{"id":101484,"date":"2026-09-06T22:33:25","date_gmt":"2026-09-06T14:33:25","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/101484.html"},"modified":"2026-09-06T22:33:25","modified_gmt":"2026-09-06T14:33:25","slug":"%e5%90%8c%e4%b8%80%e4%b8%aa%e6%98%8e%e6%96%87%e5%8f%91%e7%bb%993%e4%b8%aa%e6%9c%8d%e5%8a%a1%e5%99%a8%ef%bc%9f%e4%b8%80%e6%96%87%e6%90%9e%e6%87%82rsa%e5%b9%bf%e6%92%ad%e6%94%bb%e5%87%bbhastad","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/101484.html","title":{"rendered":"\u540c\u4e00\u4e2a\u660e\u6587\u53d1\u7ed93\u4e2a\u670d\u52a1\u5668\uff1f\u4e00\u6587\u641e\u61c2RSA\u5e7f\u64ad\u653b\u51fb(Hastad)"},"content":{"rendered":"

\u5fa1\u7f51\u676f<\/span> Crypto <\/span>\u8d5b\u9898\u8be6\u89e3<\/span><\/span><\/h2>\n

\u672c\u6587\u5c06\u8be6\u7ec6\u89e3\u6790\u5fa1\u7f51\u676f\u7f51\u7edc\u5b89\u5168\u7ade\u8d5b\u4e2d\u4e00\u9053 Crypto \u8d5b\u9898\u7684\u89e3\u9898\u601d\u8def，\u6db5\u76d6RSA \u5e7f\u64ad\u653b\u51fb，\u6b64\u9898\u76ee\u4ece\u9898\u76ee\u5206\u6790\u3001\u6f0f\u6d1e\u539f\u7406\u3001\u89e3\u9898\u811a\u672c\u5230\u6700\u7ec8 Flag \u8fdb\u884c\u5b8c\u6574\u8bb2\u89e3，\u5e0c\u671b\u5bf9\u5927\u5bb6\u7684\u5b66\u4e60\u6709\u6240\u5e2e\u52a9\u3002<\/span><\/p>\n

\u4e8c\u3001<\/span>ScatterRSA <\/span>\u2014\u2014<\/span> \u5e7f\u64ad\u653b\u51fb<\/span><\/span><\/h2>\n

\u9898\u76ee\u4fe1\u606f：<\/span>\u5206\u7c7b CRYPTO | \u96be\u5ea6 \u4e2d\u7ea7<\/span><\/p>\n

2.1 <\/span>\u9898\u76ee\u5206\u6790<\/span><\/span><\/h3>\n

\"\"<\/h3>\n

\u672c\u9898\u540c\u6837\u4f7f\u7528\u4e86 e = 3 \u7684 RSA \u52a0\u5bc6，\u4f46\u4e0e BabyRSA \u4e0d\u540c\u7684\u662f，\u672c\u9898\u63d0\u4f9b\u4e86\u4e09\u7ec4\u4e0d\u540c\u7684\u52a0\u5bc6\u53c2\u6570（n1, a1, b1, c1）\u3001（n2, a2, b2, c2）\u3001（n3, a3, b3, c3），\u6bcf\u7ec4\u53c2\u6570\u7684\u52a0\u5bc6\u65b9\u5f0f\u4e3a c = (a*m + b)^3 mod n\u3002\u7531\u4e8e\u76f8\u540c\u7684\u660e\u6587 m \u88ab\u4e09\u4e2a\u4e0d\u540c\u7684\u6a21\u6570\u52a0\u5bc6，\u8fd9\u6b63\u662f\u7ecf\u5178\u7684\u54c8\u7ef4\u5e7f\u64ad\u653b\u51fb\u7684\u53d8\u4f53\u3002\u6211\u4eec\u9700\u8981\u5148\u5c06\u6bcf\u7ec4\u53c2\u6570\u8f6c\u5316\u4e3a\u5173\u4e8e m \u7684\u7b49\u4ef7\u591a\u9879\u5f0f，\u7136\u540e\u901a\u8fc7\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406（CRT）\u5408\u5e76\u4e09\u4e2a\u591a\u9879\u5f0f，\u6700\u540e\u7528 Coppersmith \u65b9\u6cd5\u6c42\u5c0f\u6839\u5f97\u5230\u660e\u6587 m\u3002<\/span><\/p>\n

2.2 <\/span>\u89e3\u9898\u601d\u8def<\/span><\/span><\/h3>\n

\u9996\u5148，\u5c06\u6bcf\u7ec4\u52a0\u5bc6\u53c2\u6570\u8f6c\u5316\u4e3a\u5173\u4e8e m \u7684\u7b49\u4ef7\u591a\u9879\u5f0f\u3002\u5bf9\u4e8e\u7b2c i \u7ec4，\u5c06 c = (a*m + b)^3 mod n \u5c55\u5f00，\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u4ee5 m \u4e3a\u53d8\u91cf\u7684\u4e09\u6b21\u591a\u9879\u5f0f\u3002\u5177\u4f53\u505a\u6cd5\u662f：\u8ba1\u7b97 a \u7684\u6a21\u9006\u5143 inv_a，\u7136\u540e\u6784\u9020\u591a\u9879\u5f0f (x + b*inv_a)^3 – c*(inv_a)^3 = 0 \u6a21 n，\u5176\u4e2d x = m\u3002\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u4e09\u4e2a\u4e0d\u540c\u6a21\u6570\u4e0b\u7684\u540c\u4e00\u4e2a\u591a\u9879\u5f0f\u7684\u7b49\u4ef7\u5f62\u5f0f\u3002<\/span><\/p>\n

\u63a5\u4e0b\u6765，\u5229\u7528\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406\u5408\u5e76\u4e09\u4e2a\u591a\u9879\u5f0f\u7684\u7cfb\u6570\u3002\u5c06\u4e09\u4e2a\u591a\u9879\u5f0f\u7684\u5bf9\u5e94\u7cfb\u6570\u901a\u8fc7 CRT \u5408\u5e76\u5230\u4e00\u4e2a\u5927\u6a21\u6570 N = n1 * n2 * n3 \u4e0b，\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u591a\u9879\u5f0f\u3002\u6700\u540e，\u5bf9\u5408\u5e76\u540e\u7684\u591a\u9879\u5f0f\u4f7f\u7528 Coppersmith \u7684 small_roots \u65b9\u6cd5\u6c42\u89e3\u5c0f\u6839，\u5373\u53ef\u6062\u590d\u660e\u6587 m\u3002\u8be5\u65b9\u6cd5\u9700\u8981\u5728 SageMath \u73af\u5883\u4e2d\u8fd0\u884c\u3002<\/span><\/p>\n

2.3 <\/span>\u89e3\u9898\u811a\u672c<\/span><\/span><\/h3>\n

\u89e3\u9898\u811a\u672c\u9700\u8981\u5728 SageMath \u73af\u5883\u4e2d\u8fd0\u884c，\u63a8\u8350\u4f7f\u7528\u5728\u7ebf\u7f51\u7ad9 sagecell.sagemath.org：<\/span><\/p>\n

e = 3<\/p>\n

\u00a0<\/p>\n

n1 = 134590028715846226751903719587861090472772080921099504036178613989523328571021119450984313988905557385036821505121612368860436253713267192159612607745301358472400758433053304358101099945515440169672493726615122471822947908806960237079424527450364377187388434878044214423874958715687632702618242260038201552961<\/p>\n

a1 = 173235201602700035769143714622479214858<\/p>\n

b1 = 58616053309986169433995951615552358657183395855412447208282770965760612937595<\/p>\n

c1 = 50192984704229516422576705848426706185707023557654942997226592852562126143797118081125527050565529571702518296053642057549607538952767595537352818598436177372629946093076195084953301667499579328461288560942915342433209991739202980657987347584757491195480319156057361829134996133204059551472598891366688783755<\/p>\n

\u00a0<\/p>\n

n2 = 68196420818362667184273231820367250019665598198220107894027697404250798750179650739212752171893537136631632644245299647403521159317636479179387396036378651369427323164026037204640004005081098772718308292781228113356944341374614054386589378286349095152732830327900238600877850386212764491160551279660914970501<\/p>\n

a2 = 235763128007574771186749199470667788696<\/p>\n

b2 = 82833393375329622580640447653813478693484654663680708964473557677310067110872<\/p>\n

c2 = 58727033167047203506164797837999819283982869287436076252773072774355826490167620639330797956244757203726547611149611642979362714607329797512143580714003751905301065313982278186988038708909021000691905664629331025167676669052317049958574849948837597306860613961557323523558290710860835719858942854534226001532<\/p>\n

\u00a0<\/p>\n

n3 = 1041978947222515494173618665626713467184482726534999334123994405129572410542742149319993470219686631218662507052426981529041688912266920273453760904110011289684534871464563352348782346283820734447304134789063563965860082130364230372312070163983650187312362722689902212270815366758596540699197732479604574605717<\/p>\n

a3 = 289837185860108823269362666161877095653<\/p>\n

b3 = 88038264202304767250178171651729306984925900158278305985019519489525502096874<\/p>\n

c3 = 12441437714234741776087648075441633972630237216692770981303759863634569477393643678959437151894987980323367988780311101132662707086178418962311399292280866817688865303206505050428252909551141504556842671029262184318022978671849627397763537173137629868777942717009356259616195047064075877128354789683093509959<\/p>\n

\u00a0<\/p>\n

print("\u6b63\u5728\u89e3\u6790\u7b49\u4ef7\u591a\u9879\u5f0f…")<\/p>\n

\u00a0<\/p>\n

def get_poly_coeffs(a, b, c, n):<\/p>\n

\u00a0\u00a0\u00a0 R = Zmod(n)<\/p>\n

\u00a0\u00a0\u00a0 P = PolynomialRing(R, 'x')<\/p>\n

\u00a0\u00a0\u00a0 x = P.gen()<\/p>\n

\u00a0\u00a0\u00a0 inv_a = R(inverse_mod(a, n))<\/p>\n

\u00a0\u00a0\u00a0 poly = (x + R(b) * inv_a) ^ 3 – R(c) * (inv_a ^ 3)<\/p>\n

\u00a0\u00a0\u00a0 return poly.list()<\/p>\n

\u00a0<\/p>\n

coeffs1 = get_poly_coeffs(a1, b1, c1, n1)<\/p>\n

coeffs2 = get_poly_coeffs(a2, b2, c2, n2)<\/p>\n

coeffs3 = get_poly_coeffs(a3, b3, c3, n3)<\/p>\n

\u00a0<\/p>\n

print("\u6b63\u5728\u8fdb\u884c\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406(CRT)\u5408\u5e76…")<\/p>\n

\u00a0<\/p>\n

combined_coeffs = []<\/p>\n

for i in range(4):<\/p>\n

\u00a0\u00a0\u00a0 c_comb = crt([Integer(coeffs1[i]), Integer(coeffs2[i]),<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Integer(coeffs3[i])], [n1, n2, n3])<\/p>\n

\u00a0\u00a0\u00a0 combined_coeffs.append(c_comb)<\/p>\n

\u00a0<\/p>\n

N = n1 * n2 * n3<\/p>\n

P_N = PolynomialRing(Zmod(N), 'x')<\/p>\n

P = P_N(combined_coeffs)<\/p>\n

\u00a0<\/p>\n

roots = P.small_roots(X=2^600, beta=1)<\/p>\n

\u00a0<\/p>\n

if roots:<\/p>\n

\u00a0\u00a0\u00a0 m = int(roots[0])<\/p>\n

\u00a0\u00a0\u00a0 print(bytes.fromhex(hex(m)[2:]).decode())<\/p>\n

2.4 <\/span>\u89e3\u9898\u6b65\u9aa4<\/span><\/span><\/h3>\n

\u7b2c\u4e00\u6b65，\u5c06\u4e09\u7ec4\u52a0\u5bc6\u53c2\u6570\u5206\u522b\u8f6c\u5316\u4e3a\u5173\u4e8e m \u7684\u591a\u9879\u5f0f\u3002\u901a\u8fc7\u8ba1\u7b97 a \u7684\u6a21\u9006\u5143，\u5c06\u52a0\u5bc6\u7b49\u5f0f c = (a*m+b)^3 mod n \u8f6c\u5316\u4e3a\u4e00\u4e2a\u4ee5 m \u4e3a\u53d8\u91cf\u7684\u4e09\u6b21\u591a\u9879\u5f0f\u3002\u7b2c\u4e8c\u6b65，\u4f7f\u7528\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406\u5c06\u4e09\u4e2a\u591a\u9879\u5f0f\u7684\u7cfb\u6570\u5408\u5e76\u5230\u7edf\u4e00\u7684\u5927\u6a21\u6570 N = n1*n2*n3 \u4e0b\u3002\u7b2c\u4e09\u6b65，\u5bf9\u5408\u5e76\u540e\u7684\u591a\u9879\u5f0f\u4f7f\u7528 SageMath \u7684 small_roots \u65b9\u6cd5\u6c42\u89e3\u5c0f\u6839\u3002\u5c06\u811a\u672c\u590d\u5236\u5230 sagecell.sagemath.org \u5728\u7ebf\u8fd0\u884c\u5373\u53ef\u5f97\u5230 flag\u3002<\/span><\/p>\n

\"\"<\/p>\n

\n","protected":false},"excerpt":{"rendered":"

\u5fa1\u7f51\u676f Crypto \u8d5b\u9898\u8be6\u89e3\u672c\u6587\u5c06\u8be6\u7ec6\u89e3\u6790\u5fa1\u7f51\u676f\u7f51\u7edc\u5b89\u5168\u7ade\u8d5b\u4e2d\u4e00\u9053 Crypto \u8d5b\u9898\u7684\u89e3\u9898\u601d\u8def，\u6db5\u76d6RSA \u5e7f\u64ad\u653b\u51fb，\u6b64\u9898\u76ee\u4ece\u9898\u76ee\u5206\u6790\u3001\u6f0f\u6d1e\u539f\u7406\u3001\u89e3\u9898\u811a\u672c\u5230\u6700\u7ec8 Flag \u8fdb\u884c\u5b8c\u6574\u8bb2\u89e3，\u5e0c\u671b\u5bf9\u5927\u5bb6\u7684\u5b66\u4e60\u6709\u6240\u5e2e\u52a9\u3002\u4e8c\u3001ScatterRSA \u2014\u2014 \u5e7f\u64ad\u653b\u51fb\u9898\u76ee\u4fe1\u606f：\u5206\u7c7b CRYPTO | \u96be\u5ea6 \u4e2d\u7ea72.1 \u9898\u76ee\u5206\u6790\u672c\u9898\u540c\u6837\u4f7f\u7528\u4e86 e 3 \u7684 RSA \u52a0\u5bc6，\u4f46\u4e0e BabyRSA \u4e0d\u540c\u7684\u662f，\u672c\u9898<\/p>\n","protected":false},"author":2,"featured_media":101482,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[81,61,1425,190,122],"topic":[],"class_list":["post-101484","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-python","tag-61","tag-1425","tag-190","tag-122"],"yoast_head":"\n\u540c\u4e00\u4e2a\u660e\u6587\u53d1\u7ed93\u4e2a\u670d\u52a1\u5668\uff1f\u4e00\u6587\u641e\u61c2RSA\u5e7f\u64ad\u653b\u51fb(Hastad) - 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