ModularArithmeticandtheCaesarCipher

YanHuang

Objectives

• Divisibility• PrimeandCompositeNumbers• FundamentalTheoremofArithmetic• ceiling, floor, /, mod• Caesarcipher

Divisibility

• Thesetofingtegers ℤ = … , −2, −1, 0, 1, 2, … .

• ) divides* if)+ = * forsome+ ∈ ℤ.

• Wewrite)|* todenote) divides*.Wesay) isadivisorof* and* isamultiple of).

Divisibility

Forall),*,. ∈ ℤ

) ), 1 ), )| 0.

Divisibility

Forall),*,. ∈ ℤ

0|) ifandonlyif) = 0.

Divisibility

Forall),*,. ∈ ℤ

) * ⇔ −) * ⇔ )| − *.

Divisibility

Forall),*,. ∈ ℤ

) * and ) . ⇒ )|(* + .).

Divisibility

Forall),*,. ∈ ℤ

) * and * . ⇒ )|..

Divisibility

Forall),* ∈ ℤ

) * and * ) ⇔ ) = ±*.

Primality

: isaprimeif: > 1 andhasnootherpositivedivisorbesides1 and:.

: isacompositeif: > 1 andisnotaprime.

TheListofPrimes

Fundamentaltheoremofarithmetic

Everynon-zerointeger: canbewrittenas

: = ±<=>? … <@

>A.

where<= < <C < ⋯ < <@ aredistinctprimesandE=, … , E@ arenon-negativeintegers.Moreover,theexpressionisunique.

DivisionwithRemainder

Let), * ∈ ℤ with * > 0.ThenthereexistuniqueF, G ∈ ℤ suchthat

) = F* + G and0 ≤ G < *.

Floors

Thefloor function,denotedby⌊⋅⌋,isafunctionfromrealnumbersℝ toℤ.Forevery M ∈ ℝ,⌊M⌋ isthegreatestintegerN ≤ M.

⌊M⌋ isuniquelydefinedforeveryM.

Ceilings

Theceilingfunction,denotedby⌈⋅⌉,isafunctionfromrealnumbersℝ toℤ.Forevery M ∈ ℝ,⌈M⌉ isthesmallestintegerN ≥ M.

⌈M⌉ isuniquelydefinedforeveryM.

Themod operator

Let), * ∈ ℤ with * > 0,) = F* + G and0 ≤ G < *.Wedefine

) mod * ≔ G

Themod operator(GeneralizedDefinition)

Let), * ∈ ℤ,wedefine

) mod * ≔ ) − *⌊)/*⌋

DayinaWeek

September1,2016isThursday.WhatdayisOct1,2016?

MessagesEncoding&Decoding

• Percharacter:

encodeC :: Char -> IntencodeC =

decodeC :: Int -> ChardecodeC =

MessagesEncoding&Decoding

• Percharacter:

encodeC :: Char -> IntencodeC ‘A’ = 0encodeC ‘B’ = 1...encodeC ‘Z’ = 25

decodeC :: Int -> ChardecodeC 0 = ‘A’decodeC 1 = ‘B’...decodeC 25 = ‘Z’

Verytediousandunscalable.Doyouhavebetterideas?

MessagesEncoding&Decoding

• Dealingwithmulti-charactermessages

encode :: [Char] -> [Int]encode m = map encodeC m

encode :: [Char] -> [Int]encode = map encodeC

Point-freeform

CaesarCipher(ShiftCipher)

Encryption:V = W + X mod 26

Decryption:W = V − X mod 26

TheencryptionkeyInputcharacter

ImplementingCaesarCipher

caesarC :: Int -> Int -> IntcaesarC k c = (c + k) mod 26

caesar :: Int -> [Int] -> [Int]caesar = map . caesarC

ImplementingCaesarCipher

caesarDC :: Int -> Int -> IntcaesarDC k c = (c - k) mod 26

caesar :: Int -> [Int] -> [Int]caesar = map . caesarDC