amultiply x by x7 x5 x3 1 in gf28mod x8

Question

A.Multiply x by x⁷ + x⁵ + x³ + 1 in GF(2⁸)mod x⁸+ x⁴+x³+x+1.

B. i) Verify that x⁶ + x is the inverse of x⁵ + x⁴+x²+ x +1in GF (2⁸) mod x⁸+ x⁴+x³+x+1.

ii) Using the given matrices A and B for the affine transformation AY+ B, (i), and the input byte 0011 0111(37 in hex), compute the corresponding entry in the RijndaelS-box.

C.  Apply the Shift Row transformation of the Rijndael Algorithm to the following state:

87	F2	4D	97
EC	6E	4C	90
4A	C3	46	E7
8C	D8	95	A6

D.  Use the Blum-Blum-Shub pseudorandom number generator to create a sequence of 6 bits, using p = 11, q = 13 and s = 3 (seed= x₀).                     

E. Use the Chinese Remainder Theorem to solve for x if: 

x ≡ 2 (mod 5),  x ≡ 3 (mod 13), and   x ≡ 1 (mod 7).                            

F. Given p = 17, q = 11, e = 7, Using the RSA algorithm,

a) Find n and d.  Find the public key and private key.

b) Encrypt m = 6.

c)  Decrypt c = 2.

G.  Compute 6⁶⁶⁶ mod 11 using Fermat's Little Theorem.

H.  Compute 5¹²³ mod 13 using Euler's Theorem.

I.  Use Fermat's Test for primality to test the following numbers:

a) n = 31

b) n = 187

J.  Complete the following table of values of 2^x mod 21:

x	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2x

Solve for x: 

a)    2^x ≡ 8 mod 21     L₂(8) =

b) 2^x ≡ 11 mod 21   L₂(11) =

K. a) Is 2 a primitive root of 7?Explain.

x
2x

b) Is 3 a primitive root of 7? Explain.

x
3x