Special Problem
Use the exponentiation algorithm to compute 4126 mod 127
x = 4 , e = 126, m = 127
prod = 1
prod = 1
x = x2 = 42 = 16 mod 127 = 16; e = 126/2 = 63
while (e > 1) cfw_
prod = prodx = 116 = 16
x = x2 = 162 = 256 mod 127 = 2; e = 63/2 = 31
if
Problem 1
(a) Define: the prime subfield of finite field F
The smallest subfield containing the multiplicative identity of
F
(b) Define: primitive element of finite field K
An element whose first n-1 powers are distinct, where n is the
order of the field;
Problem 1
(a) Define: the prime subfield of finite field F
The smallest subfield containing the multiplicative identity of
F
(b) Define: primitive element of finite field K
An element whose first n-1 powers are distinct, where n is the
order of the field;
Advanced Discrete Exam 4 Name (Print) _
Do all work on this exam.
For questions 1 to 5, let f(x) = x3 + x2 + 1 and GF8 = GF2[x]/f(x). The table of powers of in
GF8 is given on the last page of this exam, which you should detach at this time.
1. Prove that
Public Key Homework Problems
1 The ciphertext 5859 was obtained from the RSA algorithm using
m = 11413 and e = 7457. Using the factorization 11413 = 101113,
find the plaintext.
= 100112 = 11200
d = e-1 mod = e-1 mod 11200 = 5793
decoded message = 5859579
Problem 1
(a) State Fermats Little Theorem
(b) Define (n) (Eulers Phi Function)
(c) State Eulers Theorem
See slides or text
(e) Define: primitive root of a prime p
(d) Compute (72533224)
(72533224) = (7 6) (52 4) (3 2) (23 1)
)
(You may stop here, but
=
Advanced Discrete Exam 1
Solution
1.
(a) The Optimists Society of Tampa restricts their plaintext messages to use only the
seven letters b, c, d, g, o, s and u. They represent the letters by the numbers from 0 to 6,
so that they operate over Z7. The assig
Problem 1
(a) State Fermats Little Theorem
(b) Define (n) (Eulers Phi Function)
(c) State Eulers Theorem
See slides or text
(e) Define: primitive root of a prime p
(d) Compute (72533224)
(72533224) = (76)(524)(32)(231)
(You may stop here, but )
= (732)(52
Advanced Discrete Exam 1
Solution
1.
The Critics Society of Tampa restricts their plaintext messages to use only the seven
letters a, b, c, d, s, u and t. They represent the letters by the numbers from 0 to 6, so that
they operate over Z7. The assignment