Lecture 1

Question 1

How is a positive integer denoted?

Answer 1

N

Question 2

How is a prime denoted?

Answer 2

p

Question 3

What is Zn?

Answer 3

The set of integers (0, 1, …, N-1}

Question 4

How do we return to the correct range of Zn values?

Answer 4

Add or subtract k.N to get back into the range.

Question 5

What is Zm?

Answer 5

Modulus which is:

1. An abelian group, which means all modular arithmetic will give security in consistent results.
2. A ring, meaning these operations will be well-behaved, always working correctly and maintaining the mathematical structure and environment.

Question 6

What is Bezout’s identity?

Answer 6

For all non-zero x, y in Z, there exists a,b in Z such that

ax + by = gcd(x, y)

GCD = greatest common divisor

Question 7

What is an efficient algorithm to find GCDs?

Answer 7

Extended Euclidian Algorithm

Question 8

How can GCDs be found with Sage?

Answer 8

do this