MATH/CSCI 4116
math代写 – 这是值得参考的math代写的题目
Cryptography
assignment 7
- Which of the following polynomials are irreducible in (Z/ 2 Z)[x]?
x
5
+x
3
+x
2
+ 1,
x
5
+x
2
+ 1,
x
5
+x
4
+ 1.
- (a) Show that, in (Z/ 2 Z)[x],
x
4
x+ 1 (modx
4
+x+ 1),
x
8
x
2
+ 1 (modx
4
+x+ 1),
x
16
x (mod x
4
+x+ 1).
(b) Show thatx
15
1 (modx
4
+x+ 1) in (Z/ 2 Z)[x].
- The finite field GF(
5
) can be constructed as (Z/ 2 Z)[x]/(x
5
+x
2
+ 1).
Compute (x
4
+x
3
)(x
3
+x
2
+ 1) in this field.
- Multiply the following elements of GF(
8
), represented as bytes:
00000111 10101011.
(Use the standard Rijndael irreducible polynomial. This question should
only be done after the class of Thursday, March 9).
- (a) Show thatx
2
+ 1 is irredubible in (Z/ 3 Z)[x].
(b) Use this to construct the finite field GF(
2
) with its addition and
multiplication tables.
(Careful: Here we havep= 3, and notp= 2 as usual.)
Due: Thursday, March 16, 2023