HOMEWORK 5
MATH 4910
Question 1. Use Pollards p 1 method in Sage to factor n from the data
le. You may create the list cfw_1, 2, . . . , B in Sage via range(1,B+1), so that
the exponent m = lcmcfw_1,
HOMEWORK 2
MATH 4910
Question 1. Suppose that xa 1 mod n and xb 1 mod n. Prove that
xgcd(k, ) 1 mod n. Conclude that the order of every x (Z/nZ) divides
(n).
Question 2. Find a primitive root modulo t
March 17, 2013
AAE 203, Spring 2013
Homework 9 - Solution
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Exercise 1 We are given a massless structure as seen in figure 1.
Figure 1: Ex
Fall 2016 AAE 251: Introduction to Aerospace Design
Homework Assignment # 21
Expected completion date: 25 October 2016
Be sure to use appropriate units. Show all of the important steps in your solutio
Fall 2016 AAE 251: Introduction to Aerospace Design
Homework Assignment # 27
Expected completion date: 15 November 2016
Be sure to use appropriate units. Show all of the important steps in your soluti
Fall 2016 AAE 251: Introduction to Aerospace Design
Homework Assignment # 28
Expected completion date: 17 November 2016
Be sure to use appropriate units. Show all of the important steps in your soluti
Fall 2016 AAE 251: Introduction to Aerospace Design
Homework Assignment # 22
Expected completion date: 27 October 2016
Be sure to use appropriate units. Show all of the important steps in your solutio
"
Students can put comments in this main docstring. Such comments will be
important for you (and your teammates in the group project) to remember
what your definition block does.
"
from scipy import *
"
Students can put comments in this main docstring. Such comments will be
important for you (and your teammates in the group project) to remember
what your definition block does.
"
from scipy import *
HOMEWORK 4
MATH 4910
Question 1. Suppose gcd(a1 , a2 , . . . , an ) = d. Prove that there exist integers
k1 , . . . , kn Z such that
d = a1 k1 + + an kn .
Question 2. Alice has RSA public key (N, e),
HOMEWORK 8
MATH 4910
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 cryptosystems. We wil
HOMEWORK 8
MATH 4910
In this homework we will explore the congruential cryptosystem which was
given as the toy case of NTRU in section 6.1 of the book.1 Lets review the
basic idea: We start with a cho
GRADUATE ASSIGNMENT 1
MATH 4910
In this assignment we will review the basic background needed for quantum
computing, beginning with a review of some basic quantum mechanics and
complex linear algebra.
GRADUATE ASSIGNMENT 2
MATH 4910
In this assignment we will explore the Fourier transform (in a fair bit of
its abstract glory). The reason for our interest lies in the fact that a fast implementation
GRADUATE ASSIGNMENT 3
MATH 4910
Contents
1. Introduction
2. Implementing the function f (x) on a quantum computer
2.1. Initial setup
2.2. Denition of Uf
2.3. Quantum parallelism
3. Shors period-nding
HOMEWORK 6
MATH 4910
Question 1. Let p be an odd prime and g a primitive root modulo p. Prove
that x F is a square (also known as a quadratic residue) if and only if
p
logg (x) is even.
Hint: First, r
HOMEWORK 10
MATH 4910
Alice and Bob use the GGH cryptosystem to commuicate. Alices private
key, the dimension n, the matrix of column vectors V = (v1 v2 vn ), and
public key W = V E where E is a matri
HOMEWORK 1
MATH 4910
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 that simply
HOMEWORK 3
MATH 4910
Question 1. Let x, y (Z/nZ) . Suppose that x has order a and y has order
b, where gcd(a, b) = 1. Prove that xy has order ab.
Question 2 (Variation on Hostein, Silverman, Pipher 1.
HOMEWORK 7
MATH 4910
In this assignment, we will explore an implementation of the elliptic curve El
Gamal cryptosystem. We will start with encoding function, using a construction suggested by Koblitz
"
Students can put comments in this main docstring. Such comments will be
important for you (and your teammates in the group project) to remember
what your definition block does.
"
from scipy import *