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[分享]推荐一些密码学方面的资料
p:=2^256-2^224-2^128+2^96-1+2^64*(2+1)^2*(2^2+1)^2*(2^4+1)^2*(2^8+1)^2*(2^16+1)^2; p; S:= FiniteField(115792089210356248756420345214020892766250353991924191454421193933289684991999); E:= EllipticCurve([S|0,0,0,-3,18505919022281880113072981827955639221458448578012075254857346196103069175443]); E; T := Twists(E); T; Q:=QuadraticTwists(E); Q; E1:= EllipticCurve([S|0,0,0,3246688514511153932416457520904014993\ 4860148566249859157380725452739373732484,990597980848191035130552872645729525611460241980432846609604900451864807481\ 09 ]); E1; E2:= EllipticCurve([S|0,0,0,32466885145111539324164575209040149934860\ 148566249859157380725452739373732484,99059798084819103513055287264572952561146024198043284660960490045186480748109 ]); E2; Order(E)+Order(E1); Order(E)+Order(E2); 2*p+2; IsIsomorphic(E, E1); IsIsomorphic(E, E2); IsIsomorphic(E1, E2); 115792089210356248756420345214020892766250353991924191454421193933289684991999 Elliptic Curve defined by y^2 = x^3 + 11579208921035624875642034521402089276625\ 0353991924191454421193933289684991996*x + 18505919022281880113072981827955639221458448578012075254857346196103069175443 over GF(11579208921035624875642034521402089276625035399192419145442119393328968\ 4991999) [ Elliptic Curve defined by y^2 = x^3 + 1157920892103562487564203452140208927\ 66250353991924191454421193933289684991996*x + 185059190222818801130729818279556392214584485780120752548573461961030691754\ 43 over GF(1157920892103562487564203452140208927662503539919241914544211939\ 33289684991999), Elliptic Curve defined by y^2 = x^3 + 3055598702044942576893852837786220404\ 3355264779787219082989808059694776059232*x + 512478505879505245004118884947368501017601900881206906701010266419695148991\ 2 over GF(11579208921035624875642034521402089276625035399192419145442119393\ 3289684991999) ] [ Elliptic Curve defined by y^2 = x^3 + 1157920892103562487564203452140208927\ 66250353991924191454421193933289684991996*x + 185059190222818801130729818279556392214584485780120752548573461961030691754\ 43 over GF(1157920892103562487564203452140208927662503539919241914544211939\ 33289684991999), Elliptic Curve defined by y^2 = x^3 + 1864554914043223771557427286849641712\ 6293959261198406709894762666677208384048*x + 889743930312778083094594540564041748182977088943107511240595212074532156912\ 8 over GF(11579208921035624875642034521402089276625035399192419145442119393\ 3289684991999) ] Elliptic Curve defined by y^2 = x^3 + 32466885145111539324164575209040149934860\ 148566249859157380725452739373732484*x + 99059798084819103513055287264572952561146024198043284660960490045186480748109 over GF(11579208921035624875642034521402089276625035399192419145442119393328968\ 4991999) Elliptic Curve defined by y^2 = x^3 + 32466885145111539324164575209040149934860\ 148566249859157380725452739373732484*x + 99059798084819103513055287264572952561146024198043284660960490045186480748109 over GF(11579208921035624875642034521402089276625035399192419145442119393328968\ 4991999) 231584178420712497512840690428041785532500707983848382908842387866579369984000 231584178420712497512840690428041785532500707983848382908842387866579369984000 231584178420712497512840690428041785532500707983848382908842387866579369984000 false false true |
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[分享]推荐一些密码学方面的资料
找出来了。。。超长类梅森数?! 2^256-2^224-2^128+2^96-1+2^64*(2+1)^2*(2^2+1)^2*(2^4+1)^2*(2^8+1)^2*(2^16+1)^2=P推; P推:=115792089210356248756420345214020892766250353991924191454421193933289684991999; P; Factorization(P-1); P0:=2^256-2^224-2^128+2^96-1; P0; P-P0; P1:=2^256-2^224-2^128+2^96-1+2^64*(2+1)^2*(2^2+1)^2*(2^4+1)^2*(2^8+1)^2*(2^16+1)^2; P1; Factorization(48); Factorization(P-1); Factorization(P-P0); 115792089210356248756420345214020892766250353991924191454421193933289684991999 [ <2, 1>, <43, 1>, <30223, 1>, <348253387243, 1>, <4641351449027, 1>, <417514796639753, 1>, <66013261729388519804782124120027, 1> ] 115792089210356248756420345214020892765910071625161709315967901256971295129599 340282366762482138453292676318389862400 115792089210356248756420345214020892766250353991924191454421193933289684991999 [ <2, 4>, <3, 1> ] [ <2, 1>, <43, 1>, <30223, 1>, <348253387243, 1>, <4641351449027, 1>, <417514796639753, 1>, <66013261729388519804782124120027, 1> ] [ <2, 64>, <3, 2>, <5, 2>, <17, 2>, <257, 2>, <65537, 2> ] |
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[分享]推荐一些密码学方面的资料
为了约减快,应该就下面式子加减号组合应能出推荐的P: 2^256+2^224+2^192+2^160+2^128+2^96+2^64+2^32-1; p:=2^256-2^224-2^128+2^96-1+2^64*(2+1)^2*(2^2+1)^2*(2^4+1)^2*(2^8+1)^2*(2^16+1)^2; p; p mod 4; Factorization(p-1); Factorization(p+1); Factorization(66013261729388519804782124120027-1); M :=(p-1)/2; M; Factorization(57896044605178124378210172607010446383125176995962095727210596966644842495999 ); pN:=115792089210356248762697446949407573529996955224135760342422259061068512044369; pN; pN mod 4; Factorization(pN-1); Factorization(pN+1); Factorization(2624747550333869278416773953-1); V:=(pN-1)/2; V; Factorization(57896044605178124381348723474703786764998477612067880171211129530534256022184 ); 115792089210356248756420345214020892766250353991924191454421193933289684991999 3 [ <2, 1>, <43, 1>, <30223, 1>, <348253387243, 1>, <4641351449027, 1>, <417514796639753, 1>, <66013261729388519804782124120027, 1> ] [ <2, 64>, <3, 2>, <5, 3>, <11, 1>, <17, 2>, <31, 1>, <41, 1>, <257, 2>, <61681, 1>, <65537, 2>, <414721, 1>, <4278255361, 1>, <44479210368001, 1> ] [ <2, 1>, <13, 1>, <1213, 1>, <71209, 1>, <6158099, 1>, <4773264379806847, 1> ] 57896044605178124378210172607010446383125176995962095727210596966644842495999 [ <43, 1>, <30223, 1>, <348253387243, 1>, <4641351449027, 1>, <417514796639753, 1>, <66013261729388519804782124120027, 1> ] 115792089210356248762697446949407573529996955224135760342422259061068512044369 1 [ <2, 4>, <3, 1>, <71, 1>, <131, 1>, <373, 1>, <3407, 1>, <17449, 1>, <38189, 1>, <187019741, 1>, <622491383, 1>, <1002328039319, 1>, <2624747550333869278416773953, 1> ] [ <2, 1>, <5, 1>, <1879, 1>, <176337611, 1>, <34946779280882916835155272231706129710560967816144871596921775673, 1> ] [ <2, 6>, <3, 2>, <1297, 1>, <16879, 1>, <208150935158385979, 1> ] 57896044605178124381348723474703786764998477612067880171211129530534256022184 [ <2, 3>, <3, 1>, <71, 1>, <131, 1>, <373, 1>, <3407, 1>, <17449, 1>, <38189, 1>, <187019741, 1>, <622491383, 1>, <1002328039319, 1>, <2624747550333869278416773953, 1> ] |
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[分享]推荐一些密码学方面的资料
PN=115792089210356248762697446949407573530086143415290314195533631308867097853951 =FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF P-256: p = 2^256−2^224+2^192+2^96−1, a =−3, h = 1 nN=115792089210356248762697446949407573529996955224135760342422259061068512044369= FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551 推荐使用素数域256位椭圆曲线。 椭圆曲线方程:y2 = x3 + ax + b。 曲线参数: p=FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 00000000 FFFFFFFF FFFFFFFF =115792089210356248756420345214020892766250353991924191454421193933289684991999----------->也是类梅森数,挑了条大的,P256不多几个类梅森数,能给找出来 a=FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 00000000 FFFFFFFF FFFFFFFC=-3 b=28E9FA9E 9D9F5E34 4D5A9E4B CF6509A7 F39789F5 15AB8F92 DDBCBD41 4D940E93 n=FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF 7203DF6B 21C6052B 53BBF409 39D54123 Gx=32C4AE2C 1F198119 5F990446 6A39C994 8FE30BBF F2660BE1 715A4589 334C74C7 Gy=BC3736A2 F4F6779C 59BDCEE3 6B692153 D0A9877C C62A4740 02DF32E5 2139F0A0 n=FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF 7203DF6B 21C6052B 53BBF409 39D54123= 115792089210356248756420345214020892766061623724957744567843809356293439045923 S:= FiniteField(115792089210356248756420345214020892766250353991924191454421193933289684991999); E:= EllipticCurve([S|0,0,0,-3,18505919022281880113072981827955639221458448578012075254857346196103069175443]); E; Order(E); Points(E); Elliptic Curve defined by y^2 = x^3 + 11579208921035624875642034521402089276625\ 0353991924191454421193933289684991996*x + 18505919022281880113072981827955639221458448578012075254857346196103069175443 over GF(11579208921035624875642034521402089276625035399192419145442119393328968\ 4991999) 115792089210356248756420345214020892766061623724957744567843809356293439045923 >> Points(E); ^ Runtime error in 'Points': Cardinality of set is too large |
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[分享]推荐一些密码学方面的资料
多谢! X962-2005超ECC都公布了。。。。。 看看例子对不对,高斯正规基的为何不对。。。。 a:= FiniteField(2,5); E:= EllipticCurve([a!1,1,0,0,1]); E; Order(E); Points(E); a1:= FiniteField(2,5); E1:= EllipticCurve([a1!11111,0,0,0,11111]); E1; Order(E1); Points(E1); Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over GF(2^5) 22 {@ (0 : 1 : 0), (a.1^3 : a.1^15 : 1), (a.1^3 : a.1^26 : 1), (a.1^6 : a.1^21 : 1), (a.1^6 : a.1^30 : 1), (a.1^7 : a.1^8 : 1), (a.1^7 : a.1^25 : 1), (a.1^12 : a.1^11 : 1), (a.1^12 : a.1^29 : 1), (a.1^14 : a.1^16 : 1), (a.1^14 : a.1^19 : 1), (a.1^17 : a.1^13 : 1), (a.1^17 : a.1^23 : 1), (a.1^19 : a.1^4 : 1), (a.1^19 : a.1^28 : 1), (a.1^24 : a.1^22 : 1), (a.1^24 : a.1^27 : 1), (a.1^25 : a.1^2 : 1), (a.1^25 : a.1^14 : 1), (a.1^28 : a.1 : 1), (a.1^28 : a.1^7 : 1), (0 : 1 : 1) @} Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^5) 44 {@ (0 : 1 : 0), (1 : 0 : 1), (1 : 1 : 1), (a.1 : a.1^14 : 1), (a.1 : a.1^15 : 1), (a.1^2 : a.1^28 : 1), (a.1^2 : a.1^30 : 1), (a.1^4 : a.1^25 : 1), (a.1^4 : a.1^29 : 1), (a.1^5 : a.1^9 : 1), (a.1^5 : a.1^15 : 1), (a.1^8 : a.1^19 : 1), (a.1^8 : a.1^27 : 1), (a.1^9 : a.1^10 : 1), (a.1^9 : a.1^27 : 1), (a.1^10 : a.1^18 : 1), (a.1^10 : a.1^30 : 1), (a.1^11 : a.1^15 : 1), (a.1^11 : a.1^21 : 1), (a.1^13 : a.1^22 : 1), (a.1^13 : a.1^29 : 1), (a.1^15 : a.1^19 : 1), (a.1^15 : a.1^25 : 1), (a.1^16 : a.1^7 : 1), (a.1^16 : a.1^23 : 1), (a.1^18 : a.1^20 : 1), (a.1^18 : a.1^23 : 1), (a.1^20 : a.1^5 : 1), (a.1^20 : a.1^29 : 1), (a.1^21 : a.1^23 : 1), (a.1^21 : a.1^26 : 1), (a.1^22 : a.1^11 : 1), (a.1^22 : a.1^30 : 1), (a.1^23 : a.1^25 : 1), (a.1^23 : a.1^28 : 1), (a.1^26 : a.1^13 : 1), (a.1^26 : a.1^27 : 1), (a.1^27 : a.1^14 : 1), (a.1^27 : a.1^28 : 1), (a.1^29 : a.1^7 : 1), (a.1^29 : a.1^14 : 1), (a.1^30 : a.1^7 : 1), (a.1^30 : a.1^19 : 1), (0 : 1 : 1) @} 推荐就一条P256。。。。。。。。比下: NIST的: Curve P-256 P=115792089210356248762697446949407573530086143415290314195533631308867097853951 n=115792089210356248762697446949407573529996955224135760342422259061068512044369 seed =c49d360886e704936a6678e1139d26b7819f7e90 r =7efba1662985be9403cb055c75d4f7e0ce8d84a9c5114abcaf3177680104fa0d a=-3 b =5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b G x =6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296 G y =4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5 h=1 |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
modp(3481466152989800111201974224682033569110641383479283233030^1585250141131541931598626946823341394255706600825785605187, 4270758482012793668746242066981501955116983159626304328379); Error, numeric exception: overflow mlog(4169670491829098150176828207284776989010151046910451822393,348146615298\ 98001112019742246820335691106413834792\ 83233030,4270758482012793668746242066981501955116983159626304328379); =================== ======================== p:=4270758482012793668746201955116983159626304328379; Factorisation (p); K := GF(p); K; b :=K!3481466152924682033569110641383479283233030 ; b; a:=K!416967049128207284776989010151046910451822393; a; m := K!Log (b, a); m; m1:=0x40A6C8A2464A891E99DDBFCFC967BAFD4BAFA67B3ECEDC43 ; m1; |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
X=1585250141131541931598626946823341394255706600825785605187; Modexp(3481466152989800111201974224682033569110641383479283233030, 1585250141131541931598626946823341394255706600825785605187, 4270758482012793668746242066981501955116983159626304328379); 4169670491829098150176828207284776989010151046910451822393 那个X是对的 那个计算器验证都不行,更别提反求对数了,作者可能是用别的工具算的,就象POWERMOD,Modexp之类的,找到P192,先算出X,反把Y,G贴前面,给人感觉P192类的DSA强素数都能被那个计算器求对数解 高次幂模是多项式时间问题,别说57位,57000位也用MAPLE之类能算, 反求对数57位模的在个人PC上用INDEX我试了15小时也没出来 DSA强素数反求对数全世界素数专家蹂躪多年了, 作者是个爱面子的人,不过他没想想那法国人用的数小多了 贴张图 |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
无意中发现P = AE2CCC8E5956DE7898143649944108EEFCA2C7EF909012BB =4270758482012793668746242066981501955116983159626304328379 是NIST 推荐的ECC中的P-192。。。。明年也没戏啊。。。。 Factorization(2^192-2^64-1); Factorization(2^192-2^64-2); [ <6277101735386680763835789423207666416083908700390324961279, 1> ] [ <2, 1>, <59, 1>, <149309, 1>, <11393611, 1>, <108341181769254293, 1>, <288626509448065367648032903, 1> ] memory used=102024.8MB, alloc=128.4MB, time=2732.08 memory used=102062.9MB, alloc=128.4MB, time=2733.06 memory used=102101.1MB, alloc=128.4MB, time=2734.08 memory used=102139.2MB, alloc=128.4MB, time=2735.08 memory used=102177.4MB, alloc=128.4MB, time=2736.09 memory used=102215.5MB, alloc=128.4MB, time=2737.11 memory used=102253.6MB, alloc=128.4MB, time=2738.13 。。。。。。。。。。。。。。。 还没OVERFLOW。。。 |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
在MAPLE上看看 > index(41696704918290981501768282072847\ > 76989010151046910451822393,34814661529\ > 89800111201974224682033569110641383479\ > 283233030,4270758482012793668746242066981501955116983159626304328379); memory used=53.8MB, alloc=43.9MB, time=1.23 memory used=92.0MB, alloc=43.9MB, time=2.11 memory used=130.1MB, alloc=43.9MB, time=2.97 memory used=4838.3MB, alloc=69.7MB, time=114.66 memory used=4876.4MB, alloc=69.7MB, time=115.55 memory used=9302.0MB, alloc=69.7MB, time=217.05 memory used=14795.2MB, alloc=69.7MB, time=360.97 memory used=22233.9MB, alloc=69.7MB, time=557.84 memory used=33214.1MB, alloc=101.7MB, time=851.64 memory used=46336.7MB, alloc=101.7MB, time=1213.91 memory used=72086.0MB, alloc=101.7MB, time=1934.06 memory used=72124.1MB, alloc=101.7MB, time=1935.09 。。。。。。。。。。。。。。。。。 明天不知行不行。。。。 娃娃[CCG引用的http://docs.webmx.fr/Securite%20reseaux%20et%20Cracking/shmeitcorp/6/Crack%20XVIII/pDrill3/solucekgmedrill3.html的小点的数: index(368624370336150382836561,217453\ 512678253980654092,856210455808897477316603); memory used=46801.7MB, alloc=64.6MB, time=1348.09 memory used=46839.8MB, alloc=64.6MB, time=1349.12 299376145767585197811667(D) 那法国的还真对。。。。 =3F6536A02CD18F3B67D3(H) |
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[分享]可爱头像
继续。。。。没画完。。。。。樱桃小口? |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
觉的BigInt Calc1得花数年。。。 把BigInt Calc1和ECCTOOLS再传上来,不知为何,主页上没有了。别的站有 |
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[求助]利用BigInt Calc1.2 计算器 DSA算法中私钥X是怎么算出来的
试试PARI代码吧,也可用MATHEMATICS H(x,g, s) = s=g; for(n=1,znorder(g),if(x==s, return(n), s=s*g)); 0; ? H(18,Mod(5,23)) %6 = 12 ? H(20,Mod(5,23)) %7 = 5 把对应数换上就OK,不过可能要算一年半载!?!我算了它文里CRACKERME#4的79位的368624370336150382836561都等不急,减到50位也不行。。。。 下了感到BIGint不友好,象夏普计算器。。。。文档就最后那一个例子,可我们21世纪的人要新工具 |
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[推荐]推荐PARI和SAGE和MXGMA
P进数域EC曲线: K := pAdicField(17,19); E := EllipticCurve([K!1,0,0,0,1]); E; W:=WeierstrassModel(E); Degree(E); K1 := pAdicField(137,121); E1:= EllipticCurve([K1!1,4,5,0,1]); E1; W:=WeierstrassModel(E1); Degree(E1); Discriminant(E); Discriminant(E1); PointsAtInfinity(E); W; Aut(E) ; Aut(E1) ; Iso(E,E1); QuadraticTwist(E, 10); QuadraticTwist(E, 7); Elliptic Curve defined by y^2 + x*y + O(17^19)*y = x^3 + O(17^19)*x^2 + O(17^19)*x + (1 + O(17^19)) over pAdicField(17, 19) 3 Elliptic Curve defined by y^2 + x*y + (5 + O(137^121))*y = x^3 + (4 + O(137^121))*x^2 + O(137^121)*x + (1 + O(137^121)) over pAdicField(137, 121) 3 -433 + O(17^19) -35335 + O(137^121) {@ (O(17^19) : 1 + O(17^19) : O(17^19)) @} Elliptic Curve defined by y^2 + O(137^121)*x*y + O(137^121)*y = x^3 + O(137^121)*x^2 - (4563 + O(137^121))*x + (438318 + O(137^121)) over pAdicField(137, 121) Set of all automorphisms of E Set of all automorphisms of E1 Set of all isomorphisms from E to E1 Elliptic Curve defined by y^2 + O(17^19)*x*y + O(17^19)*y = x^3 + O(17^19)*x^2 - (2700 + O(17^19))*x + (46710000 + O(17^19)) over pAdicField(17, 19) Elliptic Curve defined by y^2 + O(17^19)*x*y + O(17^19)*y = x^3 + O(17^19)*x^2 - (1323 + O(17^19))*x + (16021530 + O(17^19)) over pAdicField(17, 19) P进数域EC曲线: K := pAdicField(17,19); E := EllipticCurve([K!1,0,0,0,1]); E; W:=WeierstrassModel(E); Degree(E); K1 := pAdicField(137,121); E1:= EllipticCurve([K1!1,4,5,0,1]); E1; W:=WeierstrassModel(E1); Degree(E1); Discriminant(E); Discriminant(E1); PointsAtInfinity(E); W; Aut(E) ; Aut(E1) ; Iso(E,E1); QuadraticTwist(E, 10); QuadraticTwist(E, 7); Elliptic Curve defined by y^2 + x*y + O(17^19)*y = x^3 + O(17^19)*x^2 + O(17^19)*x + (1 + O(17^19)) over pAdicField(17, 19) 3 Elliptic Curve defined by y^2 + x*y + (5 + O(137^121))*y = x^3 + (4 + O(137^121))*x^2 + O(137^121)*x + (1 + O(137^121)) over pAdicField(137, 121) 3 -433 + O(17^19) -35335 + O(137^121) {@ (O(17^19) : 1 + O(17^19) : O(17^19)) @} Elliptic Curve defined by y^2 + O(137^121)*x*y + O(137^121)*y = x^3 + O(137^121)*x^2 - (4563 + O(137^121))*x + (438318 + O(137^121)) over pAdicField(137, 121) Set of all automorphisms of E Set of all automorphisms of E1 Set of all isomorphisms from E to E1 Elliptic Curve defined by y^2 + O(17^19)*x*y + O(17^19)*y = x^3 + O(17^19)*x^2 - (2700 + O(17^19))*x + (46710000 + O(17^19)) over pAdicField(17, 19) Elliptic Curve defined by y^2 + O(17^19)*x*y + O(17^19)*y = x^3 + O(17^19)*x^2 - (1323 + O(17^19))*x + (16021530 + O(17^19)) over pAdicField(17, 19) pari+MAGMA一起用: Qp = pAdicField(5,11) E = EllipticCurve(Qp,[7, 2]) E.pari_curve() [O(5^11), O(5^11), O(5^11), 2 + 5 + O(5^11), 2 + O(5^11), O(5^11), 4 + 2*5 + O(5^11), 3 + 5 + O(5^11), 1 + 3*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + O(5^11), 4 + 2*5 + 5^2 + 2*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + O(5^11), 2 + 4*5 + 5^3 + 2*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + O(5^11), 4*5 + 2*5^2 + 2*5^4 + 2*5^5 + 3*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + O(5^11), 5^-1 + 4 + 4*5 + 5^3 + 3*5^4 + 2*5^6 + 5^8 + O(5^9), [3 + 2*5^2 + 4*5^3 + 5^4 + 4*5^5 + 3*5^6 + 4*5^7 + 2*5^8 + 2*5^9 + 4*5^10 + O(5^11)], 3 + 5^2 + 5^3 + 4*5^4 + 4*5^5 + 3*5^6 + 5^7 + 3*5^8 + 3*5^10 + O(5^11), 0, 5 + 3*5^3 + 3*5^5 + 5^7 + 5^8 + O(5^9), 1 + 5 + 4*5^2 + 4*5^3 + 3*5^4 + 5^5 + 4*5^6 + 3*5^7 + 2*5^9 + O(5^11), 0] plot(E), pAdicField图还没实现 NotImplementedError: Plotting of curves over 5-adic Field with capped relative precision 11 not implemented yet 找到了超越咱国大学ECC叫兽们的地方: http://sagemath.org/doc/reference/plane_curves.html SAGE好像还不能直接绘ECC 2进制域图,超ECC的也不能直接绘,得把点求出才行 E = EllipticCurve(GF(2^100,'a'),[1,2,3,4,5]) E Elliptic Curve defined by y^2 + x*y + y = x^3 + 1 over Finite Field in a of size 2^100Elliptic Curve defined by y^2 + x*y + y = x^3 + 1 over Finite Field in a of size 2^100 plot(E,rgbcolor=hue(0.7)) NotImplementedError R.<t> = PolynomialRing(GF(7)) H = HyperellipticCurve(t^5 + t + 2) HF:=H.frobenius_polynomial() HF; plot(H,rgbcolor=hue(0.7)) plot(HF,rgbcolor=hue(0.7)) H.points() NotImplementedError: Plotting of curves over Finite Field of size 37 not implemented yetTraceback (most recent call last): raise NotImplementedError, "Plotting of curves over %s not implemented yet"%K NotImplementedError: Plotting of curves over Finite Field of size 37 not implemented yet |
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[推荐]老外用照相机拍芯片电磁信息
有,要介绍信才卖 |
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[推荐]推荐PARI和SAGE和MXGMA
推荐曲线余因子验证:挑了几条条,除了KM233都对,HASH和点都还没试 P:=2^192-2^64-1;-------------素域192 P; p192 := 6277101735386680763835789423207666416083908700390324961279; p192; IsPrime(p192); Ilog2(p192); Ilog(10, p192); Ilog(2, p192); 二进制位数 Ilog(16, p192); > E192 := EllipticCurve([GF(p192) | -3, 2455155546008943817740293915197451784769108058161191238065]);a=-3 E192; O192:=Order(E192); 曲线阶 FactoredOrder(E192);曲线阶分解 O192; IsPrime(O192); Trace(E192); Twists(EK192); 扭曲线,曲线阶和=2(p+1) TraceOfFrobenius(E192);Frobenius迹,可看出同构的椭圆曲线 Points(E192);求点,太大了 K163 := FiniteField(2,163); // finite field of size 2^163 Ilog2(2^163); Ilog(10, 2^163); Ilog(2, 2^163); Ilog(16, 2^163); EK163 := EllipticCurve([K163!1,1,0,0,1]); EK163; Order(EK163); Twists(EK163); FactoredOrder(EK163); Factorization(11692013098647223345629483507196896696658237148126); K233 := FiniteField(2,233); Ilog2(2^233); Ilog(10, 2^233); Ilog(2, 2^233); Ilog(16, 2^233); EK233 := EllipticCurve([K233!1,1,0,0,1]); EK233; OEK233:=Order(EK233); FactoredOrder(EK233); OEK233; IsPrime(OEK233); Twists(EK233); Factorization(13803492693581127574869511724554050767520671933232537715337748796231814); K283 := FiniteField(2,283); // finite field of size 2^83 Ilog2(2^283); Ilog(10, 2^283); Ilog(2, 2^283); Ilog(16, 2^283); EK283 := EllipticCurve([K283!1,0,0,0,1]); EK283; Order(EK283); FactoredOrder(EK283); Twists(EK283); Factorization(1554135113780583256735569525458815125313924693517224529718349999011926331881769\ 0415492 ); K409 := FiniteField(2,409); // finite field of size 409 Ilog2(2^409); Ilog(10, 2^409); Ilog(2, 2^409); Ilog(16, 2^409); EK409 := EllipticCurve([K409!1,0,0,0,1]); EK409; Order(EK409); FactoredOrder(EK409); Twists(EK409); Factorization(1322111937580497197903830616065542079656809365928562438569297580091522845156996\ 764202693033831109832056385466362470925434684); K571 := FiniteField(2,571); // finite field of size 571 Ilog2(2^571); Ilog(10, 2^571); Ilog(2, 2^571); Ilog(16, 2^571); EK571 := EllipticCurve([K409!1,0,0,0,1]); EK571; Order(EK571); FactoredOrder(EK571); Twists(EK571); Factorization(7729075046034516689390703781863974688597854659412869997314470502903038284579120\ 8490725359140908268473388268512033014058450946998962664692477187296864683700142\ 22934741106692); K109 := FiniteField(2,109); // finite field of size 109 Ilog2(2^109); Ilog(10, 2^109); Ilog(2, 2^109); Ilog(16, 2^109); EK109 := EllipticCurve([K109!1,1,0,0,1]); EK109; Order(EK109); FactoredOrder(EK109); Factorization(649037107316853402974897312922934); Points(EK109); Twists(EK109); Km163 := FiniteField(2,163); // 2m finite field of size 109 Ilog2(2^163); Ilog(10, 2^163); Ilog(2, 2^163); Ilog(16, 2^163); EKm163 := EllipticCurve([K163!1,0,0,0,2982236234343851336267446656627785008148015875581]); EKm163; Order(EKm163); FactoredOrder(EKm163); Factorization(11692013098647223345629473816263631617836683539492 ); Twists(EKm163); K4 := FiniteField(2,4); // finite field of size 16 试了2^4域上的求点,还不太对 Ilog2(2^4); Ilog(10, 2^4); Ilog(2, 2^4); Ilog(16, 2^4); EK4 := EllipticCurve([K4!0001,1000,0000,0000,1001]);专找了MENEZES书上的例子,应该20个点,可只算出了16个。。。 EK4; Order(EK4); FactoredOrder(EK4); Factorization(16); Points(EK4); Twists(EK4); ===== 6277101735386680763835789423207666416083908700390324961279 6277101735386680763835789423207666416083908700390324961279 true 191 57 191 47 Elliptic Curve defined by y^2 = x^3 + 62771017353866807638357894232076664160839\ 08700390324961276*x + 245515554600894381774029391519745178476910805816119123806\ 5 over GF(6277101735386680763835789423207666416083908700390324961279) [ <6277101735386680763835789423176059013767194773182842284081, 1> ] 6277101735386680763835789423176059013767194773182842284081 true 31607402316713927207482677199 31607402316713927207482677199 [ Elliptic Curve defined by y^2 = x^3 + 6277101735386680763835789423207666416\ 083908700390324961276*x + 2455155546008943817740293915197451784769108058161\ 191238065 over GF(627710173538668076383578942320766641608390870039032496127\ 9), Elliptic Curve defined by y^2 = x^3 + 3256071953165425550869791931684810521\ 755660057874970232744*x + 3441361442102155539006528255597946985237925423249\ 825914045 over GF(627710173538668076383578942320766641608390870039032496127\ 9) ] >> Points(E192); 太大了。。。。。 ^ Runtime error in 'Points': Cardinality of set is too large 163 49 163 40 Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over GF(2^163) 11692013098647223345629483507196896696658237148126 [ <2, 1>, <5846006549323611672814741753598448348329118574063, 1> ]---------殆素阶a=1,h=2 [ <2, 1>, <5846006549323611672814741753598448348329118574063, 1> ] 233 70 233 58 Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over GF(2^233) [ <2, 1>, <92269, 1>, <114861079, 1>, <130034039, 1>, <5062109767067236109, 1>,就这不是殆素阶。。。。。。。??? <989331137390630128765577490907, 1> ] 13803492693581127574869511724554050767520671933232537715337748796231814 false [ <2, 1>, <92269, 1>, <114861079, 1>, <130034039, 1>, <5062109767067236109, 1>, <989331137390630128765577490907, 1> ] 283 85 283 70 Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^283) 1554135113780583256735569525458815125313924693517224529718349999011926331881769\ 0415492 [ <2, 2>, <38853377844514581418389238136470378132848117337930613242958749975298\ 15829704422603873, 1> ]--------殆素阶a=0,h=4 [ <2, 2>, <38853377844514581418389238136470378132848117337930613242958749975298\ 15829704422603873, 1> ] 409 123 409 102 Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^409) 1322111937580497197903830616065542079656809365928562438569297580091522845156996\ 764202693033831109832056385466362470925434684 [ <2, 2>, <33052798439512429947595765401638551991420234148214060964232439502288\ 0711289249191050673258457777458014096366590617731358671, 1> ] 殆素阶a=0,h=4 [ Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^409),--------------扭曲线很复杂啊 Elliptic Curve defined by y^2 + x*y = x^3 + (K409.1^408 + K409.1^407 + K409.1^401 + K409.1^400 + K409.1^399 + K409.1^397 + K409.1^395 + K409.1^394 + K409.1^393 + K409.1^392 + K409.1^390 + K409.1^389 + K409.1^388 + K409.1^385 + K409.1^383 + K409.1^380 + K409.1^379 + K409.1^378 + K409.1^374 + K409.1^373 + K409.1^372 + K409.1^371 + K409.1^370 + K409.1^369 + K409.1^366 + K409.1^360 + K409.1^357 + K409.1^356 + K409.1^355 + K409.1^354 + K409.1^349 + K409.1^348 + K409.1^345 + K409.1^344 + K409.1^343 + K409.1^341 + K409.1^339 + K409.1^337 + K409.1^336 + K409.1^335 + K409.1^334 + K409.1^333 + K409.1^332 + K409.1^331 + K409.1^330 + K409.1^323 + K409.1^322 + K409.1^321 + K409.1^320 + K409.1^318 + K409.1^317 + K409.1^316 + K409.1^315 + K409.1^314 + K409.1^312 + K409.1^311 + K409.1^310 + K409.1^306 + K409.1^304 + K409.1^301 + K409.1^300 + K409.1^299 + K409.1^298 + K409.1^297 + K409.1^295 + K409.1^294 + K409.1^292 + K409.1^291 + K409.1^290 + K409.1^289 + K409.1^288 + K409.1^286 + K409.1^285 + K409.1^283 + K409.1^278 + K409.1^277 + K409.1^276 + K409.1^275 + K409.1^274 + K409.1^271 + K409.1^269 + K409.1^266 + K409.1^264 + K409.1^258 + K409.1^257 + K409.1^256 + K409.1^255 + K409.1^254 + K409.1^253 + K409.1^249 + K409.1^245 + K409.1^244 + K409.1^243 + K409.1^241 + K409.1^240 + K409.1^239 + K409.1^237 + K409.1^236 + K409.1^234 + K409.1^233 + K409.1^232 + K409.1^230 + K409.1^228 + K409.1^226 + K409.1^225 + K409.1^223 + K409.1^222 + K409.1^220 + K409.1^218 + K409.1^212 + K409.1^207 + K409.1^206 + K409.1^202 + K409.1^200 + K409.1^199 + K409.1^198 + K409.1^197 + K409.1^196 + K409.1^195 + K409.1^194 + K409.1^193 + K409.1^190 + K409.1^189 + K409.1^188 + K409.1^184 + K409.1^182 + K409.1^181 + K409.1^180 + K409.1^178 + K409.1^175 + K409.1^173 + K409.1^172 + K409.1^171 + K409.1^170 + K409.1^167 + K409.1^165 + K409.1^164 + K409.1^163 + K409.1^162 + K409.1^161 + K409.1^160 + K409.1^157 + K409.1^156 + K409.1^155 + K409.1^152 + K409.1^151 + K409.1^149 + K409.1^148 + K409.1^145 + K409.1^144 + K409.1^140 + K409.1^137 + K409.1^135 + K409.1^134 + K409.1^133 + K409.1^132 + K409.1^130 + K409.1^129 + K409.1^126 + K409.1^125 + K409.1^122 + K409.1^119 + K409.1^117 + K409.1^115 + K409.1^114 + K409.1^111 + K409.1^110 + K409.1^109 + K409.1^108 + K409.1^106 + K409.1^101 + K409.1^99 + K409.1^98 + K409.1^97 + K409.1^96 + K409.1^94 + K409.1^88 + K409.1^85 + K409.1^80 + K409.1^77 + K409.1^75 + K409.1^74 + K409.1^72 + K409.1^71 + K409.1^70 + K409.1^69 + K409.1^68 + K409.1^67 + K409.1^64 + K409.1^63 + K409.1^62 + K409.1^59 + K409.1^54 + K409.1^53 + K409.1^48 + K409.1^45 + K409.1^44 + K409.1^42 + K409.1^41 + K409.1^40 + K409.1^39 + K409.1^38 + K409.1^37 + K409.1^36 + K409.1^34 + K409.1^33 + K409.1^31 + K409.1^30 + K409.1^27 + K409.1^26 + K409.1^22 + K409.1^21 + K409.1^20 + K409.1^19 + K409.1^18 + K409.1^17 + K409.1^16 + K409.1^15 + K409.1^14 + K409.1^13 + K409.1^9 + K409.1^6 + K409.1^5 + K409.1^4 + K409.1^3 + K409.1 + 1)*x^2 + 1 over GF(2^409) ] [ <2, 2>, <33052798439512429947595765401638551991420234148214060964232439502288\ 0711289249191050673258457777458014096366590617731358671, 1> ] 殆素阶a=0,h=4 571 171 571 142 Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^409) 1322111937580497197903830616065542079656809365928562438569297580091522845156996\ 764202693033831109832056385466362470925434684 [ <2, 2>, <33052798439512429947595765401638551991420234148214060964232439502288\ 0711289249191050673258457777458014096366590617731358671, 1> ] [ <2, 2>, <19322687615086291723476759454659936721494636648532174993286176257257\ 5957114478021226813397852270671183470671280082535146127367497406661731192968242\ 1617092503555733685276673, 1> ] 109 32 109 27 Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over GF(2^109) 649037107316853402974897312922934 [ <2, 1>, <324518553658426701487448656461467, 1> ] [ <2, 1>, <324518553658426701487448656461467, 1> ] 殆素阶a=1,h=2 >> Points(EK109);---------------------too large,109的就那女解放军教授算出的,比老外慢五年。可国内首次 ^ Runtime error in 'Points': Cardinality of set is too large 163 49 163 40 Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^163) 11692013098647223345629473816263631617836683539492 [ <2, 2>, <653, 1>, <6521, 1>, <34101072914026637, 1>, <20129541232727197849723433, 1> ] [ <2, 2>, <653, 1>, <6521, 1>, <34101072914026637, 1>, <20129541232727197849723433, 1> ] 4 1 4 1 Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^4)--------------二进制域上求点,一般书上都没程序化,MAGMA就是强 T := FiniteField(2,4); E:= EllipticCurve([T!1,T.1^3,0,0,T.1^3+1]); E; Twists(E); E1:= EllipticCurve([T!1,T.1,0,0,T.1^3+1]); E1; Points(E1); O1:=Order(E1); O1; Twists(E1); E2:= EllipticCurve([T!1,T.1^12,0,0,T.1^3+1]); E2; Trace(T.1^3); Trace(T.1^4); Trace(T.1^12); IsIsomorphic(E1, E); IsIsomorphic(E1, E2); IsIsomorphic(E, E2); Points(E); Points(E2); O2:=Order(E2); O2; O:=Order(E); O; O1+O; 2*(16+1); ======================= Elliptic Curve defined by y^2 + x*y = x^3 + T.1^3*x^2 + T.1^14 over GF(2^4) [ Elliptic Curve defined by y^2 + x*y = x^3 + T.1^3*x^2 + T.1^14 over GF(2^4), Elliptic Curve defined by y^2 + x*y = x^3 + T.1^10*x^2 + T.1^14 over GF(2^4) ] Elliptic Curve defined by y^2 + x*y = x^3 + T.1*x^2 + T.1^14 over GF(2^4) {@ (0 : 1 : 0), (T.1^2 : T.1^7 : 1), (T.1^2 : T.1^12 : 1), (T.1^5 : T.1^7 : 1), (T.1^5 : T.1^13 : 1), (T.1^9 : T.1^10 : 1), (T.1^9 : T.1^13 : 1), (T.1^11 : T.1^4 : 1), (T.1^11 : T.1^13 : 1), (T.1^13 : 1 : 1), (T.1^13 : T.1^6 : 1), (0 : T.1^7 : 1) @} 12 [ Elliptic Curve defined by y^2 + x*y = x^3 + T.1*x^2 + T.1^14 over GF(2^4), Elliptic Curve defined by y^2 + x*y = x^3 + T.1^14*x^2 + T.1^14 over GF(2^4) ] Elliptic Curve defined by y^2 + x*y = x^3 + T.1^12*x^2 + T.1^14 over GF(2^4) 1 0 1 false false true {@ (0 : 1 : 0), (1 : 0 : 1), (1 : 1 : 1), (T.1 : T.1^12 : 1), (T.1 : T.1^13 : 1), (T.1^3 : 1 : 1), (T.1^3 : T.1^14 : 1), (T.1^4 : T.1^6 : 1), (T.1^4 : T.1^12 : 1), (T.1^6 : 0 : 1), (T.1^6 : T.1^6 : 1), (T.1^7 : T.1 : 1), (T.1^7 : T.1^14 : 1), (T.1^8 : 0 : 1), (T.1^8 : T.1^8 : 1), (T.1^10 : T.1^6 : 1), (T.1^10 : T.1^7 : 1), (T.1^12 : T.1^2 : 1), (T.1^12 : T.1^7 : 1), (T.1^14 : T.1^5 : 1), (T.1^14 : T.1^12 : 1), (0 : T.1^7 : 1) @} {@ (0 : 1 : 0), (1 : T.1^2 : 1), (1 : T.1^8 : 1), (T.1 : T.1^8 : 1), (T.1 : T.1^10 : 1), (T.1^3 : T.1^10 : 1), (T.1^3 : T.1^12 : 1), (T.1^4 : 0 : 1), (T.1^4 : T.1^4 : 1), (T.1^6 : T.1^8 : 1), (T.1^6 : T.1^14 : 1), (T.1^7 : T.1^3 : 1), (T.1^7 : T.1^4 : 1), (T.1^8 : T.1 : 1), (T.1^8 : T.1^10 : 1), (T.1^10 : T.1^2 : 1), (T.1^10 : T.1^4 : 1), (T.1^12 : T.1 : 1), (T.1^12 : T.1^13 : 1), (T.1^14 : T.1^2 : 1), (T.1^14 : T.1^13 : 1), (0 : T.1^7 : 1) @} 22 22 34 34 22MOD4==2 12MOD4==0 |
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