Skills for Mathematical Sciences - Cryptographyand Factorisation

Dr Craig

3 February 2017

What is cryptography?

The science of protecting information bytransforming it (encrypting it) into an unreadableformat, called ciphertext.

Plaintext is normal text which can be read byanyone. Plaintext is encrypted into ciphertext whichcan then be decrypted back into plaintext.

A cipher is the method used to encrypt anddecrypt the plaintext.

Where is cryptography used?

In 50BC:

Julius Caesar, the Roman emperor, used

encrypted messages to communicate with his

generals.

Where is cryptography used?

In World War II:

Where is cryptography used?

I internet bankingI cellphone calls and messagesI online transactionsI military and diplomatic communication

A simple cipher: the Caesar cipher

Choose any number x. Shift each letter x

places along the alphabet.

Example: x = 5:

a −→ f

b −→ g

. . .

y −→ d

z −→ e

Grahamstown −→ Lwfmfrxytbs

Using maths: how secure is your cipher?

Q: What is the problem with the Caesar

cipher?

A: Once you know the encryption of one

letter, you know the encryption of them all.

Q: How many possible Caesar ciphers are

there?

A: 26

We can break a Caesar cipher by trying all 26

possible combinations. A bit of hard work,

but not too bad (especially for a computer).

Making stronger ciphers: What if we no longerrequire that the substitution simply shifts the orderof the alphabet? That is, we could have asubstitution that looks as follows:

a −→ p

b −→ a

c −→ n

. . .

y −→ m

z −→ q

How many different substitution ciphers like thisexist? 26× 25× 24× . . .× 3× 2× 1 =

403, 291, 461, 126, 605, 635, 584, 000, 000

How to break a cipher: frequency analysis

Frequency analysis uses the fact that certain lettersoccur more often than others. The technique wasfirst developed in the 9th century by Al Kindi, anArab mathematician and philosopher.

In English, the letters that occur most often are:

E T A O I N

% 12.7 9.1 8.2 7.5 7.0 6.7

More tricks for frequency analysis:Digrams are pairs of letters which occur

together. The most common digrams are:

TH HE IN ER AN RE ED

We can look for common pairs of letters in

the ciphertext and match them to the most

common digrams.

We also know which letters are repeated

most often:

SS, EE, TT, FF, LL, MM and OO.

Frequency in English:

Frequency in Text 1:

Frequency in Text 2:

Ciphertext 1:

KA, JRA GAXGEA XD NXOJR IDHYPI, HAPXZFYNA JRA

YFBONJYPAN XD XOH GINJ; RXFXOH JRXNA KRX NODDAHAC

DXH BONJYPA IFC DHAACXU YF XOH EIFC; HANGAPJ

JRXNA KRX RITA KXHQAC JX MOYEC IFC CATAEXG XOH

PXOFJHV; IFC MAEYATA JRIJ NXOJR IDHYPI MAEXFZN JX

IEE KRX EYTA YF YJ, OFYJAC YF XOH CYTAHNYJV.

Ciphertext 2:

DS KSGU ABJGZCGR BXX UDS RNBJGW QI UDBJRD UDSNS

KSNS NBZVSUI CG DCI UBSI QGW SQZD IUNCWS DS UBBV

FCXUSW DCT CGUB UDS QCN. DS DQWGU SMSG RBGS CGUB

UDQU KDCPPCGR UBY RSQN BX DCI, KDSG UDS RNBJGW

ASZQTS AFJNNSW QGW UDS KCGW DBKFSW QGW DCI XSSU

WCWGU ISST UB AS UBJZDCGR QGLUDCGR AJU QCN.

Ciphertext 1:

KE, JRE GEXGEE XD NXOJR IDHYPI, HEPXZFYNE JRE

YFBONJYPEN XD XOH GINJ; RXFXOH JRXNE KRX NODDEHEC

DXH BONJYPE IFC DHEECXU YF XOH EIFC; HENGEPJ

JRXNE KRX RITE KXHQEC JX MOYEC IFC CETEEXG XOH

PXOFJHV; IFC MEEYETE JRIJ NXOJR IDHYPI MEEXFZN JX

IEE KRX EYTE YF YJ, OFYJEC YF XOH CYTEHNYJV.

Ciphertext 2:

DE KEGU ABJGZCGR BXX UDE RNBJGW QI UDBJRD UDENE

KENE NBZVEUI CG DCI UBEI QGW EQZD IUNCWE DE UBBV

FCXUEW DCT CGUB UDE QCN. DE DQWGU EMEG RBGE CGUB

UDQU KDCPPCGR UBY REQN BX DCI, KDEG UDE RNBJGW

AEZQTE AFJNNEW QGW UDE KCGW DBKFEW QGW DCI XEEU

WCWGU IEET UB AE UBJZDCGR QGLUDCGR AJU QCN.

Text 1:

We, the people of South Africa,Recognise the injustices of our past;Honour those who suffered for justice and freedom in our land;Respect those who have worked to build and develop ourcountry; andBelieve that South Africa belongs to all who live in it, unitedin our diversity.

Text 2:

He went bouncing off the ground as though there were rocketsin his toes and each stride he took lifted him into the air.He hadn’t even gone into that whizzing top gear of his, whenthe ground became blurred and the wind howled and his feetdidn’t seem to be touching anything but air.

Prime time

Given an integer n (that is, n ∈ Z) we say

that a ∈ Z is a factor of n if there exists

b ∈ Z such that a · b = n.

A positive integer p that is greater than 1 is

said to be prime if it has exactly two

factors: 1 and p.

The largest known prime number was

discovered in 2016. It has 22,338,618 digits!

It is a Mersenne prime, so it is easy to write:

274,207,281 − 1

Fundamental Theorem ofArithmetic: every integer n > 1 can be

written uniquely as the product of powers

of its prime factors. That is, if n has m

different prime factors, then

n = pk11 · pk22 · · · pkmm

What is the easiest/most efficient way to

factorise a number into its prime factors?

Relatively prime

Two positive integers are said to be

relatively prime if their greatest common

divisor is 1. That is, they share no common

factors other than 1. Examples are:

I 12 and 5

I 33 and 35

I 15 and 154

Exercise: find out how many numbers below

16 are relatively prime to 16. Do the same

for 18.

Euler’s Phi Function

Discoverd by Leonard Euler in 1763. If

n = pk11 · pk22 · · · pkmm

then

ϕ(n) = n

(p1 − 1

p1

)(p2 − 1

p2

)· · ·

(pm − 1

pm

)What about ϕ(n) when n = p · q for p, q

prime? Then

ϕ(n) = (p− 1)(q − 1)

Multiplying versus factorising

It is worth watching this entire video. If you

just want to see the difference in complexity

between multiplication and factorisation,

watch from 6:30 to 8:15.

https://www.youtube.com/watch?v=wXB-V_Keiu8

The main point here is that it is

computationally easy to multiply two large

numbers but very difficult to factorise a large

number.

Using prime numbers in cryptography

A very general description of RSA encryption:I Take two large prime numbers p and q. Multiply

them to get n = p× q.I Calculate ϕ(n) (easy if you know p and q!).I Choose e such that e is relatively prime to ϕ(n)

and 1 6 e 6 ϕ(n).I Make n and e public. Your counterparts will use

these to encode their messages.I Calculate d (the multiplicative inverse of emod ϕ(n), i.e. d · e = 1 + kϕ(n), k ∈ Z). NB:you can’t quickly find d without knowing ϕ(n).

I After receiving a message encrypted with n ande, you will decode it using d and ϕ(n).

More about prime numbers

Become a GIMP!

Remember the largest prime with 22 million

digits? This was found by GIMPS – Great

Internet Mersenne Prime Search. This search

method uses the computing power of idle

computers to search for prime numbers. You

can find out more and sign up for GIMPS

here:

http://www.mersenne.org/

The Goldbach Conjecture

“Every even positive integer greater than 2

can be written as the sum of two primes.”

No proof of this has yet been found but all of

the even numbers up to

4× 1017

have been checked and the conjecture holds

for all of them.

The twin prime conjecture

The pairs 17, 19 and 29, 31 are examples of

twin primes. The conjecture states:

“There are infinitely many pairs of twin

primes.”

Read this article for coverage of the progress

towards resolving the conjecture: http://www.

slate.com/articles/health_and_science/do_the_

math/2013/05/yitang_zhang_twin_primes_conjecture_

a_huge_discovery_about_prime_numbers.html