Introduction to Computer SecurityFoundations of Cryptography

Pavel LaskovWilhelm Schickard Institute for Computer Science

Secret communication

Encryption

Alice Bob

Decryption

key

unitue

plaintext

I love you

plaintext

I love you

ciphertext

C ywoy cih

key

unitue

Cryptography and security objectives

Which security objectives are addressed by cryptography?

Confidentialitysymmetric cryptographyasymmetric cryptography

Integrityhashing

Authentication and non-repudiationdigital signatures

Cryptography and security objectives

Which security objectives are addressed by cryptography?

Confidentialitysymmetric cryptographyasymmetric cryptography

Integrityhashing

Authentication and non-repudiationdigital signatures

Symmetric cryptography

Encryption

Alice Bob

Decryption

shared key

unitue

plaintext

I love you

plaintext

I love you

ciphertext

C ywoy cih

shared key

unitue

any valid key

Early permutation cipher: scytale

Encryption:Wrap a parchment strip over a woodenrod of a fixed diameter and write lettersalong the rod.

Decryption:Wrap a received strip over a wooden rodof the same diameter and read off thetext.

Example:troopsheadingnorthsendmorefood

−→ thgsr renee oaonf odrdo pitmo snhod

Monoalphabetic substitution cipher: Caesar

EncryptionReplace each letter with the one threepositions to the right in the alphabet.

DecryptionReplace each letter with the one threepositions to the left in the alphabet.

Example:

HABES OPINIONIS MEAE TESTIMONIUM

MDEHV RSNQNRQNV PHDH XHVXNPRQNZP

Polyalphabetic substitution cipher: Vigenere

EncryptionWrite the key over a message,repeating as necessary.Substitute each letter with the onefrom an appropriate column in theVigenere tableau.

DecryptionSame as encryption, use a rowinstead of a column.

Example:unitueuniloveyou

−→ cywoycih

Polyalphabetic substitution: Enigma

Operating principle: electromechanicalvarying map substitutionMain components:

3–5 rotors with pre-defined connectivityinter-rotor rings: mapping between letters andconnectionsletter swap by jumper cables

Key definition: rotor types, ring positions,jumper settingsTag UKW Walzenlage Ringstellung ---- Steckerverbindungen ----

31 B I IV III 16 26 08 AD CN ET FL GI JV KZ PU QY WX

30 B II V I 18 24 11 BN DZ EP FX GT HW IY OU QV RS

29 B III I IV 01 17 22 AH BL CX DI ER FK GU NP OQ TY

One-time pad ciphers

Encryption:Generate a random key sequence.Add a key to a message usingmodular arithmetic.

Decryption:Subtract a key from a message usingmodular arithmetic.

Example:7 (H) 4 (E) 11 (L) 11 (L) 14 (O) message

+ 23 (X) 12 (M) 2 (C) 10 (K) 11 (L) key

= 30 16 13 21 25 message + key

= 4 (E) 16 (Q) 13 (N) 21 (V) 25 (Z) mod 26

Feistel cipher: S and P boxes

S-boxComplex substitution controlled by a keySecure if enough internal statesUnrealizable for a large number of states

P-box

Block-wise permutation of digitsSimple transformation with maximalentropyInsecure against a “tickling attack”

Feistel network

Revival of the idea of a product cipherStrong polyalphabetic substitution via multiple roundsFollows theoretical principles of Shannon

A practical Feistel cipher

A multiple-round scheme withseparate keysEncryption:

Li+1 = Ri Ri+1 = Li ⊕ f (Ki, Ri)

Decryption: reverse the key order

Li+1 = Ri Ri+1 = Li⊕ f (Kn−i, Ri)

3 rounds suffice to achieve apseudorandom permutation

DES: Digital Encryption Standard

Adopted in 1977 after two rounds of proposalsWon by IBM’s Lucifer cipher based on Feistel’s designKey length reduced by NIST from 128 bits to 56 bitsSubject to extensive cryptanalysis research in 1990s’Broken by specialized hardware crackers in 1997–1999(fastest result: 22 hours 15 minutes by Deep Crack)Still widely used in practice (as 3DES)Replaced by Advanced Encryption Standard (AES) in 2000

DES overview

PermutedChoice 1

64 bit plaintext 56 bit key

InitialPermutation

Iteration 1PermutedChoice 2

Left CircularShift / 2

K1

Iteration 2PermutedChoice 2

Left CircularShift / 2

K2

Iteration 16PermutedChoice 2

Left CircularShift / 2

K16

32 bit Swap

Inverse InitialPermutation

64 bit ciphertext

...

DES round structure

Li-1 Ri-1

ExpansionPermutation

Ci-1 Di-1

Left Shift Left Shift

Permutation Contraction(Perm. Choice 2)+

4848 Ki

S-Box: Choice Substitution

Permutation

+

Li Ri Ci Di

48

32

32 bit 32 bit 28 bit 28 bit

32

Data to be encrypted Key used for encryption

Other symmetric ciphers

Block ciphersAlgorithm Key size Block size Rounds Applications3DES 112/168 64 48 Finance, PGP, S/MIMEAES 128/192/256 128 10/12/14 Repl. for DES/3DESIDEA 128 64 8 PGPBlowfish up to 448 64 16 Various softwareRC5 up to 2048 64 up to 255 Various software

Stream ciphersAlgorithm Key size IV State ApplicationsA5/2 54 114 64 GSMRC4 40-256 8 2064 WEP, WPA, SSL, SSH, Kerberos, etc.

Resume of symmetric cryptography

Provides (with some exceptions) a reliable means forenforcing confidentialityHighly efficientKey distribution is a major problem!

Asymmetric cryptography

Encryption

Alice Bob

Decryption

Bob’s public key

unitue

plaintext

I love you

plaintext

I love you

ciphertext

C ywoy cih

Bob’s private key

zxtr9y

specially generatedkeypair

Prime numbers

An integer p is a prime number if its only divisors are ±1 and±p.A positive integer c is said to be the greatest common divisorof a and b if

c is a divisor of a and of b;any divisor of a and of b is a divisor of c.

Integers a and b are said to be relatively prime if

gcd(a, b) = 1.

Euler’s totient function

A totient φ(n) of an integer n is the number of integers lessthan n that are relatively prime to n.Example:

φ(9) = 6 : {1, 2, 4, 5, 7, 8}

Two integers a and b are congruent modulo n, written asa ≡ b mod n, if

(a mod n) = (b mod n)

Euler’s Theorem: If a and n are relatively prime, then

aφ(n) ≡ 1 mod n.

RSA overview

Alice sends her love message to Bob via RSA:

Alice BobGenerate a keypair Ku / Kr

Send Ku to AliceEncrypt plaintext M with Ku

Send ciphertext C to BobDecrypt C with Kr

RSA key generation

Step ConditionSelect p, q p, q prime, p 6= qCompute n = p× qCompute φ(n) = (p− 1)(q− 1)Select 1 < e < φ(n) gcd(φ(n), e) = 1Compute d (de) mod φ(n) = 1 (∗)Public key Ku = {e, n}Private key Kr = {d, n}

RSA encryption and decryption

Encryption:

Plaintext: M < nCiphertext: C = Me mod n

Decryption:

Ciphertext: CPlaintext: M = Cd mod n

Correctness of RSA encryption

By the property (∗),

(de) mod φ(n) = 1 ⇒ ∃k : (de) = 1 + kφ(n).

Then,

M?≡ Cd mod n

≡ (Me)d mod n

≡ M(ed) mod n

≡ M1+kφ(n) mod n?≡ M mod n

Correctness of RSA encryption (ctd.)

For prime numbers p,

φ(p) = (p− 1).

By the key generation algorithm and the multiplicative property ofthe totient function,

φ(n) = φ(p) · φ(q) = (p− 1) · (q− 1).

By Euler’s Theorem, if p does not divide M,

M(p−1) = 1 mod p

and since (p− 1) divides φ(n)

M1+kφ(n) ≡ M mod p.

Similar argument holds for q and hence for n = pq.

What’s secret in RSA?

An attacker needs to know d to decrypt C.To find d, an attacker needs to solve (∗):

(de) mod φ(n) = 1.

For this, he needs to know φ(n).If p and q are known, then finding φ(n) is trivial:

φ(n) = (p− 1) · (q− 1)

However p and q are discarded during key generation.Factoring n into a product of two prime numbers is anintractable problem!Finding φ(n) directly is likewise intractable.

Other asymmetric ciphers

Algorithm E/D D.S. KEX HardnessRSA Yes Yes Yes FactorizationElGamal Yes No No DLPDSS No Yes No DLPDiffie-Hellmann No No Yes DLPElliptic curve Yes Yes Yes EC DLP

Summary

Cryptographic methods provide solutions for variousconfidentiality, integrity and authentication tasks.Symmetric cryptography is based on a single key that mustbe shared between the communication parties and keptsecret.Asymmetric cryptography is based two related keys; onlyone of them (private key) must be kept secret, the other one(public key) can be distributed over insecure media.

Next lecture

Cryptographic hash functionsDigital signatures