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
Question 2. Find a primitive root modulo the prime p = 197 by hand. Prove
your answer is correct.
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), provided in the hw4-data.txt
le. Her key is large enoug
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 will start with some basic examples.
Question 1. Solve for
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 in his paper on using elliptic curves in cryptography.
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 1.31). Let p be an odd
prime and suppose q is a prime tha
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 maps single letters to other letters (a simple
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 matrix of elementary column operations, is
given in the le h
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
logg (x) is even.
Hint: First, remember that since p 1 is even, the notion of even or o
GRADUATE ASSIGNMENT 3
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 algorithm and the Fourier transform
3.1. The No-Cloning
GRADUATE ASSIGNMENT 2
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 of the discrete Fourier transform, that is, the Fourier
GRADUATE ASSIGNMENT 1
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.
A bit classically is a system that is always in one of
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 choice of a large positive integer q, and we choose
e, f w
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, . . . , B is computed as m = lcm(range(1,B+1).