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[分享]LFSR和m序列

发表于: 2011-12-20 14:40 5219

[分享]LFSR和m序列

lilianjie

2011-12-20 14:40

5219

众多的序列产生法：
http://www.newwaveinstruments.com/products/index.htm

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Direct Sequence Generators
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PN Code Generators
PN Sequence Generators
PRBS Generators
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Pseudo-Noise Code Generators
Pseudo-Noise Sequence Generators
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Pseudonoise Code Generators
Pseudonoise Sequence Generators
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An Awesome Generator

http://www.xilinx.com/support/documentation/application_notes/xapp210.pdf
http://homepage.mac.com/afj/lfsr.html

http://homepage.mac.com/afj/lfsr.html

B-M算法是流密码中用的一种求取LSRE，具体就是，对于二进制序列a，计算求得f（x），能够产生a序列并且级数最小的线性移位寄存器的反馈多项式

把5个寄存器的试了两种，都不是32-1=31，都不是m序列
S:= [GF(2)| 1,1,0,0,1];
S;

BerlekampMassey(S);
ConnectionPolynomial(S);
CharacteristicPolynomial(S);

C<D>, L := BerlekampMassey(S);
C;
LFSRStep(C, S);
LFSRSequence(C, S, 100);

[ 1, 1, 0, 0, 1 ]
$.1^3 + $.1^2 + 1
3
$.1^3 + $.1^2 + 1
3
$.1^3 + $.1^2 + 1
3
D^3 + D^2 + 1
[ 1, 0, 0, 1, 0 ]
[ 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1,
0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0,
1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0,
0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1 ]

S:= [GF(2)| 1,0,0,1,1];
S;

BerlekampMassey(S);
ConnectionPolynomial(S);
CharacteristicPolynomial(S);

C<D>, L := BerlekampMassey(S);
C;
LFSRStep(C, S);
LFSRSequence(C, S, 100);

[ 1, 0, 0, 1, 1 ]
$.1^3 + $.1 + 1
3
$.1^3 + $.1 + 1
3
$.1^3 + $.1 + 1
3
D^3 + D + 1
[ 0, 0, 1, 1, 1 ]
[ 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1,
1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1,
1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0,
1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0 ]

[ 1, 0, 0, 1, 0 ]
$.1^3 + 1
3
$.1^3 + 1
3
$.1^3 + 1
3
D^3 + 1
[ 0, 0, 1, 0, 0 ]
[ 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 ]

[ 1, 0, 1, 1, 0 ]
$.1^2 + $.1 + 1
2
$.1^2 + $.1 + 1
2
$.1^2 + $.1 + 1
2
D^2 + D + 1
[ 0, 1, 1, 0, 1 ]
[ 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1 ]

只有这个31步=2^5-1 周期最大可分圆多项式=特征多项式D^2 + D + 1。。。最高次不是5？？？？

[ 1, 1, 0, 1, 0 ]
$.1^3 + $.1 + 1
3
$.1^3 + $.1 + 1
3
$.1^3 + $.1 + 1
3
D^3 + D + 1
[ 1, 0, 1, 0, 0 ]
[ 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0,
0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1,
0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0,
1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1 ]

[ 1, 1, 1, 1, 0 ]
$.1^4 + $.1 + 1
4
$.1^4 + $.1 + 1
4
$.1^4 + $.1 + 1
4
D^4 + D + 1
[ 1, 1, 1, 0, 1 ]
[ 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0,
1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1,
1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0,
1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0 ]

=============

多项式$x^n-1$分解,它所分解得到的不可约多项式称为分圆多项式.事实上,分圆多项式的定义可以用以下的方式来得到:设ε是$x^n-1=0$的一个根,即ε是n次单位根,如果对任意的自然数k<n,ε都不是$x^k-1=0$的根,那么称ε为n次本原单位根.由所有n次本原单位根构成的多项式就称为n次分圆多项式.

n次分圆多项式=(x^n-1)/LCM(x^d-1),       ,0d<n ,d|n

Zp[x]中p^n-1次分圆多项式的每个不可约式都是是n次本原的，数量为（p^n-1次分圆多项式/n）

难就是p可为2,3。。。进制上的域

试求上面周期31的反馈函数：假设已知5个寄存器（当成m）序列。。。0,  1,  1,  0, 1,  1, 0,  1, 1,  0,  1, 1, 0, 1, 1, 0, 1
                                                            a1a2a3 a4  a5a6 a7a8a9a10

[ 0, 1, 1, 0, 1 ]
[ 1, 0, 1, 1, 0, 1, 1, 0, 1,1,0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1 ]

(a6,a7,a8,a9,a10)=(c5,c,4,c3,c2,c1)= Matrix(GF(2), 5, 5, [0, 1, 1, 0, 1, 1,1,0,1,1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,1]);
X;
X^-1;

[0 1 1 0 1]
[1 1 0 1 1]
[1 0 1 1 0]
[0 1 1 0 1]
[1 1 0 1 1]

>> X^-1;
^
Runtime error in '^': Argument 1 is not invertible

没逆距阵？？？肯定不是m序列-----》周期能最大，可还是可能不是m序列

次数对应的分圆多项式
1 x-1
2 x+1
3 x2+x+1 4 x2+1
5 x4+x3+x2+x+1
6 x2-x+1
7 x6+x5+x4+x3+x2+x+1
8 x4+1
9 x6+x3+1
10 x4-x3+x2-x+1
11 x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1
12 x4-x2+1

[招生]科锐逆向工程师培训(2024年11月15日实地，远程教学同时开班, 第51期)

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lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 2 楼 p<x> := PolynomialRing(Integers()); > f := x^38-1; > f; Rank(p); a:=Factorization(f) ; a; b:=Factorisation(f^27); b; c:=Factorisation(f^279f); c; f1:=x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 -x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1; f1; IsIrreducible(f); IsIrreducible(f1); Degree(f); Degree(f1); Discriminant(f) ; Discriminant(f1) ; CompanionMatrix(f) ; CompanionMatrix(f1) ; F<i> := PolynomialRing(GF(5)); i := x^2-2x+2; i; IsSeparable(F!i) ; Factorization(F!i); F<k> := PolynomialRing(GF(5)); k := x^2-3x+2; k; IsSeparable(F!k) ; Factorization(F!k); F<g> := PolynomialRing(GF(2)); g := x^2-2x+2; g; IsSeparable(F!g) ; Factorization(F!g); F<m> := PolynomialRing(GF(2^2)); m := x^2-2x+2; m; IsSeparable(F!m) ; Factorization(F!m); F<n> := PolynomialRing(GF(2^103)); n := x^2-2x+2; n; IsSeparable(F!n) ; Factorization(F!n); x^38 - 1 1 [ <x - 1, 1>, <x + 1, 1>, <x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1>, <x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1> ] [ <x - 1, 27>, <x + 1, 27>, <x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 27>, <x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 27> ] [ <x - 1, 280>, <x + 1, 280>, <x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 280>, <x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 280> ] x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 false true 38 18 1075911801979993982060429252856123779115487368830416064610304 -5480386857784802185939 [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1] x^2 - 2x + 2 true [ <i + 1, 1>, <i + 2, 1> ] x^2 - 3x + 2 true [ <k + 3, 1>, <k + 4, 1> ] x^2 - 2x + 2 false [ <g, 2> ] x^2 - 2x + 2 false [ <m, 2> ] x^2 - 2x + 2 false [ <n, 2> ] P<x> := PolynomialRing(GF(2)); PrimePolynomials(P,2); NumberOfPrimePolynomials(P, 2); NumberOfPrimePolynomials(P, 3); P<Z> := PolynomialRing(GF(2^4)); PrimePolynomials(P,2); NumberOfPrimePolynomials(P, 2); NumberOfPrimePolynomials(P, 3); f3 := x^3+x^2+13; f3; HasPolynomialFactorization(P) ; f30 := x^30-1; f30; HasPolynomialFactorization(P) ; [ x^2 + x + 1 ] 1 2 [ Z^2 + $.1Z + $.1^8, Z^2 + $.1^2Z + $.1, Z^2 + $.1Z + $.1^9, Z^2 + $.1^4Z + $.1^2, Z^2 + $.1^6Z + $.1^10, Z^2 + $.1^2Z + $.1^3, Z^2 + Z + $.1^11, Z^2 + $.1^8Z + $.1^4, Z^2 + $.1^12Z + $.1^12, Z^2 + $.1^12Z + $.1^5, Z^2 + $.1^7Z + $.1^13, Z^2 + $.1^4Z + $.1^6, Z^2 + $.1^10Z + $.1^14, Z^2 + Z + $.1^7, Z^2 + $.1^9Z + 1, Z^2 + $.1^9Z + $.1^9, Z^2 + $.1^10Z + $.1^2, Z^2 + $.1^9Z + $.1^10, Z^2 + $.1^12Z + $.1^3, Z^2 + $.1^14Z + $.1^11, Z^2 + $.1^10Z + $.1^4, Z^2 + $.1^8Z + $.1^12, Z^2 + $.1Z + $.1^5, Z^2 + $.1^5Z + $.1^13, Z^2 + $.1^5Z + $.1^6, Z^2 + Z + $.1^14, Z^2 + $.1^12Z + $.1^7, Z^2 + $.1^3Z + 1, Z^2 + $.1^8Z + $.1^8, Z^2 + $.1^2Z + $.1^10, Z^2 + $.1^3Z + $.1^3, Z^2 + $.1^2Z + $.1^11, Z^2 + $.1^5Z + $.1^4, Z^2 + $.1^7Z + $.1^12, Z^2 + $.1^3Z + $.1^5, Z^2 + $.1Z + $.1^13, Z^2 + $.1^9Z + $.1^6, Z^2 + $.1^13Z + $.1^14, Z^2 + $.1^13Z + $.1^7, Z^2 + $.1^8Z + 1, Z^2 + $.1^5Z + $.1^8, Z^2 + $.1^11Z + $.1, Z^2 + $.1^10Z + $.1^11, Z^2 + $.1^11Z + $.1^4, Z^2 + $.1^10Z + $.1^12, Z^2 + $.1^13Z + $.1^5, Z^2 + Z + $.1^13, Z^2 + $.1^11Z + $.1^6, Z^2 + $.1^9Z + $.1^14, Z^2 + $.1^2Z + $.1^7, Z^2 + $.1^6Z + 1, Z^2 + $.1^6Z + $.1^8, Z^2 + $.1Z + $.1, Z^2 + $.1^13Z + $.1^9, Z^2 + $.1^3Z + $.1^12, Z^2 + $.1^4Z + $.1^5, Z^2 + $.1^3Z + $.1^13, Z^2 + $.1^6Z + $.1^6, Z^2 + $.1^8Z + $.1^14, Z^2 + $.1^4Z + $.1^7, Z^2 + $.1^2Z + 1, Z^2 + $.1^10Z + $.1^8, Z^2 + $.1^14Z + $.1, Z^2 + $.1^14Z + $.1^9, Z^2 + $.1^9Z + $.1^2, Z^2 + $.1^11Z + $.1^13, Z^2 + $.1^12Z + $.1^6, Z^2 + $.1^11Z + $.1^14, Z^2 + $.1^14Z + $.1^7, Z^2 + $.1Z + 1, Z^2 + $.1^12Z + $.1^8, Z^2 + $.1^10Z + $.1, Z^2 + $.1^3Z + $.1^9, Z^2 + $.1^7Z + $.1^2, Z^2 + $.1^7Z + $.1^10, Z^2 + $.1^4Z + $.1^14, Z^2 + $.1^5Z + $.1^7, Z^2 + $.1^4Z + 1, Z^2 + $.1^7Z + $.1^8, Z^2 + $.1^9Z + $.1, Z^2 + $.1^5Z + $.1^9, Z^2 + $.1^3Z + $.1^2, Z^2 + $.1^11Z + $.1^10, Z^2 + Z + $.1^3, Z^2 + $.1^12Z + 1, Z^2 + $.1^13Z + $.1^8, Z^2 + $.1^12Z + $.1, Z^2 + Z + $.1^9, Z^2 + $.1^2Z + $.1^2, Z^2 + $.1^13Z + $.1^10, Z^2 + $.1^11Z + $.1^3, Z^2 + $.1^4Z + $.1^11, Z^2 + $.1^5Z + $.1, Z^2 + $.1^6Z + $.1^9, Z^2 + $.1^5Z + $.1^2, Z^2 + $.1^8Z + $.1^10, Z^2 + $.1^10Z + $.1^3, Z^2 + $.1^6Z + $.1^11, Z^2 + $.1^4Z + $.1^4, Z^2 + $.1^13Z + $.1^2, Z^2 + $.1^14Z + $.1^10, Z^2 + $.1^13Z + $.1^3, Z^2 + $.1Z + $.1^11, Z^2 + $.1^3Z + $.1^4, Z^2 + $.1^14Z + $.1^12, Z^2 + $.1^6Z + $.1^3, Z^2 + $.1^7Z + $.1^11, Z^2 + $.1^6Z + $.1^4, Z^2 + $.1^9Z + $.1^12, Z^2 + $.1^11Z + $.1^5, Z^2 + $.1^14Z + $.1^4, Z^2 + Z + $.1^12, Z^2 + $.1^14Z + $.1^5, Z^2 + $.1^2Z + $.1^13, Z^2 + $.1^7Z + $.1^5, Z^2 + $.1^8Z + $.1^13, Z^2 + $.1^7Z + $.1^6, Z^2 + Z + $.1^6, Z^2 + $.1Z + $.1^14, Z^2 + $.1^8*Z + $.1^7 ] 120 1360 x^3 + x^2 + 1 true x^30 + 1 true 2011-12-21 12:33 0
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lilianjie

雪币： 433

活跃值： (45)

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( LV4，RANK：50 )

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lilianjie 1: 2 楼

p<x> := PolynomialRing(Integers());
> f := x^38-1;
> f;

Rank(p);
a:=Factorization(f) ;
a;
b:=Factorisation(f^27);
b;
c:=Factorisation(f^279*f);
c;
f1:=x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 -x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1;
f1;
IsIrreducible(f);
IsIrreducible(f1);

Degree(f);
Degree(f1);

Discriminant(f) ;
Discriminant(f1) ;

CompanionMatrix(f) ;
CompanionMatrix(f1) ;

F<i> := PolynomialRing(GF(5));
i := x^2-2*x+2;
i;
IsSeparable(F!i) ;
Factorization(F!i);
F<k> := PolynomialRing(GF(5));
k := x^2-3*x+2;
k;
IsSeparable(F!k) ;
Factorization(F!k);

F<g> := PolynomialRing(GF(2));
g := x^2-2*x+2;
g;
IsSeparable(F!g) ;
Factorization(F!g);
F<m> := PolynomialRing(GF(2^2));
m := x^2-2*x+2;
m;
IsSeparable(F!m) ;
Factorization(F!m);
F<n> := PolynomialRing(GF(2^103));
n := x^2-2*x+2;
n;
IsSeparable(F!n) ;
Factorization(F!n);

x^38 - 1
1
[
<x - 1, 1>,
<x + 1, 1>,
<x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 -
      x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1>,
<x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 +
      x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1>
]
[
<x - 1, 27>,
<x + 1, 27>,
<x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 -
      x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 27>,
<x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 +
      x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 27>
]
[
<x - 1, 280>,
<x + 1, 280>,
<x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 -
      x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 280>,
<x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 +
      x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 280>
]
x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 +
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
false
true
38
18
1075911801979993982060429252856123779115487368830416064610304
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[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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[ 0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0]
[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1]
[-1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1]
x^2 - 2*x + 2
true
[
<i + 1, 1>,
<i + 2, 1>
]
x^2 - 3*x + 2
true
[
<k + 3, 1>,
<k + 4, 1>
]
x^2 - 2*x + 2
false
[
<g, 2>
]
x^2 - 2*x + 2
false
[
<m, 2>
]
x^2 - 2*x + 2
false
[
<n, 2>
]

P<x> := PolynomialRing(GF(2));
PrimePolynomials(P,2);
NumberOfPrimePolynomials(P, 2);
NumberOfPrimePolynomials(P, 3);
P<Z> := PolynomialRing(GF(2^4));

PrimePolynomials(P,2);
NumberOfPrimePolynomials(P, 2);
NumberOfPrimePolynomials(P, 3);

f3 := x^3+x^2+13;
f3;
HasPolynomialFactorization(P) ;
f30 := x^30-1;
f30;
HasPolynomialFactorization(P) ;

[
x^2 + x + 1
]
1
2
[
Z^2 + $.1*Z + $.1^8,
Z^2 + $.1^2*Z + $.1,
Z^2 + $.1*Z + $.1^9,
Z^2 + $.1^4*Z + $.1^2,
Z^2 + $.1^6*Z + $.1^10,
Z^2 + $.1^2*Z + $.1^3,
Z^2 + Z + $.1^11,
Z^2 + $.1^8*Z + $.1^4,
Z^2 + $.1^12*Z + $.1^12,
Z^2 + $.1^12*Z + $.1^5,
Z^2 + $.1^7*Z + $.1^13,
Z^2 + $.1^4*Z + $.1^6,
Z^2 + $.1^10*Z + $.1^14,
Z^2 + Z + $.1^7,
Z^2 + $.1^9*Z + 1,
Z^2 + $.1^9*Z + $.1^9,
Z^2 + $.1^10*Z + $.1^2,
Z^2 + $.1^9*Z + $.1^10,
Z^2 + $.1^12*Z + $.1^3,
Z^2 + $.1^14*Z + $.1^11,
Z^2 + $.1^10*Z + $.1^4,
Z^2 + $.1^8*Z + $.1^12,
Z^2 + $.1*Z + $.1^5,
Z^2 + $.1^5*Z + $.1^13,
Z^2 + $.1^5*Z + $.1^6,
Z^2 + Z + $.1^14,
Z^2 + $.1^12*Z + $.1^7,
Z^2 + $.1^3*Z + 1,
Z^2 + $.1^8*Z + $.1^8,
Z^2 + $.1^2*Z + $.1^10,
Z^2 + $.1^3*Z + $.1^3,
Z^2 + $.1^2*Z + $.1^11,
Z^2 + $.1^5*Z + $.1^4,
Z^2 + $.1^7*Z + $.1^12,
Z^2 + $.1^3*Z + $.1^5,
Z^2 + $.1*Z + $.1^13,
Z^2 + $.1^9*Z + $.1^6,
Z^2 + $.1^13*Z + $.1^14,
Z^2 + $.1^13*Z + $.1^7,
Z^2 + $.1^8*Z + 1,
Z^2 + $.1^5*Z + $.1^8,
Z^2 + $.1^11*Z + $.1,
Z^2 + $.1^10*Z + $.1^11,
Z^2 + $.1^11*Z + $.1^4,
Z^2 + $.1^10*Z + $.1^12,
Z^2 + $.1^13*Z + $.1^5,
Z^2 + Z + $.1^13,
Z^2 + $.1^11*Z + $.1^6,
Z^2 + $.1^9*Z + $.1^14,
Z^2 + $.1^2*Z + $.1^7,
Z^2 + $.1^6*Z + 1,
Z^2 + $.1^6*Z + $.1^8,
Z^2 + $.1*Z + $.1,
Z^2 + $.1^13*Z + $.1^9,
Z^2 + $.1^3*Z + $.1^12,
Z^2 + $.1^4*Z + $.1^5,
Z^2 + $.1^3*Z + $.1^13,
Z^2 + $.1^6*Z + $.1^6,
Z^2 + $.1^8*Z + $.1^14,
Z^2 + $.1^4*Z + $.1^7,
Z^2 + $.1^2*Z + 1,
Z^2 + $.1^10*Z + $.1^8,
Z^2 + $.1^14*Z + $.1,
Z^2 + $.1^14*Z + $.1^9,
Z^2 + $.1^9*Z + $.1^2,
Z^2 + $.1^11*Z + $.1^13,
Z^2 + $.1^12*Z + $.1^6,
Z^2 + $.1^11*Z + $.1^14,
Z^2 + $.1^14*Z + $.1^7,
Z^2 + $.1*Z + 1,
Z^2 + $.1^12*Z + $.1^8,
Z^2 + $.1^10*Z + $.1,
Z^2 + $.1^3*Z + $.1^9,
Z^2 + $.1^7*Z + $.1^2,
Z^2 + $.1^7*Z + $.1^10,
Z^2 + $.1^4*Z + $.1^14,
Z^2 + $.1^5*Z + $.1^7,
Z^2 + $.1^4*Z + 1,
Z^2 + $.1^7*Z + $.1^8,
Z^2 + $.1^9*Z + $.1,
Z^2 + $.1^5*Z + $.1^9,
Z^2 + $.1^3*Z + $.1^2,
Z^2 + $.1^11*Z + $.1^10,
Z^2 + Z + $.1^3,
Z^2 + $.1^12*Z + 1,
Z^2 + $.1^13*Z + $.1^8,
Z^2 + $.1^12*Z + $.1,
Z^2 + Z + $.1^9,
Z^2 + $.1^2*Z + $.1^2,
Z^2 + $.1^13*Z + $.1^10,
Z^2 + $.1^11*Z + $.1^3,
Z^2 + $.1^4*Z + $.1^11,
Z^2 + $.1^5*Z + $.1,
Z^2 + $.1^6*Z + $.1^9,
Z^2 + $.1^5*Z + $.1^2,
Z^2 + $.1^8*Z + $.1^10,
Z^2 + $.1^10*Z + $.1^3,
Z^2 + $.1^6*Z + $.1^11,
Z^2 + $.1^4*Z + $.1^4,
Z^2 + $.1^13*Z + $.1^2,
Z^2 + $.1^14*Z + $.1^10,
Z^2 + $.1^13*Z + $.1^3,
Z^2 + $.1*Z + $.1^11,
Z^2 + $.1^3*Z + $.1^4,
Z^2 + $.1^14*Z + $.1^12,
Z^2 + $.1^6*Z + $.1^3,
Z^2 + $.1^7*Z + $.1^11,
Z^2 + $.1^6*Z + $.1^4,
Z^2 + $.1^9*Z + $.1^12,
Z^2 + $.1^11*Z + $.1^5,
Z^2 + $.1^14*Z + $.1^4,
Z^2 + Z + $.1^12,
Z^2 + $.1^14*Z + $.1^5,
Z^2 + $.1^2*Z + $.1^13,
Z^2 + $.1^7*Z + $.1^5,
Z^2 + $.1^8*Z + $.1^13,
Z^2 + $.1^7*Z + $.1^6,
Z^2 + Z + $.1^6,
Z^2 + $.1*Z + $.1^14,
Z^2 + $.1^8*Z + $.1^7
]
120
1360
x^3 + x^2 + 1
true
x^30 + 1
true

2011-12-21 12:33

lilianjie

141

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