MATC16
CODING THEORY AND CRYPTOGRAPHY
Administrivia
Instructor: Professor John Scherk
Oce: IC496
Email: [email protected]
(emails will be answered within 24 hours, but may not be answered immediately)
Oce Hour: Monday, 1:30-2:30
Description
The stud
ASSIGNMENT 4
MATC16
due: November 6, 2015
(1) (a) List the points on the elliptic curve E : y 2 x3 2
(mod 7).
(b) Find the sum (3, 2) + (5, 5) on E.
(c) Find the sum (3, 2) + (3, 2) on E.
(2) (a) Show that P = (0, 1) on y 2 = x3 + 1 satisfies 6P = . Do
th
ASSIGNMENT 3
MATC16
due: October 19, 2015
(1) (a) Suppose that the primes used in the RSA cryptosystem are
consecutive primes. How would you factor n?
(b) The ciphertext 10787770728 was encrypted using e = 113
and n = 10993522499. The factors p and q were
ASSIGNMENT 1
MATC16
due: September 25, 2015
(1) The ciphertext 5859 was obtained from the RSA algorithm using n = 11413 and e = 7467. Using the factorization 11413 =
101 113 , nd the plaintext.
(2) Naive Nelson uses RSA to receive a single ciphertext c, c
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATC16H
Coding Theory and Cryptography
Examiner: E. Mendelsohn
Date: February 13, 2004
Room: BV 516
Duration: 100 minutes
1. Dene
(a) [2 points] A private key cryptosystem
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATC16 - Coding Theory and Cryptography
Examiner: T. Pham
Date: October 28, 2006
Duration: 120 minutes
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
NOTES:
There
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Term Test
MATC16H
Coding Theory and Cryptography
Examiner: E. Mendelsohn
Date: Friday, March 4, 2005
Time: 10:1012:00 PM
1. [each part worths 5 points]
(a) Find gcd (217 3
MATC16 Cryptography and Coding Theory
Gbor Pete
a
University of Toronto Scarborough
gpete at utsc dot utoronto dot ca
Solutions for Assignment 1
The solutions below are sometimes sketchy, skipping computational details.
Problem 1. What is 491 (mod 50)? An
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATC16H Coding Theory and Cryptography
Examiner: P. Selick
Date: October 21, 2005
Duration: 120 minutes
1. [13 points] Find an integer x such that x 4(mod 9),
Let and be ordered bases for a nite dimensional vector space V
and let I : V V be the identity map. Then for v V ,
v
The matrix I
= I(v)
= I
v
.
is called the change of coordinate matrix
from to .
This matrix is unique.
Two n n matrices are similar if and
Denition: Let V be a nite dimensional vector space and let T :
V V be linear. If is a basis for V such that T
is a Jordan
form, then we say that is a Jordan (canonical) basis.
Assume that J is a Jordan form for a n n matrix A. We have previously seen that
Chapter 2 Compactness
Denition 2.0.7 A topological space X is called compact if it has the property that every open cover of X has a nite subcover. Theorem 2.0.8 Heine-Borel A subset X Rn is closed and bounded if and only if every open cover of X has a ni
Chapter 5 Paracompactness
Let cfw_W I be a cover of X . (We do not assume W is open.) Denition 5.0.27 A cover cfw_T J is called a renement of cfw_W I if J, I s.t. T W . Denition 5.0.28 A collection cfw_W I of subsets of X is called locally nite if each x
Chapter 6 Connectedness
Denition 6.0.40 A pair of nonempty open subsets A and B of a topological space X is called a disconnection of X if A B = and A B = X . Note: If A, B is a disconnection of X then A and B are also closed since A = B c and B = Ac . Pr
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set I (Solution Sketch)
MAT C27F
1. Since A is closed in Y , a set B which is closed in X such that A = B Y .
Therefore A is the intersection of two closed sets in X , and so it is closed in X .
2.
a) Let h
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set III (Solution Sketch)
MAT C27F
1. To show X is homeomorphic to S 1 S 1 :
The canonical projection q : [0, 1] [0, 1] X is surjective.
[0, 1] [0, 1] is compact, since it is a closed bounded subset of R3 .
1. Dene f : X S 1 by f (t) = (cos 2t, sin 2t)
[0, 1]
(cos 2t, sin 2t)
q
S1
f
X
If q : [0, 1] X is the quotient map (taking each element to its equivalence class) then f q is the continous function t (cos 2t, sin 2t)
on [0, 1].
It follows that f is continu
Chapter 2
Topological Spaces
2.1
Metric spaces
Denition 2.1.1 A metric space consists of a set X together with a function d : X X R +
s.t.
1. d(x, y) = 0 x = y
2. d(x, y) = d(y, x)
x, y
3. d(x, z) d(x, y) + d(y, z)
x, y, z
triangle inequality
Example 2.1.
Chapter 3
Separation axioms
Let X be a topological space.
Denition 3.0.12 X has the following names if it has the following properties:
1. X is T0 if x = y X either open U s.t. x U, y U x U, y U or open
/
/
U s.t. x U, y U
/
2. X is T1 if x = y X open U s
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set IV (Solution Sketch)
MAT C27F
1. Suppose X R is connected.
Let x, y, belong to X and assume x < z < y.
If z X then (, z) X and (z, ) X forms a disconnection of X, contradicting
X connected.
Therefore x