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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 # 3
Due: Friday, February 11, 2011 in class
Question 1: Bob uses a RSA cryptosystem with n = 2183 and e = 5.
a) Encrypt 27
b) Find the decryption exponent and decrypt 4.
Question 2: Bobs RSA cryptosystem is (n, e), which Alice uses t
MATH 3CY3, ASSIGNMENT # 2
Due: Wednesday, February 2, 2011 in class
Question 1: Find the greatest common divisor d of a = 2613 and b = 2171 and express
d as an integral combination of a and b using the Extended Euclidean Algorithm.
Question 2: Find the sm
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 # 3
Due: Friday, February 11, 2011 in class
Question 1: Bob uses a RSA cryptosystem with n = 2183 and e = 5.
a) Encrypt 27
b) Find the decryption exponent and decrypt 4.
Question 2: Bobs RSA cryptosystem is (n, e), which Alice uses t
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 # 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 # 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
MATH 3CY3, ASSIGNMENT # 1
Due: Monday, September 24, in class
Question 1: Alice uses the following ane cipher to send messages to Bob:
x 21x + 7.
a) Encrypt the following message: comenow
b) 10 minutes later Bob receives the encrypted message UPEHQNA from
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
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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
Assignment 1 Solutions
March 19, 2011
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.
Solution: Given : du GD and x ax + k = y. So 3, 20 6, 3
MATH 3CY3 - Assignment #6 Solutions
Problem 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)
Solution:
(a) For a F17 , we compute a2 and a3 + 1. The results of these computat
Math 3CY3 Assignment 4 Solutions
1
Question 1
We wish to show that the polynomial x5 +x3 +x2 +x+1 F2 is irreducible. (We
denote this polynomial by f (x).) To see that f (x) is irreducible, we suppose,
for a contradiction, that it is reducible. We then hav
Math 3CY3 Assignment 3 Solutions
1
Question 1
(a)
We wish to show that 3 is a primitive root modulo 31. That is, we wish to
show that ord3 (31) = 30. Since 3 and 31 are relatively prime, we have by
Lemma 2.18 that ord3 (31)|30. We notice that the non-triv
Math 3CY3 Assignment 1 Solutions
1
Question 1
We are given an ane cipher with decryption key (, k) = (11, 4), and we wish
to decrypt the ciphertext PCJORS (corresponding to 15 2 9 14 17 18). Recall
that decryption of an ane cipher is done via y (y k) wher
Math 3CY3 Assignment 2 Solutions
1
Question 1
We wish to nd gcd(1547, 427) and express it as an integral combination of 1547
and 427. We recall the extended Euclidean algorithm:
a
= q1 b + r1 , 0 r1 < b
b
= q2 r1 + r2 , 0 r2 < r1
r1
= q3 r2 + r3 , 0 r3 <