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Notes

1. Why is Modular Arithmetic Important for Crpytography

We are used to dealing with problems in math classes that deal with infinite sets of numbers, such as the real numbers and integers. Cryptosystems however are often based on finite and discrete sets and modular arithmetic is at the heart of many cryptosystems. We need to know about modular arithmetic in order to compute within these systems. However, everyone has some experience with modular arithmetic whether they are aware of it or not. Telling time, though time possibly is infinite, is a finite system. If we are talking about the time, we don't have an infinite number of hours to work with, we only have 12 (or 24 in military time). So this is a finite system and we need modular arithmetic to deal with it. Let's look at an easy example.

Example 1.1: Say it's 10 am and your friend wants to meet up with you after lunch. They tell you to meet them in 5 hours. What time are you going to meet them? Obviously, you will meet your friend at 1 pm. But $ \ 10+5=15 $. How we write this is $ 15 \equiv 1 \bmod 12 $.

Definition: Modulus Operation Let a,r,m $ \in \Z <\math> and m>0. We write <math> a \equiv r \bmod m <\math> if m divides a-r. m is called the modulus and r is called the remainder. ==[[2015_Summer_Cryptography_Paar_Introduction to Cryptography_Kathryn Marsh and Divya Agarwal _comments | Questions and comments]]== If you have any questions, comments, etc. please post them [[2015_Summer_Cryptography_Paar_Introduction to Cryptography_Kathryn Marsh and Divya Agarwal _comments|here]]. ---- [[2015 Summer Cryptography Paar|Back to 2015 Summer Cryptography Paar]] ---- $