vi
a
MATH 3CY3, ASSIGNMENT # 1
Due: Wednesday, January 19, in class
ha
re
d
Question 1:
The ciphertext GDJO is intercepted by Eve. She knows that an affine cipher is being
used and that the plaintext starts with du. Determine the complete plaintext.
as
s
MATH 3CY3, ASSIGNMENT # 5
Due: Wednesday, March 23, 2011 in class
Question 1: In an ElGamal digital signature scheme (Fq , g, b) Bob chooses a random
k, gcd(k, q 1) = 1 and signs a message m using the following variation (r, s) of the usual
signature sche
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 material from sections 1,2 and 3 of the script except for sect
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 since the second midterm is
strongly emphasized.
No books/
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 7
S AMPLE SOLUTIONS
Required problems (to be handed in):
1. Let F be a field. Let f (x) = x4 + x3 + x2 + 1 and g(x) = x3 + 1 in F [x]. Use the Extended Euclidean
Algorithm for polynomials
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 3
S AMPLE SOLUTIONS
1.
(a) Use the Extended Euclidean Algorithm to find integers x and y such that 17x + 101y = 1. Show
your work.
Sample solution: The Extended Euclidean Algorithm comes i
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 6
DUE : M ONDAY, 29 F EBRUARY 2016, BY 5:00PM IN THE 3CY03 HOMEWORK LOCKER
Required problems (to be handed in):
1. Suppose you discover that
8805252 2,
20572022 3,
6485812 6,
6686762 77, a
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 2
DUE : M ONDAY, 18 J ANUARY 2016, BY 5:00PM IN THE 3CY03 HOMEWORK LOCKER
(The 3CY03 homework lockers are located in the hallway near Hamilton Halls lecture hall 109 and is
clearly labelle
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 6
DUE : M ONDAY, 7 M ARCH 2016, BY 5:00PM IN THE 3CY03 HOMEWORK LOCKER
Required problems (to be handed in):
1. Let F be a field. Let f (x) = x4 + x3 + x2 + 1 and g(x) = x3 + 1 in F [x]. Us
M ATH 3CY03: I NTRODUCTION TO C RYPTOGRAPHY
W INTER 2016
I N - CLASS T EST # 1: S AMPLE SOLUTIONS AND C OMMENTS
1.
(a) Let p be a prime number. Suppose a, b Z and ab 0 modulo p. Prove that either a 0 modulo
p or b 0 modulo p. Justify all steps.
Sample sol
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 5
S AMPLE SOLUTIONS
Required problems (to be handed in):
1. Let p andq be odd primes and let n = pq. Let r be the least common multiple of p 1 and q 1. Let e
be an encryption exponent for
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 2
S AMPLE SOLUTIONS
Required problems (to be handed in):
1. The ciphertext U CR was encrypted using the affine cipher 9x + 2 mod 26. Find the plaintext. Justify
your answer with clear reas
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
H OMEWORK 8
S AMPLE SOLUTIONS
It was noticed by some students that Problem # 3 in the original problem set had a major error, causing
the problem to be unsolvable. As a consequence, we have deleted
M ATH 3CY03
I NTRODUCTION TO C RYTPOGRAPHY, W INTER 2016
S AMPLE E SSAY
Essay question. Elliptic curves are not finite fields. Why can we still use them to do analogues of cryptographic schemes (e.g. ElGamal) which require finite fields?
Sample essay. In
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. Determine the complete plaintext.
Question 2:
A message
MATH 3CY3, ASSIGNMENT # 6
Due: Friday, April 1, 2011 in class
Question 1: Let E denote the elliptic curve
E : y 2 = x3 + 1
over the eld F17 .
a) Compute E(F17 ).
b) Find the order of the point (0, 1).
Question 2: SmartBob uses an elliptic curve cryptosyst
vi
a
MATH 3CY3, ASSIGNMENT # 3
Due: Friday, February 11, 2011 in class
ha
re
d
Question 1: Bob uses a RSA cryptosystem with n = 2183 and e = 5.
a) Encrypt 27
b) Find the decryption exponent and decrypt 4.
w
Question 3: Use Fermat factorization to factor
a
MATH 3CY3, ASSIGNMENT # 6
Due: Friday, April 1, 2011 in class
Question 1: Let E denote the elliptic curve
E : y 2 = x3 + 1
over the eld F17 .
a) Compute E(F17 ).
b) Find the order of the point (0, 1).
Question 2: SmartBob uses an elliptic curve cryptosyst
MATH 3CY3, ASSIGNMENT # 5
Due: Wednesday, March 23, 2011 in class
Question 1: In an ElGamal digital signature scheme (Fq , g, b) Bob chooses a random
k, gcd(k, q 1) = 1 and signs a message m using the following variation (r, s) of the usual
signature sche
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. Determine the complete plaintext.
Question 2:
A message
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
in F28 = F2 [x]/(x8 + x4 + x3 + x + 1).
Question 2: Bo
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)g(x)
with a(x), b(x) F2 [x].
Question 2: Let p(x) = x5
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 sends all three of them the time
m, when the four of t
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 by the pair (n, e). Alice sends the
message m to Bob. I
MATH 3CY3, ASSIGNMENT # 2
Due: Wednesday, October 10, in class
Question 1: Find the greatest common divisor d of a = 1547 and b = 427 and express
d as an integral combination of a and b using the Extended Euclidean Algorithm.
Question 2: Find the smallest
MATH 3CY3, ASSIGNMENT # 6
Due: Thursday, November 29, in class
Question 1: What is the probability that in the Group of Seven two members had
birthdays in the same month?
Question 2: Let E denote the elliptic curve
E:
y 2 = x3 + x,
dened over Fp , where p