u540d\u672c\u8eab\u7531\u4e24\u90e8\u5206\u7ec4\u6210，\u79f0\u4e3a r \u548c s \u3002\u901a\u8fc7 \u79c1\u94a5k（\u968f\u673a\u6570） \u548c\u6587\u4ef6\u7684 \u54c8\u5e0c \u7ec4\u6210\u4e00\u4e2a\u9b54\u6cd5\u6570\u5b66\u65b9\u7a0b，\u8fd9\u5c06\u7ed9\u51fa\u4f60\u7684\u7b7e\u540d\u7684 s \u90e8\u5206\u3002\u53d6 \u516c\u94a5 P \u7684 x \u8f74\u5373\u4e3a\u7b7e\u540d\u7684 r \u90e8\u5206\u3002\u4e3a\u4e86\u9a8c\u8bc1\u7b7e\u540d\u7684\u6b63\u786e\u6027，\u4f60\u9700\u8981 \u516c\u94a5 P \u548c\u7b7e\u540d s\u3001r\u7ec4\u6210\u4e00\u4e2a\u9b54\u6cd5\u6570\u5b66\u65b9\u7a0b，\u8be5\u65b9\u7a0b\u8ba1\u7b97\u4f1a\u5f97\u5230\u4e00\u4e2a\u5750\u6807\u70b9，\u5982\u679c\u8be5\u5750\u6807\u70b9\u7684 x \u8f74\u521a\u597d\u4e3a\u7b7e\u540d\u4e2d\u7684 r，\u90a3\u4e48\u5373\u53ef\u8ba4\u4e3a\u6539\u7b7e\u540d\u662f\u6709\u6548\u7684\u3002<\/p>\n

2.\u692d\u5706\u66f2\u7ebf\u5bc6\u94a5\u751f\u6210<\/h4>\n

\u2003\u2003\u50cf <\/p>\n

y <\/p>\n

2 <\/p>\n

= <\/p>\n

x <\/p>\n

3 <\/p>\n

+ <\/p>\n

a <\/p>\n

x <\/p>\n

+ <\/p>\n

b <\/p>\n

y^2 = x^3 +ax+b <\/p>\n

<\/span><\/span>y<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u8fd9\u6837\u7684\u5f0f\u5b50\u901a\u5e38\u753b\u51fa\u6765\u662f\u4e2a\u692d\u5706\u66f2\u7ebf，\u5982\u4e0b\u56fe\u6240\u793a：<\/p>\n


\n \u2003\u2003\u753b\u4e00\u6761\u76f4\u7ebf\u4e0e\u692d\u5706\u66f2\u7ebf\u4ea7\u751f\u4e09\u4e2a\u4ea4\u70b9（P\u3001Q\u3001-R），\u6211\u4eec\u79f0 P + Q = R，R \u5373\u4e3a -R \u5173\u4e8ex\u8f74\u7684\u5bf9\u79f0\u70b9（\u8bf7\u6ce8\u610f\u8fd9\u91cc\u7684 + \u5b9e\u9645\u6307\u7684\u662f\u7b2c\u4e09\u4e2a\u4ea4\u70b9\u7684 x \u8f74\u5bf9\u79f0\u70b9）\u3002 <\/p>\n


\n \u2003\u2003\u82e5\u4ee5\u692d\u5706\u66f2\u7ebf\u7684\u67d0\u4e00\u5207\u70b9 G \u505a\u4e00\u76f4\u7ebf，\u5219\u76f4\u7ebf\u4e0e\u692d\u5706\u66f2\u7ebf\u7684\u53e6\u4e00\u4ea4\u70b9\u5373\u4e3a -2G，\u5176\u5173\u4e8ex\u8f74\u5bf9\u79f0\u70b9\u5373\u4e3a 2G \u70b9，\u82e5 2G \u70b9\u4e0e G \u70b9\u8fde\u63a5\u5373\u53ef\u5f97\u5230 3G \u70b9，\u4ee5\u6b64\u7c7b\u63a8，\u5373\u53ef\u5f97\u5230 kG \u70b9\u3002 <\/p>\n


\n \u2003\u2003\u5f15\u5165 G \u70b9\u7684\u597d\u5904\u662f\u53ef\u4ee5\u5b9e\u73b0\u5feb\u901f\u5bfb\u627e，\u6211\u4eec\u4ee5 G \u70b9\u505a\u5207\u7ebf\u5373\u53ef\u5f97\u5230 2G \u70b9，\u4ee5 2G\u70b9\u4e3a\u5207\u7ebf\u5373\u53ef\u5f97\u5230 4G \u70b9，\u4ee5\u6b64\u7c7b\u63a8，\u8fd9\u6837\u7684\u5bfb\u627e\u8fc7\u7a0b，\u5927\u5927\u7684\u51cf\u5c11\u4e86\u5bfb\u627e\u6b21\u6570\u3002 <\/p>\n

\u2003\u2003\u692d\u5706\u66f2\u7ebf\u8fd8\u6709\u4e00\u4e2a\u7279\u6027\u5c31\u662f，\u6211\u4eec\u4ee5 G \u4e3a\u8d77\u70b9\u7ecf\u8fc7 k \u6b21\u5bfb\u627e\u540e，\u5f97\u5230 kG \u70b9\u8fd9\u4e00\u987a\u5e8f\u8ba1\u7b97\u8fc7\u7a0b\u662f\u6bd4\u8f83\u7b80\u5355\u7684，\u4f46\u5982\u679c\u6211\u4eec\u5df2\u77e5 G \u70b9\u8981\u5f97\u5230 kG \u70b9\u662f\u7ecf\u8fc7\u591a\u5c11\u6b21\u5bfb\u627e\u5f97\u5230\u7684\u662f\u6bd4\u8f83\u56f0\u96be\u7684，\u6211\u4eec\u53ea\u80fd\u5bf9 k \u4e00\u4e2a\u4e00\u4e2a\u5c1d\u8bd5，\u5f53 k \u6bd4\u8f83\u5927\u65f6，k \u7684\u5bfb\u627e\u8fc7\u7a0b\u662f\u53ca\u5176\u56f0\u96be\u7684，\u56e0\u6b64，\u8fd9\u4e00\u8fc7\u7a0b\u662fECDSA\u7b97\u6cd5\u80cc\u540e\u5b89\u5168\u6027\u7684\u57fa\u7840，\u800c\u8fd9\u4e00\u539f\u5219\u4e5f\u88ab\u79f0\u4e3a \u5355\u5411\u9677\u95e8\u51fd\u6570\u3002<\/p>\n

\u6bd4\u7279\u5e01\u7684\u692d\u5706\u66f2\u7ebf\u4e00\u822c\u662f\u91c7\u7528\u4ee5\u4e0b\u51fd\u6570： <\/p>\n

y <\/p>\n

2 <\/p>\n

= <\/p>\n

x <\/p>\n

3 <\/p>\n

+ <\/p>\n

7 <\/p>\n

， <\/p>\n

a <\/p>\n

= <\/p>\n

0 <\/p>\n

, <\/p>\n

b <\/p>\n

= <\/p>\n

7 <\/p>\n

y^2 = x^3+7，a=0,b=7 <\/p>\n

<\/span><\/span>y<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>7<\/span>，<\/span>a<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span>,<\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span><\/span><\/span> \u5f00\u59cb\u7684\u8282\u70b9 Generator(G) \u5750\u6807\u4e3a： <\/p>\n

G <\/p>\n

x <\/p>\n

= <\/p>\n

0 <\/p>\n

x <\/p>\n

79 <\/p>\n

B <\/p>\n

E <\/p>\n

667 <\/p>\n

E <\/p>\n

F <\/p>\n

9 <\/p>\n

D <\/p>\n

C <\/p>\n

B <\/p>\n

B <\/p>\n

A <\/p>\n

C <\/p>\n

55 <\/p>\n

A <\/p>\n

06295 <\/p>\n

C <\/p>\n

E <\/p>\n

870 <\/p>\n

B <\/p>\n

07029 <\/p>\n

B <\/p>\n

F <\/p>\n

C <\/p>\n

D <\/p>\n

B <\/p>\n

2 <\/p>\n

D <\/p>\n

C <\/p>\n

E <\/p>\n

28 <\/p>\n

D <\/p>\n

959 <\/p>\n

F <\/p>\n

2815 <\/p>\n

B <\/p>\n

16 <\/p>\n

F <\/p>\n

81798 <\/p>\n

Gx = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798 <\/p>\n

<\/span><\/span>G<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span>x<\/span>79<\/span>BE<\/span>667<\/span>EF<\/span>9<\/span>D<\/span>CBB<\/span>A<\/span>C<\/span>55<\/span>A<\/span>06295<\/span>CE<\/span>870<\/span>B<\/span>07029<\/span>BFC<\/span>D<\/span>B<\/span>2<\/span>D<\/span>CE<\/span>28<\/span>D<\/span>959<\/span>F<\/span>2815<\/span>B<\/span>16<\/span>F<\/span>81798<\/span><\/span><\/span><\/span><\/span><\/span> <\/p>\n

G <\/p>\n

y <\/p>\n

= <\/p>\n

0 <\/p>\n

x <\/p>\n

483 <\/p>\n

A <\/p>\n

D <\/p>\n

A <\/p>\n

7726 <\/p>\n

A <\/p>\n

3 <\/p>\n

C <\/p>\n

4655 <\/p>\n

D <\/p>\n

A <\/p>\n

4 <\/p>\n

F <\/p>\n

B <\/p>\n

F <\/p>\n

C <\/p>\n

0 <\/p>\n

E <\/p>\n

1108 <\/p>\n

A <\/p>\n

8 <\/p>\n

F <\/p>\n

D <\/p>\n

17 <\/p>\n

B <\/p>\n

448 <\/p>\n

A <\/p>\n

68554199 <\/p>\n

C <\/p>\n

47 <\/p>\n

D <\/p>\n

08 <\/p>\n

F <\/p>\n

F <\/p>\n

B <\/p>\n

10 <\/p>\n

D <\/p>\n

4 <\/p>\n

B <\/p>\n

8 <\/p>\n

Gy = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8 <\/p>\n

<\/span><\/span>G<\/span>y<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span>x<\/span>483<\/span>A<\/span>D<\/span>A<\/span>7726<\/span>A<\/span>3<\/span>C<\/span>4655<\/span>D<\/span>A<\/span>4<\/span>FBFC<\/span>0<\/span>E<\/span>1108<\/span>A<\/span>8<\/span>F<\/span>D<\/span>17<\/span>B<\/span>448<\/span>A<\/span>68554199<\/span>C<\/span>47<\/span>D<\/span>08<\/span>FFB<\/span>10<\/span>D<\/span>4<\/span>B<\/span>8<\/span><\/span><\/span><\/span><\/span><\/span> \u79c1\u94a5 k \u4e00\u822c\u662f\u9009\u62e9\u6bd4\u8f83\u5927\u7684\u968f\u673a\u6570，\u901a\u8fc7\u5f00\u59cb\u8282\u70b9 G \u4e0e\u79c1\u94a5 k，\u6211\u4eec\u5373\u53ef\u5f97\u5230\u516c\u94a5\u8282\u70b9 P<\/p>\n

3.\u692d\u5706\u66f2\u7ebf\u6570\u5b57\u7b7e\u540d\u5b9e\u73b0<\/h4>\n

\u2003\u2003\u5047\u8bbeAlice\u8981\u7ed9Bob\u53d1\u9001\u6d88\u606f M，Alice \u6839\u636e \u8d77\u59cb\u70b9 G \u4e0e\u9009\u62e9\u7684 \u968f\u673a\u6570（\u79c1\u94a5） k \u53ef\u5f97\u5230 \u516c\u94a5 P，\u7136\u540e\u7528 \u79c1\u94a5 k \u4e0e \u8981\u53d1\u9001\u7684\u6d88\u606fM\u7684\u54c8\u5e0c\u503c HASH(M) \u76f8\u4e58\u5f97\u5230 s，\u7136\u540e\u5c06\u8981\u53d1\u9001\u7684\u4fe1\u606f M \u4e0e s \u4e00\u540c\u53d1\u9001\u7ed9 Bob\u3002Bob\u5f97\u5230\u4fe1\u606f\u540e，\u901a\u8fc7Alice\u7684 \u516c\u94a5 P\u4e0e Hash(M) \u76f8\u4e58\u5f97\u5230 Y，\u7136\u540e\u5c06 s \u4e0e G \u76f8\u4e58\u5f97\u5230X，\u5982\u679cX=Y\u5219\u6539\u7b7e\u540d\u5373\u4e3a\u6709\u6548，\u5177\u4f53\u8fc7\u7a0b\u5982\u4e0b\u56fe\u6240\u793a： \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" X=Y\u5408\u7406\u6027\u8bc1\u660e： <\/p>\n

X <\/p>\n

= <\/p>\n

H <\/p>\n

a <\/p>\n

s <\/p>\n

h <\/p>\n

( <\/p>\n

M <\/p>\n

) <\/p>\n

\u2217 <\/p>\n

P <\/p>\n

= <\/p>\n

H <\/p>\n

a <\/p>\n

s <\/p>\n

h <\/p>\n

( <\/p>\n

M <\/p>\n

) <\/p>\n

\u2217 <\/p>\n

k <\/p>\n

\u2217 <\/p>\n

G <\/p>\n

= <\/p>\n

s <\/p>\n

\u2217 <\/p>\n

G <\/p>\n

= <\/p>\n

Y <\/p>\n

X=Hash(M) * P=Hash(M)*k*G=s*G=Y <\/p>\n

<\/span><\/span>X<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>a<\/span>s<\/span>h<\/span>(<\/span>M<\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>a<\/span>s<\/span>h<\/span>(<\/span>M<\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>k<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>G<\/span><\/span>=<\/span><\/span><\/span><\/span>s<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>G<\/span><\/span>=<\/span><\/span><\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span><\/span> \u2003\u2003\u867d\u7136\u4ee5\u4e0a\u8fc7\u7a0b\u5b9e\u73b0\u4e86\u6570\u5b57\u7b7e\u540d，\u4f46\u662f\u4ee5\u4e0a\u7684\u7b7e\u540d\u8fc7\u7a0b\u662f\u5b58\u5728\u4e00\u5b9a\u6f0f\u6d1e\u7684，\u56e0\u4e3a Bob \u5f97\u5230\u7684\u6570\u636e\u6709 Alice \u7684\u516c\u94a5 P\u3001s\u3001\u4ee5\u53ca\u8d77\u59cb\u5750\u6807 G，\u6839\u636e G \u4e0e P \u662f\u63a8\u65ad\u4e0d\u51fa\u79c1\u94a5 k \u7684，\u4f46\u662f k \u53ef\u7531 s \u8ba1\u7b97\u5f97\u5230 <\/p>\n

k <\/p>\n

= <\/p>\n

s <\/p>\n

\/ <\/p>\n

H <\/p>\n

a <\/p>\n

s <\/p>\n

h <\/p>\n

( <\/p>\n

M <\/p>\n

) <\/p>\n

k = s\/Hash(M) <\/p>\n

<\/span><\/span>k<\/span><\/span>=<\/span><\/span><\/span><\/span>s<\/span>\/<\/span>H<\/span>a<\/span>s<\/span>h<\/span>(<\/span>M<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u56e0\u6b64，\u79d1\u5b66\u5bb6\u4e3a\u5176\u53c8\u60f3\u4e86\u65b0\u7684\u529e\u6cd5\u3002<\/p>\n

\u7b7e\u540d\u8fc7\u7a0b：<\/p>\n