Where’s the missing square?

Dang! It’s Math again… I know how you feel. Really. But Math can be fun, very fun.

Math, fun? Really? › Let’s start with a classic.

Choose a 3 digit number, ABC. Form a new number by repeating your number twice, i.e. ABCABC is my new number. Divide it by 7, 11, then 13

What is the final number you’ve got?

MAGIC SQUARES? LOOK AGAIN.

What is all this about? An introduction to recreational Math

This is so fun!

Melodies Sim, 406’14Cogitare: 20/7/14

A mysterious invention: Mobius Strip

Polynominoes

Polynominoes› Shapes made by connecting certain number of

equal-sized squares, each joined together with at least 1 other square along an edge

Polynominoes: So what? › Another classic

More Polynominoes› Consider this: › Given 2 squares, we can form 1 distinct shape of

dominoes.

› Given 3 squares, we can form 2 distinct shapes of trominoes.

More Polynominoes› Given 4 squares, we can form 5 distinct shapes of

tetrominoes

› Given 5 squares, we can form 12 distinct shapes of pentominoes

More Polynominoes› So, how many distinct shapes can we form with

squares?

Paradoxes They are everywhere! “Don't believe anything you read on the net. Except this. Well, including this, I suppose.” ― Douglas Adams“Good judgment comes from experience, and experience comes from bad judgment.” ― Rita Mae Brown, Alma Mater

Paradoxes in Math › Zeno’s Paradoxes › Birthday Paradox In a room of just 23 people there’s a 50-50 chance of two people having the same birthday. In a room of 75 there’s a 99.9% chance of two people matching.

“Education is the kindling of a flame, not the filling of a vessel.” ― Socrates› Some recommended topics for reading: - Game Theory (e.g. Prisonner’s Dilemma) - Paradoxes! (e.g. Zeno’s, Newcomb’s, Birthday, Friends, Missing Square) -Recreational Math (Martin Gardner!)

Math is beautiful

Teaching should be such that what is offered is perceived as a valuable gift rather than a hard duty.- Albert EinsteinThank you, and hope you have enjoyed this talk!

Quiz Time: This is really ingenius I would say

Cryptography An Introduction into the field, its history, its present and advancement

What, Why, so What? (WWW.) What is it? › Cryptography is the practice and study of

techniques for secure communication in the presence of third parties

Why do we need it? › It is an indispensable tool for protecting information

in computer systems!

Really?

Here’s why we need Cryptography.

Terminology› Plaintext: Whatever your message is› Ciphertext: Your encrypted message

› Encryption: The process of converting ordinary info (plaintext) into code-like text (ciphertext)

› Decryption: The reverse process of encryption

› Cipher: A pair of algorithm that creates the encryption and decryption

› Cryptanalysis: The analysis of a cryptosystem

History of Cryptography What is it for?› Cryptography was concerned solely with message

confidentiality› To ensure secrecy in communications

Who uses them? › E.g. Spies, Military leaders, Diplomats

Classic Cryptography

Or fher gb qevax lbhe BinygvarBe sure to drink your Ovaltine

Classic Cryptography

What was that again? › An early substitution cipher was the Caesar cipher, in which each letter in the plaintext was replaced by a letter some fixed number of positions further down the alphabet.

Cryptography in the Computer ERA

Modern Cryptography

Symmetric Asymmetric

Binary Numbers

0, 1

Binary Numbers and ASCIIA recap and link › Our number system is in base 10

› Binary Numbers = Numbers in Base 2

Symmetric Cryptography: We share the same key

Alice Bob

Symmetric Cryptography: We share the same key

Alice Bob

Stream Ciphers › Encrypt bits (characters, e.g. letters) individually › Achieved by adding a bit from the key stream to the

plaintext bit

› Modulus 2 Arithmetic

Brief on Modulus 2› Using modulus, we are only interested in finding the

remainder

Stream Cipher

Add

Key Stream

Sfs4334.d,fmaso;kdfj,masm,

d

Encrypted Message

Stream Cipher› For example Alice sent a message <A> › A = 01000001› Let say our 1st bit of key streamis 00100111Then to get our encrypted message:

Add0 1

0 1 0 0 0 0 0

0 1 0 0 1

1

1

0 1 1 0 0 1 1 0

Block Cipher› Similar to stream cipher› But encrypts a block of plaintext bits at a time, not

individually

Symmetric Cryptography: We share the same key

Alice

Bob

Calvin

Dora

1 32

2 31

1

2

3

Disadvantages of Symmetric Cryptograpy

› Need many copies of keys and locks

› Hard to implement in the setting of the World Wide Web

(too many people using the Internet/visiting the sites)

› Both parties (client and server) may not initially share the same key

Symmetric Cryptography: We share the same key

Alice Bob

Asymmetric Cryptography: Only 1 key is needed.

Alice

Bob

Calvin

DoraSecret key

Public key

Asymmetric Cryptography: Only 1 key is needed.› SummaryThe lock is public (known as the public key). This is for encryption.

Only Alice has the key to open the lock (known as the secret key.) This is for decryption.

Asymmetric Cryptography: Only 1 key is needed.› In real life, how do we model the key and lock? › We need a mathematical function (the lock) that is

easy to encrypt but difficult to encrypt without a piece of secret information (the secret key)

› Only then can we make the lock public to everyone (public key)

RSA: Rivest Shamir Adleman Rationale: › It is easy to multiply numbers. 5 x 3 = ?

› It’s difficult to factorise. What are the prime factors of 323?

RSA: Rivest Shamir Adleman › Consider this: › We have 2 big primes (hundreds of digits long), › It is easy to multiply to get the product, say

› But given only, it is very difficult to get its 2 prime factors

› Based on this principle, a one-way function is created (easy to encrypt but difficult to decrypt without key)

RSA

Recommanded Books/Author › Martin Garder (Recreational Math) › The mathematics of ciphers : number theory and RSA cryptography / S.C. Coutinho

› The Universe in Zero Words› How to Cut a Cake: And Other Mathematical Conundrums

› Polyominoes: Puzzles, Patterns, Problems, and PackingsSolomon W. Golomb

This is the end. Any questions?