HOMEWORK 10 SOLUTIONS
MATH 4713
Question 1 (What do points at innity look like?). Let S 2 = cfw_(x, y, z ) R3 :
x2 + y 2 + z 2 = 1 denote the usual unit sphere in 3-space. Let
H = cfw_(x, y, z ) S 2 :
HOMEWORK 11 SOLUTIONS
MATH 4713
Question 1. Recall from lecture that the torsion subgroup E (Q)tors of an elliptic
curve E consists of all points P E (Q) such that nP = O for some n N. As we
discussed
HOMEWORK 3 SOLUTIONS
MATH 4713
Question 1. Suppose that p is a prime and ap + bp = cp for some a, b, c Z.
Show that p|a + b c.
Proof. Fermats Little Theorem tells us that
ap a mod p,
bp b mod p,
cp c
HOMEWORK 3
MATH 4713
Question 1. Suppose that p is a prime and ap + bp = cp for some a, b, c Z.
Show that p|a + b c.
Question 2. Let p > 2 be a prime. Recall that on Question #6 (2) of HW2
we proved t
HOMEWORK 1 SOLUTIONS
MATH 4713
Question 1 (Stein 1.1). Compute gcd(455, 1235) by hand.
Question 2 (Stein 1.2). Use the Sieve of Eratosthenes to make a list of all
primes up to 100.
Question 3 (Stein 1
HOMEWORK 1
MATH 4713
Question 1 (Stein 1.1). Compute gcd(455, 1235) by hand.
Question 2 (Stein 1.2). Use the Sieve of Eratosthenes to make a list of all
primes up to 100.
Question 3 (Stein 1.3). Prove
HOMEWORK 2 SOLUTIONS
MATH 4713
Question 1.
(1) Prove that gcd(6n + 5, 7n + 6) = 1 for all n Z.
Solution. Observe that
6(7n + 6) 7(6n + 5) = 1
This is now in the form of Bezouts identity, and allows us
HOMEWORK 6 SOLUTIONS
MATH 4713
Question 1. Suppose there is a primitive root g modulo n. We demonstrated in
lecture that there are in fact (n) primitive roots. Suppose h is another primitive
root.1 Pr
HOMEWORK 4
MATH 4713
Question 1. Recall Wilsons theorem states that (p 1)! 1 mod p for
p a prime. Prove that if n > 4 is composite, then (n 1)! 0 mod n. (If
n = 4, then in fact (n 1)! = 6 2 mod 4, the
HOMEWORK 4
MATH 4753
Question 1. Alice has RSA public key (N, e), provided in the hw4-data.txt
file. Her key is large enough to encode 41 ASCII characters at once (rounding
down log256 N yields 41). B
HOMEWORK 2 SOLUTIONS
MATH 4753
Warning on terminology: These solutions use the term
order for what we called exact order in lecture. I will leave
these solutions in their original form, particularly s
HOMEWORK 5
MATH 4713
Question 1. If a single primitive root exists modulo n, then there are in fact (n)
primitive roots. (Hint: If h g k mod n, and g, h are both primitive, what can you
say about k? Y
HOMEWORK 4
MATH 4713
MITCHELL LANKERT
For these questions, please print out your Sage worksheet when
you are done showing your solutions. If you have trouble installing or
do not wish to install Sage,
HOMEWORK 8 SOLUTIONS
MATH 4753
In this homework we will practice taking square roots of elements in Fp in
Fp2 , and study the encoding scheme suggested by Koblitz for use in elliptic
curve cryptosyste
HOMEWORK 6
MATH 4753
Question 1. Let p be an odd prime and g a primitive root modulo p. Prove
that x F
p is a square (also known as a quadratic residue) if and only if
logg (x) is even. Conclude that
HOMEWORK 5 SOLUTIONS
MATH 4753
Midterm Review Questions
You will find it helpful to study each of these questions, but you are only
required for this homework assignment to answer and submit solutions
HOMEWORK 3 SOLUTIONS
MATH 4753
Question 1. Use Shanks Babystep-Giantstep algorithm to solve by hand for
the exponent k in
2k 10 mod 29.
Where is k defined?
Proof. As in the algorithm, we let
n = b 29
HOMEWORK 11
MATH 4753
Recall that R = Z[x]/(xN 1) where N > 1. For p > 1 any modulus (not
necessarily prime), Rp = (Z/pZ)[x]/(xN 1). We do not assume p, q are prime
below unless otherwise stated.
Ques
HOMEWORK 1 SOLUTIONS
MATH 4753
In lecture we discussed some of the issues with the security of the Caesar
cipher, starting with the small key space. However, there are even more
clever ways a cipher t
HOMEWORK 7 SOLUTIONS
MATH 4713
Question 1. Let z = x + yi be a complex number (x, y R). Dene the complex
conjugate to be z = x yi, and the absolute value to be |z | = z z = x2 + y 2 .
(1) Prove that
z
HOMEWORK 9 SOLUTIONS
MATH 4713
Question 1. Recall that 22/7 is a partial convergent to . How close an approximation is it, especially for such a small denominator as 7? Prove that if < a/b < 22/7,
the
HOMEWORK 8 SOLUTIONS
MATH 4713
Question 1. Perform the following evaluations.
(1) Find the continued fraction expansion of 25/41 using the process outlined
in class.
Solutions.
25
= 1,
41
41
= 2,
16
1
HOMEWORK 9 SOLUTIONS
MATH 4713
Question 1. Recall that 22/7 is a partial convergent to . How close an approximation is it, especially for such a small denominator as 7? Prove that if < a/b < 22/7,
the
Math 4713 Assignment 8
Due Monday, Nov. 8, in class, or 2:00 AM Tues. Nov. 9 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Math 4713 Assignment 9
Due Monday, Nov. 22, in class, or 2:00 AM Tues. Nov. 23 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on thes
Math 4713 Assignment 10
Due Friday, Dec. 10, in class, or 2:00 AM Sat. Dec. 11 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on thes
Math 4713 Assignment 7
Due Monday, Nov. 1, in class, or 2:00 AM Tues. Nov. 2 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Math 4713 Assignment 6
Due Friday, Oct. 22, in class, or 2:00 AM Sat. Oct. 23
in D2L
Problems to Read, think about and read the solutions in the back
of the textbook: Do not turn in your work on these
Math 4713 Assignment 3
Due Wednesday, Sep. 22, in class, or 2:00 AM
Thurs. Sep. 23 in D2L
Problems to read, think about and read the solutions in the back of the textbook:
Do not turn in your work on
Math 4713 Assignment 4
Due Friday, Oct. 1, in class, or 2:00 AM Saturday, Oct. 2 in
D2L
Problems to read, think about and read the solutions in the back of the textbook:
Do not turn in your work on th