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[原创]magma,/PARI/GP中文文档ECC初步:
SAGE: 几个数学家椭圆曲线的各种数据全包里了, cremona,Stein-Watkins 的有理域,John Jones数域多项式 http://www.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html 前导子N分类的,默认前10000,可加载大的到130000 from sage.databases.cremona import LargeCremonaDatabase 随机选一条: CremonaDatabase().random() Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 324*x - 18068 over Rational FieldElliptic Curve defined by y^2 + x*y = x^3 - x^2 + 324*x - 18068 over Rational Field 新老版cremona字母转换: old_cremona_letter_code(5221) 'UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU\ UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU\ UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU' 标签分离: c = CremonaDatabase() parse_cremona_label('37a2') import sage.databases.cremona as cremona cremona.split_code('37a2') 求前导子21的所有曲线: c = CremonaDatabase() c.allcurves(21) {'a1': [[1, 0, 0, -4, -1], 0, 8], 'a3': [[1, 0, 0, -39, 90], 0, 8], 'a2': [[1, 0, 0, -49, -136], 0, 4], 'a5': [[1, 0, 0, -784, -8515], 0, 2], 'a4': [[1, 0, 0, 1, 0], 0, 4], 'a6': [[1, 0, 0, -34, -217], 0, 2]}{'a1': [[1, 0, 0, -4, -1], 0, 8], 'a3': [[1, 0, 0, -39, 90], 0, 8], 'a2': [[1, 0, 0, -49, -136], 0, 4], 'a5': [[1, 0, 0, -784, -8515], 0, 2], 'a4': [[1, 0, 0, 1, 0], 0, 4], 'a6': [[1, 0, 0, -34, -217], 0, 2]} 从5个不变量定义曲线:要CremonaData理的才行: c = CremonaDatabase() sage: c.elliptic_curve_from_ainvs(11, [0, -1, 1, -10, -20]) 列出给定前导子的所有曲线: c = CremonaDatabase() CremonaDatabase().list([3171]) [Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7*x - 9 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 49*x - 58 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 454*x + 3506 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 7249*x + 234536 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 139*x + 8672 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 14*x + 3677 over Rational Field][Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7*x - 9 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 49*x - 58 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 454*x + 3506 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 7249*x + 234536 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 139*x + 8672 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 14*x + 3677 over Rational Field] 列出给定前导子的所有曲线数: c = CremonaDatabase() c.number_of_curves(9920) 70 FSD猜想。千年问题: c = CremonaDatabase() c.allbsd(11) {'a1': ['5', '1.2692093042795534217', '0.25384186085591068434', '1', '1.00000000000000000000'], 'a3': ['1', '6.3460465213977671084', '0.25384186085591068434', '1', '1'], 'a2': ['1', '0.25384186085591068434', '0.25384186085591068434', '1', '1.00000000000000000000']}{'a1': ['5', '1.2692093042795534217', '0.25384186085591068434', '1', '1.00000000000000000000'], 'a3': ['1', '6.3460465213977671084', '0.25384186085591068434', '1', '1'], 'a2': ['1', '0.25384186085591068434', '0.25384186085591068434', '1', '1.00000000000000000000']} SAGE连千年问题黎曼零点都整合了,10万个,精度10^9,图灵,冯诺曼依都算过黎曼零点,《美丽心灵》纳什就是想解决黎曼零点问题发疯的 zz = zeta_zeros() zz zz[100] --------第一百个 [14.134725142000001, 21.022039638999999, 25.01085758, 30.424876126000001, 32.935061588000003, 37.586178158999999, 40.918719011999997, 43.327073280999997, 48.005150880999999, 49.773832478000003, 52.970321478000002, 56.446247696999997, 59.347044003000001, 60.831778524999997, 65.112544048000004, 67.079810529, 69.546401711000001, 72.067157674000001, 75.704690698999997, 77.144840068999997, 79.337375019999996, 82.910380853999996, 84.735492980999993, 87.425274612999999, 88.809111208000004, 92.491899270999994, 94.651344041000002, 95.870634228, 98.831194217999993, SAGE把在线整数列也包了 http://oeis.org/ 按编号找数列: SloaneEncyclopedia[6083] [1, 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, 1, 3, 7, 3, 1, 7, 1, 3, 9, 3, 1, 9, 1, 3, 11, 3, 1, 11, 1, 3, 13, 3, 1, 13, 1, 3, 15, 3, 1, 15, 1, 3, 17, 3, 1, 17, 1, 3, 19, 3, 1, 19, 1, 3, 21, 3, 1, 21, 1, 3, 23, 3, 1, 23, 1, 3, 25, 3, 1, 25, 1, 3, 27, 3, 1, 27, 1, 3, 29, 3, 1, 29, 1, 3, 31, 3][1, 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, 1, 3, 7, 3, 1, 7, 1, 3, 9, 3, 1, 9, 1, 3, 11, 3, 1, 11, 1, 3, 13, 3, 1, 13, 1, 3, 15, 3, 1, 15, 1, 3, 17, 3, 1, 17, 1, 3, 19, 3, 1, 19, 1, 3, 21, 3, 1, 21, 1, 3, 23, 3, 1, 23, 1, 3, 25, 3, 1, 25, 1, 3, 27, 3, 1, 27, 1, 3, 29, 3, 1, 29, 1, 3, 31, 3] 按数列找: SloaneEncyclopedia.find([1,2,3,5,8], 1) Searching Sloane's online database... [[45, 'Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1, F(2) = 1, ...', [0, 1, 1, 2, 3, 5, 按编号找名称: sequence_name(12) SAGE: 康威多项式。。。。。还活着就命名多项式了。。。。 c = ConwayPolynomials() c.polynomial(3, 21) 约翰·何顿·康威,26个散单群里有板有3个他找的 http://www.hudong.com/wiki/%E7%BA%A6%E7%BF%B0%C2%B7%E4%BD%95%E9%A1%BF%C2%B7%E5%BA%B7%E5%A8%81 |
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[求助]MORDELL定理问题
E00:=EllipticCurve( x^3-4*x^2+16); E00; MordellWeilShaInformation(E00); DescentInformation(E00); Generators(E00) ; Q00, reps := IntegralPoints(E00); Q00; TorsionSubgroup(E00); Elliptic Curve defined by y^2 = x^3 - 4*x^2 + 16 over Rational Field mod17 0 0 0 Torsion Subgroup = Z/5 Analytic rank = 0 ==> Rank(E) = 0 [ 0, 0 ] [] [] Torsion Subgroup = Z/5 Analytic rank = 0 ==> Rank(E) = 0 [ 0, 0 ] [] [] [ (0 : 4 : 1) ] [ (0 : 4 : 1), (4 : -4 : 1) ] Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*$.1 = 0 ==================== Qx<x> := PolynomialRing(Rationals()); E0:=EllipticCurve( x^3-2*x+16); E0; MordellWeilShaInformation(E0); DescentInformation(E0); Generators(E0) ; Q0, reps := IntegralPoints(E0); Q0; TorsionSubgroup(E0); Elliptic Curve defined by y^2 = x^3 - 2*x + 16 over Rational Field mod17 1 1 1 Torsion Subgroup is trivial Analytic rank = 1 ==> Rank(E) = 1 The 2-Selmer group has rank 1 Found a point of infinite order. After 2-descent: 1 <= Rank(E) <= 1 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 1, 1 ] [ (0 : 4 : 1) ] [ <2, [ 0, 0 ]> ] Torsion Subgroup is trivial Analytic rank = 1 ==> Rank(E) = 1 The 2-Selmer group has rank 1 Found a point of infinite order. After 2-descent: 1 <= Rank(E) <= 1 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 1, 1 ] [ (0 : 4 : 1) ] [ <2, [ 0, 0 ]> ] [ (0 : 4 : 1) ] [ (0 : 4 : 1) ] Abelian Group of order 1 =========================================== Qx<x> := PolynomialRing(Rationals()); E0:=EllipticCurve( x^3-x+16); mod17 mod21 E0; MordellWeilShaInformation(E0); DescentInformation(E0); Generators(E0) ; Q0, reps := IntegralPoints(E0); Q0; TorsionSubgroup(E0); Elliptic Curve defined by y^2 = x^3 - x + 16 over Rational Field 2 2 2 Torsion Subgroup is trivial Analytic rank = 2 The 2-Selmer group has rank 2 Found a point of infinite order. Found 2 independent points. After 2-descent: 2 <= Rank(E) <= 2 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 2, 2 ] [ (0 : 4 : 1), (1 : -4 : 1) ] [ <2, [ 0, 0 ]> ] Torsion Subgroup is trivial Analytic rank = 2 The 2-Selmer group has rank 2 Found a point of infinite order. Found 2 independent points. After 2-descent: 2 <= Rank(E) <= 2 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 2, 2 ] [ (0 : 4 : 1), (1 : -4 : 1) ] [ <2, [ 0, 0 ]> ] [ (1 : 4 : 1), (-1 : 4 : 1) ] [ (-1 : -4 : 1), (0 : 4 : 1), (1 : -4 : 1), (16 : -64 : 1), (63 : 500 : 1), (65 : -524 : 1), (262016 : 134119436 : 1) ] Abelian Group of order 1 |
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[推荐]户照里的密码知识扫盲
4个,传了3。。。另一个自各下吧 |
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[求助]MORDELL定理问题
Qx<x> := PolynomialRing(Rationals()); E1:=EllipticCurve( x^3-39); E1; MordellWeilRank(E1); RankBounds(E1); MordellWeilShaInformation(E1); F:=FaltingsHeight(E1) ; F; HasComplexMultiplication(E1); DescentInformation(E1); Generators(E1) ; Q1, reps := IntegralPoints(E1); Q1; TorsionSubgroup(E1); Elliptic Curve defined by y^2 = x^3 - 39 over Rational Field 2 2 2 Torsion Subgroup is trivial Analytic rank = 2 The 2-Selmer group has rank 2 Found a point of infinite order. Found 2 independent points. After 2-descent: 2 <= Rank(E) <= 2 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 2, 2 ] [ (22 : -103 : 1), (4 : 5 : 1) ] [ <2, [ 0, 0 ]> ] -0.204821688386087651093858399724 true -3 Torsion Subgroup is trivial Analytic rank = 2 The 2-Selmer group has rank 2 Found a point of infinite order. Found 2 independent points. After 2-descent: 2 <= Rank(E) <= 2 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 2, 2 ] [ (22 : -103 : 1), (4 : 5 : 1) ] [ <2, [ 0, 0 ]> ] [ (4 : 5 : 1), (10 : 31 : 1) ] [ (4 : 5 : 1), (10 : 31 : 1), (22 : -103 : 1) ] Abelian Group of order 1 |
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[求助]MORDELL定理问题
多谢! 先贴点慢慢理解 Qx<x> := PolynomialRing(Rationals()); E1:=EllipticCurve( x^3-39);- E1; jInvariant(E1); Genus(E1); Q1, reps := IntegralPoints(E1); Q1; TorsionSubgroup(E1);-------------------------------------RANK=2,T=1 Mordell's curve E: y^2 = x^3 - 39 %%% %%% %%% %%% E_-00039: r = 2 t = 1 #III = 1 %%% %%% E(Q) = <(4, 5)> x <(10, 31)> %%% %%% R = 2.7347941297 %%% %%% 6 integral points %%% %%% 1. (4, 5) = 1 * (4, 5) %%% %%% 2. (4, -5) = -(4, 5) %%% %%% 3. (22, 103) = 1 * (4, 5) - 1 * (10, 31) %%% %%% 4. (22, -103) = -(22, 103) %%% %%% 5. (10, 31) = 1 * (10, 31) %%% %%% 6. (10, -31) = -(10, 31) Elliptic Curve defined by y^2 = x^3 - 39 over Rational Field 0 1 [ (4 : 5 : 1), (10 : 31 : 1), (22 : -103 : 1) ] Abelian Group of order 1 ========================== E2:=EllipticCurve(x^3 +1331);---------------------------------RANK=1,T=2 E2; jInvariant(E2); Genus(E2); TorsionSubgroup(E2); Q2, reps := IntegralPoints(E2); Q2; E_+01331: r = 1 t = 2 #III = 1 E(Q) = <(37, 228)> x <(-11, 0)> R = 3.2472475292 3 integral points 1. (-11, 0) = (-11, 0) 2. (37, 228) = 1 * (37, 228) 3. (37, -228) = -(37, 228) Elliptic Curve defined by y^2 = x^3 + 1331 over Rational Field 0 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 [ (-11 : 0 : 1), (37 : -228 : 1) ] ========================= E3:=EllipticCurve(x^3 - 9998);------RANK=3,T=1 无整点 E3; jInvariant(E3); Genus(E3); TorsionSubgroup(E3); Q3, reps := IntegralPoints(E3); Q3; r = 3 t = 1 #III = 1 E(Q) = <(-71/4, 531/8)> x <(19/9, 2701/27)> x <(283/9, 5473/27)> R = 114.5499715889 0 integral points Elliptic Curve defined by y^2 = x^3 - 9998 over Rational Field 0 1 Abelian Group of order 1 [] ========================== E6:=EllipticCurve(x^3 +1521);----------RANK=1,T=3,有三对整点 E6; jInvariant(E6); Genus(E6); TorsionSubgroup(E6); Q6, reps := IntegralPoints(E6); Q6; E_+01521: r = 1 t = 3 #III = 1 E(Q) = <(12, 57)> x <(0, 39)> R = 1.9164245639 6 integral points 1. (0, 39) = (0, 39) 2. (0, -39) = -(0, 39) 3. (12, 57) = 1 * (12, 57) 4. (12, -57) = -(12, 57) 5. (52, 377) = (0, 39) - 1 * (12, 57) 6. (52, -377) = -(52, 377) Elliptic Curve defined by y^2 = x^3 + 1521 over Rational Field 0 1 Abelian Group isomorphic to Z/3 Defined on 1 generator Relations: 3*$.1 = 0 [ (0 : 39 : 1), (12 : -57 : 1), (52 : 377 : 1) ] E5:=EllipticCurve(x^3 -4*x^2+16);-----RANK=?,T=1,整点? E5; jInvariant(E5); Genus(E5); TorsionSubgroup(E5); Q5, reps := IntegralPoints(E5); Q5; E7:=EllipticCurve(x^3 + 7388); E7; jInvariant(E7); Genus(E7); TorsionSubgroup(E7); Q7, reps := IntegralPoints(E7); Q7; E8:=EllipticCurve(x^3 +7428); E8; jInvariant(E8); Genus(E8); TorsionSubgroup(E8); Q8, reps := IntegralPoints(E8); Q8; E9:=EllipticCurve(x^3 - 7416); E9; jInvariant(E9); Genus(E9); TorsionSubgroup(E9); Q9, reps := IntegralPoints(E9); Q9; |
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[分享]亿群大康,挑一个试试啊
http://www.equn.com/forum/thread-17011-1-1.html 非主流数学、密码类项目 - 文章索引 原帖为其他非主流数学、密码类项目 - 文章索引 本帖对原帖有所增删修改。 版权归中国分布式计算总站所有。 BinSYS-SZDG —— 搜索高阶广义二进制数系 官方站点:http://szdg.lpds.sztaki.hu/szdg/ 中文站点:http://boinc.equn.com/szdg/desc_numsys_es.htm 项目介绍:SZDG 中的数学 项目指南: 项目新闻:http://www.equn.com/forum/viewth ... =page%3D1#pid207584 相关帖子:SZDG、数系与其他、成功完成对11阶二进制数系的搜索 Enigma-M4-Project —— 破解四转子之谜 官方站点:http://www.bytereef.org/ 中文站点:http://www.equn.com/m4/ 项目介绍:家用电脑破译“二战”纳粹密码,破译者都是业余的、分布式计算破解 Enigma 代码 项目指南:Enigma项目(M4-Project)参加说明 相关帖子: Enigma@home —— 破解四转子之谜(BOINC 平台) 官方站点:http://www.enigmaathome.net/ 项目介绍:见“Enigma-M4-Project —— 破解四转子之谜” 相关帖子:[新项目]Enigma@Home The 3x + 1 class record search —— 3x+1 问题 官方站点:http://www.ericr.nl/wondrous/index.html 中文站点:http://www.equn.com/3x+1/ 项目介绍:http://www.equn.com/distributed/ap-math.html#3xplus1 项目指南:3x+1问题 相关帖子:3x+1Problem参加全攻略 PrimeGrid —— 质数网格 官方站点:http://www.primegrid.com 项目介绍:http://www.equn.com/distributed/ap-crypto.html#primegrid 项目指南:http://www.equn.com/forum/viewthread.php?tid=8534 项目新闻:http://www.equn.com/forum/viewthread.php?tid=14688&pid=207572&page=1&extra=page%3D1#pid207572 相关帖子:[PrimeGrid 相关新闻] RSA-640 被分解、PrimeGrid FAQ 中英文对照、RieselSieve & PrimeGrid这两个项目有什么关系吗?、PrimeGrid子项目比较 RieselSieve —— Riesel 质数搜索 官方站点(传统):http://www.rieselsieve.com 官方站点(BOINC):http://boinc.rieselsieve.com 项目介绍:http://www.equn.com/distributed/ap-math.html#rieselsieve 项目新闻(BOINC):http://www.equn.com/forum/viewth ... =page%3D1#pid207573 相关帖子:RieselSieve加入BOINC平台、RieselSieve & PrimeGrid这两个项目有什么关系吗? Rectilinear Crossing Number —— 寻找平面化完全图最小交叉数 官方站点:http://dist.ist.tugraz.at/cape5/ 项目新闻:http://www.equn.com/forum/viewthread.php?tid=14688&pid=207584&page=1&extra=page%3D1#pid219468 相关帖子:[新项目][数学类]Rectilinear Crossing Number Sudoku —— 寻找最小唯一解数独 官方站点:http://dist2.ist.tugraz.at/sudoku/ 项目新闻:http://www.equn.com/forum/viewthread.php?tid=14688&pid=225535&page=1&extra=page%3D1#pid225535 相关帖子:[新项目][数学类]SUDOKU NQueens Project —— 枚举 N 皇后问题的解 官方站点:http://nqueens.ing.udec.cl/ 相关帖子:[新项目] [算法类] NQueens Project SHA-1 Collision Search Graz —— 寻找SHA-1算法的碰撞 官方站点:http://boinc.iaik.tugraz.at/sha1_coll_search/ 相关帖子:SHA-1 Collision Search Graz - 寻找SHA-1算法的碰撞 - 新手指南 |
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[求助]MORDELL定理问题
没人啊。。。刷墙涂鸦 |
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[原创]magma,/PARI/GP中文文档ECC初步:
超椭圆曲线: 有理域下定义超椭圆曲线: P<x> := PolynomialRing(RationalField()); > C := HyperellipticCurve(x^6+x^2+1); C; 复域下定义超椭圆曲线: P<x> := PolynomialRing(ComplexField()); > C := HyperellipticCurve(x^6+x^2+10); C; Hyperelliptic Curve defined by y^2 = x^6 + x^2 + 10.0000000000000000000000000000 over Complex field of precision 30 有限狱模17: P1<x> := PolynomialRing(GF(17)); C17 := HyperellipticCurve(x^6+x^2+1); C17; P2<x> := PolynomialRing(GF(17)); C1717 := HyperellipticCurve(x^16+x^5+x^2+1); C1717; 扭曲线: Twists(C17); 同构吗?: IsIsomorphic(C17, C1717); tws,auts := Twists(C17); tws; 是椭圆曲线吗? IsEllipticCurve(C) 亏格数: Genus(C); Genus(C17); Genus(C1717); 阶: Degree(C); Degree(C17); Degree(C1717); 判别式: Discriminant(C); Discriminant(C17); Discriminant(C1717); P3<x> := PolynomialRing(GF(17)); > C2 := HyperellipticCurve(x^6+x^2+x+1); J不变量: JInvariants(C2); 绝对不变量: AbsoluteInvariants(C2); 无限远点座标: PointsAtInfinity(C); PointsAtInfinity(C2); PointsAtInfinity(C17); 求点: Points(C : Bound := 1); Points(C17 : Bound := 1); Points(C2: Bound := 1); 自同构: Aut(C); Aut(C2); Aut(C17) ; Iso(C17, C2); 变为雅格布式 J1:=Jacobian(C); J1; J2:=Jacobian(C2); J2; J17:=Jacobian(C17); J17; 变为雅格布式后求阶 Order(J2); Order(J17); Curve(C17) 外尔下降算法分解: PP<x>:=PolynomialRing(GF(2)); > h := PP!1; > f := x^7 + x^4 + x^3 + 1; > J := Jacobian(HyperellipticCurve(f,h)); // a supersingular curve > Jext := BaseExtend(J, 41); > Factorization(#Jext); > m := 177722253954175633; // some big subgroup order > cofact := 3887047*7; > P := cofact*Random(Jext); > Q := 876213876263897634*P; // Q in <P> |
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[推荐]推荐PARI和SAGE和MXGMA
NFS筛法分解,80位的要64 M of RAM , 0.5 G of disk space 有个http://www.distributed.net/Download_clients,下了一路YES安好就可挣钱。。。。 要四部才能完成: 1.Sieving 2.Auxiliary data gathering 3.Linear algebra 4.Factorization 二次筛选法的工作原理 http://bbs.emath.ac.cn/thread-1439-1-1.html 用二次筛选法对一个数字 n 进行因数分解,就是要找到两个数字 x 和 y ,它们模 n 之后不相等,并且 x 和 -y 模 n 之后也不相等,但是 x 2 =y 2 (mod n) 。如果找到了这两个数字,那么就可以说 ( x+ y) ( x- y) = 0 (mod n) 。因此 x+ y 和 x- y 就一定与 n 具有相同的非平凡因数。 用二次筛选法进行因数分解有赖于是否能够找到一组数字,这组数的因数可以表示为一些预先选择的素数的乘积。然后用幂向量的形式将这些因数记录下来。一旦具备了足够多的向量,就可以构造一个包含线性依赖关系的集合。用这种线性依赖关系就可以找到两个平方后模 n 相等的数字。 为实现这一目标,二次筛选方法使用了一个素数集合,称作因数基( factor base)。然后,搜索出那些可以完全分解成这个因数基中的素数的数字。如果因数基中有 k 个素数,那么每一个可分解为因数基中素数的数字就存储为一个 k 维的向量,向量 y 中的第 i 个项表示因数基中第 i 个素数在 y 因数分解结果当中的幂。 最后进行筛选操作,找出 f( r) = r 2 - n 的那些值,而这些值因数分解的结果完全包含在这个因数基中。然后像 Dixon 因数分解法中那样应用高斯消除法,找出是某数的完全平方的一组 f( r) 。 难,贴着慢慢琢磨。。。 http://magma.maths.usyd.edu.au/magma/handbook/text/166 |
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[推荐]推荐PARI和SAGE和MXGMA
可用time 看速度: Z := IntegerRing(); n := 123456789011111222222222455555555555999999999999999999999999999999999999999999999999999999999999999999999999999999999; time Factorization(n); time PollardRho(n) ; time Divisors(n); time PartialFactorization([n]); time CoprimeBasis(([ 1, 8961811, 170036731, 1523837046279841, 81017054489197218714231866471, 726059530108887015842608997490338981, 13775875100591969884460009750757346301, 123456789011111222222222444444444444411111111 ] )); [ <4493, 1>, <1866358739047092109, 1>, <3233066218483497821, 1>, <4553746329843002811944884231128063084716920766552412595314945012903616003987, 1> ] Time: 5.690 [ <4493, 1> ] [ 27477584912332789277147219130993891831738259514800801246383262853327398174938\ 793679056309815268194969953260627643 ] Time: 0.010 [ 1, 4493, 1866358739047092109, 3233066218483497821, 8385549814538584845737, 14526166519646355709753, 6034061390784611392275850413087794489, 27111037828795258985495395906003460639077, 4553746329843002811944884231128063084716920766552412595314945012903616003987, 2045998225998461163406836485045838743963312500411998979075004794297594670591359\ 1, 8498924258106110314598224912416571356509386353570039615682567850288675773618\ 059783746900238583, 14722563426558624062843851217253269088818827496734439080768\ 194963873985715008238232114441812327, 38185666691670753643489824531487655104796\ 672886590187993261777351347020250865942608374822771953419, 6614847747552789791435742351911893801606299194282783478989149997268581781753201\ 4376890187062785211, 2747758491233278927714721913099389183173825951480080124638\ 3262853327398174938793679056309815268194969953260627643, 1234567890111112222222224555555555559999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999 ] Time: 0.010 [ [ [], [ <12345678901111122222222245555555555599999999999999999999999999999999\ 9999999999999999999999999999999999999999999999999, 1> ] ] ] Time: 0.000 [ <8961811, 4>, <170036731, 4>, <81017054489197218714231866471, 4> ] Time: 0.000 |
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[推荐]推荐PARI和SAGE和MXGMA
整数分解:PollardRho和Cremona 算法,快100位了 Z := IntegerRing(); n := 1234567890111112222222224555555555559999999999999999999999999999999999999999999999999999999999999999999999999999999994444444444441122222222222222222222111111777; Factorization(n); PollardRho(n) ; Divisors(n); PartialFactorization([n]); [ <37, 1>, <3336669973273276276276282582582582594594594594594594594594594594594\ 5945945945945945945945945945945945945945945945945795795795795706006006006006006\ 006003003021, 1> ] [ <37, 1>, <3336669973273276276276282582582582594594594594594594594594594594594\ 5945945945945945945945945945945945945945945945945795795795795706006006006006006\ 006003003021, 1> ] [] [ 1, 37, 3336669973273276276276282582582582594594594594594594594594594594594594\ 5945945945945945945945945945945945945945945945795795795795706006006006006006006\ 003003021, 12345678901111122222222245555555555599999999999999999999999999999999\ 9999999999999999999999999999999999999999999999999444444444444112222222222222222\ 2222111111777 ] [ [ [], [ <12345678901111122222222245555555555599999999999999999999999999999999\ 99999999999999999999999999999999999999999999999994444444444441122222222\ 222222222222111111777, 1> ] ] ] [ <8961811, 4>, <170036731, 4>, <81017054489197218714231866471, 4> ] |
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[推荐]推荐PARI和SAGE和MXGMA
整数分解:PollardRho和Cremona Z := IntegerRing(); n := 1234567890111112222211111111888888888888888888888888888888888888888888888888888888888888882222244444444444441122222222222222222222111111; Factorization(n); PollardRho(n) ; Divisors(n); PartialFactorization([n]); CoprimeBasis(([ 1, 8961811, 170036731, 1523837046279841, 81017054489197218714231866471, 726059530108887015842608997490338981, 13775875100591969884460009750757346301, 123456789011111222222222444444444444411111111 ] )); |
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