## What does modulo do

### 6.3 Modular Exponentiation - Penn Math

6.3 Modular Exponentiation Most technological applications of modular arithmetic involve exponentials with very large numbers. For example, a typical problem related to encryption might involve solving one of the following two equations: 6793032319 ⌘ a (mod 103969) (70) 67930b ⌘ 48560 (mod 103969). (71)

December 2nd,2019

### Everything You Need to Know About Modular Arithmetic

5 does not have an inverse modulo 10. If 5 · b ≡ 1(mod 10) then this means that 5 · b − 1 = 10 · k for some k. In other words 5·b = 10·k −1 which is impossible. Conditions for an inverse of a to exist modulo m Deﬁnition Two numbers are relatively prime if their prime factorizations have no factors in common.

http://pi.math.cornell.edu/~morris/135/mod.pdf

December 2nd,2019

### Verilog - Operators

Verilog - Operators I Verilog operators operate on several data types to produce an output I Not all Verilog operators are synthesible (can produce gates) I Some operators are similar to those in the C language I Remember, you are making gates, not an algorithm (in most cases)

http://web.engr.oregonstate.edu/~traylor/ece474/beamer_lectures/verilog_operators.pdf

December 2nd,2019

### Congruence and Congruence Classes

Congruence and Congruence Classes Definition 11.1. An equivalence relation ~ on a set S is a rule or test applicable to pairs of elements of S such that (i) a ˘a ; 8a 2S (re exive property) (ii) a ˘b ) b ˘a (symmetric property) (iii) a ˘b and b ˘c ) a ˘c (transitive property) :

https://math.okstate.edu/people/binegar/3613/3613-l11.pdf

December 2nd,2019

### CS 70 Discrete Mathematics and Probability Theory Modular ...

Theorem several times we can write: (13+11)¢18 =(6+4)¢4 mod 7 =10¢4 mod 7 =3¢4 mod 7 =12 mod 7 =5 mod 7: In summary, we can always do calculations modulo m by reducing intermediate results modulo m. Inverses Addition and multiplication modm is easy. To add two numbers a and b modulo m, we just add the numbers and then subtract m if necessary to reduce the result to a number between 0 and m¡1.

http://web.stanford.edu/~dntse/classes/cs70_fall09/cs70_fall09_5.pdf

December 2nd,2019

### What kind of treasures English do you know about ...

• Do you think the way you use your money speaks about your character? • Do you think the things you spend your money on can also tell a bit of a story about who you are? Why /Why not? Mineduc Freepik / izakowski 1. Discuss the questions. There is a common expression in English: ‘Money talks’. What does this mean? When do people say it?

December 2nd,2019

### 6.3 Modular Exponentiation - Penn Math

6.3 Modular Exponentiation Most technological applications of modular arithmetic involve exponentials with very large numbers. For example, a typical problem related to encryption might involve solving one of the following two equations: 6793032319 ⌘ a (mod 103969) (70) 67930b ⌘ 48560 (mod 103969). (71)

https://www.math.upenn.edu/~mlazar/math170/notes06-3.pdf

December 2nd,2019

### Modulo Arithmetic - Mathematics

an understanding of modulo arithmetic, people won’t appreciate the many things that come as a result of it, such as programs, calendars, time, and the many tricks and theorems found in Number Theory. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n.

http://math.sfsu.edu/cm2/papers/dido.pdf

December 2nd,2019

### Modular Arithmetic and Cryptography!

Modular Arithmetic and Cryptography! Math Circle Thursday January 22, 2015 What is Modular Arithmetic? In modular arithmetic, we select an integer, n, to be our \modulus".

https://www.math.uci.edu/~mathcircle/materials/Modular_Arithmetic_and_Cryptography_Jan22_2015.pdf

December 2nd,2019

### Math 110 Homework 2 Solutions

k does reduce to [c] modulo k, and conversely that every class that reduces to cmodulo kmust be among this set. The rst claim follows since c+ tk c(mod k) for any t2Z, so all of these congruence classes do reduce to [c] modulo k. Conversely, suppose that a c(mod k). Then by de nition a= c+ ktfor some t2Z.

http://www.math.lsa.umich.edu/~jchw/2015Math110Material/Homework2-Math110-W2015-Solutions.pdf