密码学Chapter4、5作业

Chapter4 · 思考题

4.2

分组密码：加密一个明文分组，通常得到的是与明文等长的密文分组
流密码：每次加密数据流得到一位或一字节

4.4

乘积密码：采用多个密码复合的方法，增强密码系统的安全性

4.5

混淆：增加密钥与密文之间关系的复杂性
扩散：小变动的影响可波及到全局

4.7

雪崩效应：保证明文或密钥微小的输入变化导致密文大的输出变化

Chapter5 · 思考题

5.1

一个集合以及定义在这个集合上的二元运算，满足封闭性、结合律、单位元。

5.2

一个集合以及定义在这个集合上的二元运算，满足封闭性、结合律、单位元、交换律。

5.3

定义在几何上具有2个二元运算的代数系统，满足若干公理。

5.4

例：
f(x)=x3+x2+2；g(x)=x2−x+1f(x)=x^3+x^2+2； g(x)=x^2-x+1 f(x)=x3+x2+2；g(x)=x2−x+1
则有
f(x)+g(x)=x3+2x2−x+3f(x)+g(x)=x^3+2x^2-x+3 f(x)+g(x)=x3+2x2−x+3

f(x)−g(x)=x3+x+1f(x)-g(x)=x^3+x+1 f(x)−g(x)=x3+x+1

f(x)⋅g(x)=x5+3x2−2x+2f(x)·g(x)=x^5+3x^2-2x+2 f(x)⋅g(x)=x5+3x2−2x+2

f(x)g(x)=(x+2)⋅(x2−x+1)⋯x\frac{f(x)}{g(x)}=(x+2)·(x^2-x+1) \cdots x g(x)f(x)=(x+2)⋅(x2−x+1)⋯x

Chapter5 · 习题

5.4

GF(5)加法GF(5)加法 GF(5)加法

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

GF(5)乘法GF(5)乘法 GF(5)乘法

*	1	2	3	4
0	0	0	0	0
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

5.7

Z10={0,1,2,3,4,5,6,7,8,9}Z_{10}=\{0,1,2,3,4,5,6,7,8,9\} Z10={0,1,2,3,4,5,6,7,8,9}

a.

(7x+2)−(x2+5)=7x+2−x2−5=[(−1)mod10]x2+7x+[(−3)mod10]=9x2+7x+7\begin{align*} (7x+2)-(x^2+5)&=7x+2-x^2-5 \\&=[(-1)mod10]x^2+7x+[(-3)mod10] \\&=9x^2+7x+7 \end{align*} (7x+2)−(x2+5)=7x+2−x2−5=[(−1)mod10]x2+7x+[(−3)mod10]=9x2+7x+7

b.

(6x2+x+3)⋅(5x2+2)=30x4+5x3+27x2+2x+6=(30mod10)x4+5x3+(27mod10)x2+2x+6=5x3+7x2+2x+6\begin{align*} (6x^2+x+3)·(5x^2+2)&=30x^4+5x^3+27x^2+2x+6 \\&=(30mod10)x^4+5x^3+(27mod10)x^2+2x+6 \\&=5x^3+7x^2+2x+6 \end{align*} (6x2+x+3)⋅(5x2+2)=30x4+5x3+27x2+2x+6=(30mod10)x4+5x3+(27mod10)x2+2x+6=5x3+7x2+2x+6

5.8

GF(2)加法GF(2)加法 GF(2)加法

+	0	1
0	0	1
1	1	0

GF(2)乘法GF(2)乘法 GF(2)乘法

*	0	1
0	0	0
1	0	1

求逆求逆求逆

w−ww−100−111\begin{array}{c|lcr} w & -w & w^{-1} \\ \hline 0 & 0 & - \\ 1 & 1 & 1 \\ \end{array} w01−w01w−1−1

a.

x3+1=(x+1){x2+[(−1)mod2]x+1}=(x+1)(x2+x+1)\begin{align*} x^3+1&=(x+1)\{x^2+[(-1)mod2]x+1\} \\&=(x+1)(x^2+x+1) \end{align*} x3+1=(x+1){x2+[(−1)mod2]x+1}=(x+1)(x2+x+1)

b.

该式无法因式分解
x3+x2+1=x2⋅(x+1)+1\begin{align*} x^3+x^2+1&=x^2·(x+1)+1 \end{align*} x3+x2+1=x2⋅(x+1)+1

c.

x4+1=x4+(2mod2)x2+1=(x2+1)2=[(x2+1+(2mod2)x]2=[(x+1)2]2=(x+1)4\begin{align*} x^4+1&=x^4+(2mod2)x^2+1 \\&=(x^2+1)^2 \\&=[(x^2+1+(2mod2)x]^2 \\&=[(x+1)^2]^2 \\&=(x+1)^4 \end{align*} x4+1=x4+(2mod2)x2+1=(x2+1)2=[(x2+1+(2mod2)x]2=[(x+1)2]2=(x+1)4

5.9

a.

w−ww−100−111\begin{array}{c|lcr} w & -w & w^{-1} \\ \hline 0 & 0 & - \\ 1 & 1 & 1 \\\end{array} w01−w01w−1−1

x3+x+1=(x+1)⋅(x2+x+1)+x\begin{align*} x^3+x+1=(x+1)·(x^2+x+1)+x \end{align*} x3+x+1=(x+1)⋅(x2+x+1)+x

x3+x2+1=x2⋅(x+1)+1\begin{align*} x^3+x^2+1=x^2·(x+1)+1 \end{align*} x3+x2+1=x2⋅(x+1)+1

gdc(x3+x+1,x3+x2+1)=1gdc(x^3+x+1,x^3+x^2+1)=1 gdc(x3+x+1,x3+x2+1)=1

b.

x3−x+1=(x+1)⋅(x2+2x)+1\begin{align*} x^3-x+1=(x+1)·(x^2+2x)+1 \end{align*} x3−x+1=(x+1)⋅(x2+2x)+1

x2+1=(x+1)2+x\begin{align*} x^2+1=(x+1)^2+x \end{align*} x2+1=(x+1)2+x

gcd(x3−x+1,x2+1)=1gcd(x^3-x+1,x^2+1)=1 gcd(x3−x+1,x2+1)=1

c.

x5+x4+x3−x2−x+1=(x+1)⋅{x4+x3+[(−2)modx]+1}=(x+1)⋅(x4+x3+x+1)=(x+1)2⋅(x3+1)=(x+1)3⋅(x2+2x+1)=(x+1)5\begin{align*} x^5+x^4+x^3-x^2-x+1&=(x+1)·\{x^4+x^3+[(-2)modx]+1\} \\&=(x+1)·(x^4+x^3+x+1) \\&=(x+1)^2·(x^3+1) \\&=(x+1)^3·(x^2+2x+1) \\&=(x+1)^5 \end{align*} x5+x4+x3−x2−x+1=(x+1)⋅{x4+x3+[(−2)modx]+1}=(x+1)⋅(x4+x3+x+1)=(x+1)2⋅(x3+1)=(x+1)3⋅(x2+2x+1)=(x+1)5

x3+x2+x+1=(x+1)⋅(x2+1)\begin{align*} x^3+x^2+x+1&=(x+1)·(x^2+1) \end{align*} x3+x2+x+1=(x+1)⋅(x2+1)

gcd(x5+x4+x3−x2−x+1,x3+x2+x+1)=x+1gcd(x^5+x^4+x^3-x^2-x+1,x^3+x^2+x+1)=x+1 gcd(x5+x4+x3−x2−x+1,x3+x2+x+1)=x+1

d.

(x5+88x4+73x3+83x2+51x+67)mod(x3+97x2+40x+38)=18x2+20x+28\begin{align*} &(x^5+88x^4+73x^3+83x^2+51x+67)mod(x^3+97x^2+40x+38) \\&=18x^2+20x+28 \end{align*} (x5+88x4+73x3+83x2+51x+67)mod(x3+97x2+40x+38)=18x2+20x+28

(x3+97x2+40x+38)mod(18x2+20x+28)=x+78\begin{align*} (x^3+97x^2+40x+38)mod(18x^2+20x+28)=x+78 \end{align*} (x3+97x2+40x+38)mod(18x2+20x+28)=x+78

gcd(x5+88x4+73x3+83x2+51x+67,x3+97x2+40x+38)=x+78gcd(x^5+88x^4+73x^3+83x^2+51x+67,x^3+97x^2+40x+38)=x+78 gcd(x5+88x4+73x3+83x2+51x+67,x3+97x2+40x+38)=x+78

5.10

以x2+x+1为模的有限域GF(4)对应表以x^2+x+1为模的有限域GF(4)对应表以x2+x+1为模的有限域GF(4)对应表

+		0	01	10	11
		0	1	x	x+1
00	0	0	1	x	x+1
01	1	1	0	x+1	x
10	x	x	x+1	0	1
11	x+1	x+1	x	1	0

*****		01	10	11
		1	x	x+1
00	0	0	0	0
01	1	1	x	x+1
10	x	x	x+1	1
11	x+1	x+1	1	x

5.11

x3+x+1的乘法逆元，以m(x)=x4+x+1为模x^3+x+1的乘法逆元，以m(x)=x^4+x+1为模 x3+x+1的乘法逆元，以m(x)=x4+x+1为模

生成元
幂多项式二进制十六进制0000000x0100011x1x00102x2x+100113x3x201004x4x2+101015x5x2+x01106x6x2+x+101117x7x310008x8x3+110019x9x3+x1010Ax10x3+x+11011Bx11x3+x21100Cx12x3+x2+11101Dx13x3+x2+x1110Ex14x3+x2+x+11111F\begin{array}{|c|l|c|r|} \hline幂 &多项式 &二进制 &十六进制 \\\hline 0 & 0 &0000 &0 \\\hline x^0 & 1 & 0001 & 1 \\\hline x^1 & x & 0010 & 2 \\\hline x^2 & x+1 & 0011 & 3 \\\hline x^3 & x^2& 0100 & 4 \\\hline x^4 & x^2+1& 0101 & 5 \\\hline x^5 & x^2+x& 0110 & 6 \\\hline x^6 & x^2+x+1 & 0111 & 7 \\\hline x^7 & x^3& 1000 & 8 \\\hline x^8 & x^3+1& 1001 & 9 \\\hline x^9 & x^3+x& 1010 & A \\\hline x^{10} & x^3+x+1 & 1011 & B \\\hline x^{11} & x^3+x^2 & 1100 & C \\\hline x^{12} & x^3+x^2+1 & 1101 & D \\\hline x^{13} & x^3+x^2+x & 1110 & E \\\hline x^{14} & x^3+x^2+x+1 & 1111 & F \\\hline \end{array} 幂0x0x1x2x3x4x5x6x7x8x9x10x11x12x13x14多项式01xx+1x2x2+1x2+xx2+x+1x3x3+1x3+xx3+x+1x3+x2x3+x2+1x3+x2+xx3+x2+x+1二进制0000000100100011010001010110011110001001101010111100110111101111十六进制0123456789ABCDEF

∗00000001001000110100010101100111100010011010101111001101111011110x0x1x2x3x4x5x6x7x8x9x10x11x12x13x1400000−0000000000000000001x001xx+1x2x2+1x2+xx2+x+1x3x3+1x3+xx3+x+1x3+x2x3+x2+1x3+x2+xx3+x2+x+10010x10xx2x2+xx3x3+xx3+x2+x1xx2x2+1x3x3+xx3+x2x3+x2x3+x2+x0011x20x+1x2+10100x30x2x2+10101x40x2+1x2+10110x50x2+xx2+10111x60x2+x+1x2+11000x70x3x2+11001x80x3+1x2+11010x90x3+xx2+11011x100x3+x+11100x110x3+x21101x120x3+x21110x130x3+x2+x1111x140x3+x2+x+1\begin{array}{cl|cr} * && 0000 & 0001 & 0010 &0011 & 0100 & 0101 & 0110 &0111 &1000& 1001 &1010&1011&1100&1101 & 1110 & 1111\\&&0 &x^0 &x^1 &x^2 &x^3 & x^4 & x^5 & x^6 & x^7 & x^8 & x^9 & x^{10} & x^{11} & x^{12} & x^{13} & x^{14}\\ \hline0000 & 0 & - & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 \\0001 & x^0 & 0 & 1 &x&x+1 &x^2 & x^2+1 & x^2+x & x^2+x+1 & x^3 & x^3+1 & x^3+x & x^3+x+1 & x^3+x^2 & x^3+x^2+1 & x^3+x^2+x & x^3+x^2+x+1\\0010 & x^1 & 0 & x & x^2 &x^2+x&x^3&x^3+x&x^3+x^2+x&1&x&x^2&x^2+1&x^3&x^3+x&x^3+x^2&x^3+x^2&x^3+x^2+x\\0011 & x^2 & 0 &x+1 &&x^2+1&&&&&&&&&&&\\0100 & x^3 & 0 &x^2 &&&&x^2+1&&&&&&&&\\0101 & x^4 & 0 & x^2+1 &&&x^2+1&&&&&&&&\\0110 & x^5 & 0 & x^2+x &&&&&x^2+1&&&&&\\0111 & x^6 & 0 & x^2+x+1 &&&&&&&&&x^2+1\\1000 & x^7 & 0 &x^3 &&&&&&&&&x^2+1\\1001 & x^8 & 0 & x^3+1 &&&&&&&&x^2+1\\1010 & x^9 & 0 &x^3+x &&&&&&&x^2+1\\1011 & x^{10} & 0 &x^3+x+1 &&&&&\\1100 & x^{11} & 0 &x^3+x^2 &&&&\\1101 & x^{12} & 0 & x^3+x^2 &&&\\1110 & x^{13} & 0 &x^3+x^2+x &&\\1111 & x^{14} & 0 &x^3+x^2+x+1&\end{array} ∗00000001001000110100010101100111100010011010101111001101111011110x0x1x2x3x4x5x6x7x8x9x10x11x12x13x1400000−0000000000000000001x001xx+1x2x2+1x2+xx2+x+1x3x3+1x3+xx3+x+1x3+x2x3+x2x3+x2+xx3+x2+x+10010x10xx20011x20x+1x2+xx2+10100x30x2x3x2+10101x40x2+1x3+xx2+10110x50x2+xx3+x2+xx2+10111x60x2+x+111000x70x3xx2+11001x80x3+1x2x2+11010x90x3+xx2+1x2+1x2+11011x100x3+x+1x31100x110x3+x2x3+x1101x120x3+x2+1x3+x21110x130x3+x2+xx3+x21111x140x3+x2+x+1x3+x2+x

gcd(x3+x+1,x4+x+1)=x2+1x2+1gcd(x^3+x+1,x^4+x+1)=x^2+1 \\x^2+1 gcd(x3+x+1,x4+x+1)=x2+1x2+1

5.12

幂多项式二进制十六进制0000000x0100011x1x00102x2x+100113x3x201004x4x2+101015x5x2+x01106x6x2+x+101117x7x310008x8x3+110019x9x3+x1010Ax10x3+x+11011Bx11x3+x21100Cx12x3+x2+11101Dx13x3+x2+x1110Ex14x3+x2+x+11111F\begin{array}{|c|l|c|r|} \hline幂 &多项式 &二进制 &十六进制\\\hline 0 & 0 &0000 &0\\\hline x^0 & 1 & 0001 & 1\\\hline x^1 & x & 0010 & 2\\\hline x^2 & x+1 & 0011 & 3\\\hline x^3 & x^2& 0100 & 4\\\hline x^4 & x^2+1& 0101 & 5\\\hline x^5 & x^2+x& 0110 & 6\\\hline x^6 & x^2+x+1 & 0111 & 7\\\hline x^7 & x^3& 1000 & 8\\\hline x^8 & x^3+1& 1001 & 9\\\hline x^9 & x^3+x& 1010 & A\\\hline x^{10} & x^3+x+1 & 1011 & B\\\hline x^{11} & x^3+x^2 & 1100 & C\\\hline x^{12} & x^3+x^2+1 & 1101 & D\\\hline x^{13} & x^3+x^2+x & 1110 & E\\\hline x^{14} & x^3+x^2+x+1 & 1111 & F\\\hline\end{array} 幂0x0x1x2x3x4x5x6x7x8x9x10x11x12x13x14多项式01xx+1x2x2+1x2+xx2+x+1x3x3+1x3+xx3+x+1x3+x2x3+x2+1x3+x2+xx3+x2+x+1二进制0000000100100011010001010110011110001001101010111100110111101111十六进制0123456789ABCDEF