July 13 2013

CryptographyChapter 1: Early Classic Ciphers

Instructor: Abbas Naderi Afooshteh (aka AbiusX)

Cryptography Usages

✤ Sending Encrypted Messages (e.g Attack At Dawn)

✤ Storing Sensitive Data (e.g Treasure Passwords)

✤ What else?

Cryptography is a Science

✤ Information Security is a Tech, not a science

✤ Cryptography is a science, very close to pure sciences

✤ Crypto is mostly based on Mathematics,

✤ But nothing like the mathematics you’ve studied so far :D

✤ It’s based on Discrete Mathematics (no Algebra)

Cryptography is a Hard

✤ Crypto is one of the hardest available sciences,

✤ You have to combine rigorous math with practice!

✤ It’s secret-like nature makes it interesting

✤ Is mostly supported by militaries

✤ Is the main weapon of nowadays wars

✤ Was the second main weapon in World War II !

Classic Cryptography

Scope of Classic Crypto

✤ Everything before the age of computers

✤ We don’t have bits and bytes, we have English alphabet

✤ We want to encrypt and decrypt messages in English

✤ So our input set is the set of English alphabet {A,B,C,...,Z}

✤ Our input set has 26 elements

✤ Every letter is equal to a number, A=0, B=1, C=2, ... , Z=25

Terminology

✤ Plaintext: the human-readable text in English

✤ Ciphertext: the encrypted text without meaning, also in English

✤ Cipher: the machine that converts plaintext to ciphertext or vice versa

✤ Encryption: the operation that converts plaintext to ciphertext

✤ Decryption: the operation that converts ciphertext to plaintext

✤ Key: the (cryptographically) holy magic maker

✤ Cryptanalysis: breaking a cipher by analyzing it

Cipher

✤ Key is always an input

✤ Either plaintext or ciphertext as input

✤ The other one as output

✤ Invalid keys make invalid outputs

Cipher

Plaintext

Ciphertext

Key

100 BC

Caesar CipherThe earliest known cryptosystem

Caesar Cipher

✤ Key: a single English letter

✤ Plaintext: any English text

✤ Cipher: Add key to every letter of plaintext

✤ If its >=26, reduce by 26

✤ Example:This is a secret message (+D)=wklv lv d vhfuhw phvvdjh

Caesar Cipher (2)

✤ Encryption:

✤ E(p,k) = c = p+k mod 26

✤ Decryption:

✤ p = D(c,k) = c-k mod 26

✤ Key Space:

✤ 26 (why?)

✤ A larger example:

Wb qfmdhcufodvm, o Qosgof qwdvsf, ozgc ybckb og Qosgof'g qwdvsf, hvs gvwth qwdvsf, Qosgof'g qcrs cf Qosgof gvwth, wg cbs ct hvs gwadzsgh obr acgh kwrszm ybckb sbqfmdhwcb hsqvbweisg. Wh wg o hmds ct gipghwhihwcb qwdvsf wb kvwqv soqv zshhsf wb hvs dzowbhslh wg fsdzoqsr pm o zshhsf gcas twlsr biapsf ct dcgwhwcbg rckb hvs ozdvopsh. Tcf sloadzs, kwhv o zsth gvwth ct 3, R kcizr ps fsdzoqsr pm O, S kcizr psqcas P, obr gc cb. Hvs ashvcr wg boasr othsf Xizwig Qosgof, kvc igsr wh wb vwg dfwjohs qcffsgdcbrsbqs.

Caesar Cryptanalysis

✤ First Attack: Always Brute Force

✤ Depends on Key Space

✤ Is a Known Plaintext Attack!

✤ But every sufficiently long message has some known words

✤ Is infeasible, with 100 BC literacy

✤ Is very slow by hand

✤ A moment of a CPU

Attack Terminology

✤ There are 3 types of attacks against cryptosystems:

✤ Ciphertext-only Attack: the attacker only has access to the ciphertext

✤ Known-plaintext Attack: the attacker has some plaintext, along with all ciphertext

✤ Chosen-plaintext Attack: the attacker has any plaintext, and its equivalent ciphertext available (e.g a crypto machine at hand)

Caesar Cryptanalysis (2)

✤ Is Caesar Linear? Lets find out:

✤ E(p,k) = (p + k) mod 26

✤ Ek(p) = (p + k) mod 26 (for a constant k)

✤ Ek1( Ek2(p)) = (( p + k2 mod 26) + k1) mod 26 =

✤ (p + k2 + k1) mod 26 = (p + k3) mod 26 =

✤ Ek3 (p) (k3 = k1 + k2)

✤ Caesar is Linear. What does it mean?

Caesar Cryptanalysis (3)

✤ Brute-Force is slow and brute! Any better methods?

✤ Frequency Analysis: Some letters are more frequent in English text than others.

✤ Requires a sufficiently large text

✤ In worst case, reduces key-space

✤ Only one run through the whole text (Counting Sort)

Frequency AnalysisSome letters are more frequent in English text than others

Caesar Cryptanalysis (4)

✤ How to employ Frequency Analysis?

✤ If our ciphertext’s counting sort result has 100 x and 80 i

✤ What is the key?

✤ If result had 70 k, 70 z and 70 o,

✤ What would the key be this time?

✤A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Caesar Cryptanalysis (5)

✤ Assignment 1:

✤ I) Write a program that receives a text and a letter, and encrypts or decrypts it based on user decision via Caesar cipher.

✤ II) Write a program (in your desired PL) that receives a Caesar ciphertext, a part of the plaintext (at least 5 letters), and outputs the key. Break the sample on page 11 using that.

✤ III) Write a program (in your desired PL) that receives a Caesar ciphertext (at least 1000 characters long) and breaks the code using Frequency Analysis. Test it on shorter passages, and add features to reduce program mistakes.

Submit To: crypto-class@abiusx.com

Crypto Epoch

Affine CipherThe twisted Caesar!

Affine Cipher

✤ Key: two single English letter

✤ one of them needs to be a coprime of 26 (i.e no 2’s or 13’s in it)

✤ We will know why later.

✤ Plaintext: any English text

✤ Cipher: Multiply every letter by first key letter, then add with second key letter

✤ If its >=26, modulo (not reduce) by 26

Affine Cipher (2)

✤ Key = (ka,kb)

✤ Encryption:

✤ E(p,k) = (ka*p+kb) mod 26

✤ Decryption:

✤ D(c,k) = ???

✤ Key Space:

✤ 26*12=312 (why?)

✤ A large example:

Uwd vsslid zlmwdq lf v urmd xs txixvemwvkdulz fjkfulujulxi zlmwdq, nwdqdli dvzw eduudq li vi vemwvkdu lf tvmmdo ux luf ijtdqlz dbjlyvediu, dizqrmudo jflih v fltmed tvuwdtvulzve sjizulxi, vio zxiydqudo kvzp ux v eduudq. Uwd sxqtjev jfdo tdvif uwvu dvzw eduudq dizqrmuf ux xid xuwdq eduudq, vio kvzp vhvli, tdvilih uwd zlmwdq lf dffdiulveer v fuviovqo fjkfulujulxi zlmwdq nluw v qjed hxydqilih nwlzw eduudq hxdf ux nwlzw. Vf fjzw, lu wvf uwd ndvpidffdf xs vee fjkfulujulxi zlmwdqf. Dvzw eduudq lf dizlmwdqdo nluw uwd sjizulxi (vc+k)\txo(26), nwdqd k lf uwd tvhilujod xs uwd fwlsu.

Affine Cipher (3)

✤ What is the decryption algorithm?

✤ We need the inverse operation for a module-multiplication (finite-field arithmetic)

✤ a * b mod x = c => a = c/b mod x (?)

✤ No division in discrete mathematics (no fractions!)

✤ We need to multiply by MMI (modular multiplicative inverse)

Affine Cipher (4)

✤ MMI of a mod x is:

✤ a-1 if and only if a * a-1 mod x = 1

✤ x is 26 in our cipher

✤ MMI of 5 mod 26 is:

✤ 5 * ? mod 26 = 1 => 21 (5*21 = 105, 105-26*4 =1)

✤ MMI is calculated using the Extended Euclidean Algorithm

Affine Cryptanalysis

✤ Affine Key Space is 312:

✤ Large enough for hand

✤ Still too small for a CPU, but we made it 10 times slower easily!

✤ Brute-Force without a computer is futile

✤ What about Frequency Analysis?

Affine Cryptanalysis (2)

✤ We can still use frequency analysis, but we have to discover both key components (ka & kb)

✤ Example: assume we had 100 x and 80 q, how can we find out keys?

Affine Cryptanalysis (3)

✤ Is Affine Linear?

✤ Ek1(Ek2(p))= Ek1( k2a* p + k2b) mod 26 (both mul and sum are modulo safe)

✤ = (k1a* ( k2a* p + k2b) + k1b) mod 26

✤ = (k1a * k2a * p + k1a*k2b + k1b) mod 26

✤ = (k3a * p + k3b) mod 26 (Where k3a = k1a*k2a and k3b= k1a*k2b + k1b)

✤ = Ek3(p)

✤ Affine is Linear too.

Affine Cryptanalysis (4)

✤ How to make Affine or Caesar more resistant to Frequency Analysis?

✤ Why were both of them prone to Frequency Analysis?

✤ Was the ciphertext letters, the same with plaintext?

Caesar Cryptanalysis (5)

✤ Assignment 2:

✤ I) Write a program that receives a text and two letters, and encrypts or decrypts it based on user decision via Affine cipher.

✤ II) (bonus) Create a program that breaks Affine ciphertext using frequency analysis (more than 1000 characters).

Submit To: crypto-class@abiusx.com

End of Chapter 1