M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
F INAL E XAM R EVIEW S HEET
1
What to expect on the final exam
The final exam is cumulative, covering the entire course, but the material sinc
MATH 3CY3, ASSIGNMENT #3
Due: Friday, February 10, 2017, before noon
Each question is worth 5 marks
Question 1: Compute
2100000
mod 77.
Question 2: Let a, b be two positive integers with a > b, and le
Assignment 4 Solutions
1. By the Fermat factor system for x = [ n] + 3, we get x2 = 1528452
which gives us y 2 = x2 n = 8042 . We therefore have that n =
(x + y)(x y) = 153649 152041
2. (a) By decompo
MATH 3CY3, ASSIGNMENT # 5
Due: Tue, March 24, noon, in boxes C62/C63 in the basement of HH.
Each question is worth 4 marks.
Question 1: Bobs favourite ElGamal cryptosystem is (F23 , 7, 4). Bob signs t
MATH 3CY3, ASSIGNMENT # 1
Due: Tue, January 20, noon, in boxes C62/C63 in the basement of HH.
Each question is worth 4 marks.
Question 1: The ciphertext CRWWZ was encrypted using an affine cipher. The
Answers to the Assignment 2
1. gcd(1547, 427) = 7 and 7 = (8)(1547) + (29)(427)
2. First solution: From the first equation we have x = 3k + 2. The second equation
implies that k = 5l + 2. Then the thi
MATH 3CY3, ASSIGNMENT # 4
Due: Tue, March 10, noon, in boxes C62/C63 in the basement of HH.
Each question is worth 4 marks.
Question 1: Consider the finite field F25 = F5 [x]/(x2 + 2).
1. Show that th
MATH 30Y3, ASSIGNMENT # Q
DUE: TUE, FEBRUARY 3. NOON, IN BOXES (362/063 IN THE BASEMENT OF HH.
Each question is worth 4 marks.
Question 1: Find the greatest common divisor d of a = 1547 and b : 427 an
Answers to the Assignment 4
1. a) Since the order has to divide (25) = 24, enough to check xn and (x + 2)n for
n cfw_1, 2, 3, 4, 6, 8, 12, 24. We can see n = 8 and n = 3 are the smallest choices such
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Assignment 6 Solutions
1. Part 1 It is enough to show that no irreducible polynomial in F2 [x] of
degree less than or equal to 4 divides p(x) = x8 + x4 + x3 + x + 1.
The irreducible polynomials of deg
MATH 3CY3, ASSIGNMENT # 4
Due: Tuesday, March 7
Question 1: Use Fermat factorization (section 3.2 in the script) to factor
n = 23360947609.
Question 2: Let n = 1829 and let B = cfw_1, 2, 3, 5, 7, 11,
Assignment 5 Solutions
1. The polynomial has no roots in F2 , and therefore the only way it can
factor is into a product of an irreducible quadratic polynomial g(x) and
an irreducible cubic polynomial
Assignment 3 Solutions
1. We have from Eulers formula that (77) = 60 = 22 35 and furthermore
100000 0 (mod 20) and 100000 1 (mod 3).
From this, we have that 100000 40 (mod 60) and thus
2100000 240 (mo
MATH 3CY3, ASSIGNMENT #2
Due: Tuesday, January 31, 2017, before noon
Each question is worth 5 marks
Question 1: Find the greatest common divisor d of a = 1547 and b = 427 and express
d as an integral
MATH 3CY3, ASSIGNMENT # 6
Due: Tuesday, March 28
Question 1:
1. Show that x8 + x4 + x3 + x + 1 F2 [x] is irreducible.
2. Use the shift method to compute
(x7 + x6 + x3 + x + 1) x2
in F28 = F2 [x]/(x8 +
MATH 3CY3, ASSIGNMENT # 7
Due: Tuesday, April 4
Question 1:
Let E denote the elliptic curve
E : y 2 = x3 + 1
over the field F17 .
a) Compute the points of E(F17 ).
b) Find the order of the point (0, 1
MATH 3CY3 ASSIGNMENT #1
Due: Monday, Jan. 23, before noon, in the box 15
Each question is worth 5 marks
1. The ciphertext CRWWZ has been encrypted using an affine cipher. The plaintext starts
with ha.
MATH 3CY3, ASSIGNMENT # 5
Due: Monday, March 13
Question 1:
Show that
x5 + x3 + x2 + x + 1 F2 [x]
is irreducible.
Question 2:
Use the Extended Euclidean Algorithm to find the gcd of the polynomials
f
Assignment 1 Solutions
1. Since CR decrypts to ha, the decryption function x y+l (mod 26)
has to satisfy 2 + l 7 (mod 26) and 17 + l 0 (mod 26). So,
15 7 (mod 26) and since (7, 26) = 1, we get 7 15
7
Assignment 7 Solutions
1. (a) For a F17 , we compute a2 and a3 + 1. The results of these
computations (all done modulo 17) are given in the following table:
a a2 a3 + 1
0 0
1
1 1
2
9
2 4
3 9
11
14
4 1
INFORMATION ABOUT THE MIDTERM
The midterm will take place on
February 16,
7 9pm,
in JHE 264.
Please note that every other seat and every other row have to remain unoccupied.
The midterm covers materia
MATH 3CY3, ASSIGNMENT # 3
Due: Thursday, October 18, in class
Question 1: a) Show that 3 is a primitive root modulo 31.
b) Find b, so that
3b 22 mod 31.
Question 2: Bob uses an RSA cryptosystem given
MATH 3CY3, ASSIGNMENT # 4
Due: Thursday, November 8, in class
Question 1: Bob, Bill and Bradd use the same encryption exponents e = 3 in their
individual cryptosystems (51, 3), (65, 3), (77, 3). Alice
MATH 3CY3, ASSIGNMENT # 4
Due: Wednesday, March 9, 2011 in class
Question 1: Find the gcd of the polynomials
f (x) = x5 + x4 + x + 1
and g(x) = x3 + 1 F2 [x].
and write it in the form
a(x)f (x) + b(x)
MATH 3CY3, ASSIGNMENT # 4
Due: Thursday, November 21, in class
Question 1:
1. Show that x8 + x4 + x3 + x + 1 F2 [x] is irreducible.
2. Use the shift and XOR method to compute
(x7 + x6 + x3 + x + 1) x2
MATH 3CY3, ASSIGNMENT # 1
Due: Wednesday, January 19, in class
Question 1:
The ciphertext GDJO is intercepted by Eve. She knows that an ane cipher is being
used and that the plaintext starts with du.