Short Overview of Cryptography (Lecture II)

John C. MitchellStanford University

Some philosophy (my opinions)

Do something useful with your life Computers can do many things Have fun! Do something that matters

Learn something about the problems you solve If you are going to do graphics, study visual art If you work on computational biology, try to learn a

little organic chemistry If we are going to analyze security protocols, we

should learn a few things about cryptography

Some security objectives

Secrecy Info not revealed

Authentication Know identity of

individual or site Data integrity

Msg not altered Message

Authentication Know source of msg

Receipt Know msg received

Access control Revocation Anonymity Non-repudiation

Some Basic Concepts

Encryption scheme: encrypt(plaintext,key) decrypt(ciphertext,key )

Secret vs. public keyPublic key: publishing key does not reveal key

Secret key: more efficient; can have key = key Hash function

map long text to short hash key; ideally, no collision

Signature schemepublic key and private key provide “authentication”

-1

-1

-1

-1

Cryptosystem

A cryptosystem consists of five parts A set P of plaintexts A set C of ciphertexts A set K of keys A pair of functions encrypt: KPC decrypt: KCP such that for every key kK and plaintext pP decrypt(k, encrypt(k, p)) = p

Good def’n for now, but doesn’t include key generation or prob encryption.

Primitive Example: Shift Cipher

Shift letters using mod 26 arithmetic Set P of plaintexts {a, b, c, … , x, y, z} Set C of ciphertexts {a, b, c, … , x, y, z} Set K of keys {1, 2, 3, … , 25} Encryption and decryption functions encrypt(key, letter) = letter + key (mod 26)

decrypt(key, letter) = letter - key (mod 26)

Example encrypt(3, marktoberdorf) = pdunwrehugrui

Evaluation of Shift Cipher

Advantages Easy to encrypt, decrypt Ciphertext does look garbled

Disadvantages Not very good for long sequences of English words

Few keys -- only 26 possibilities Regular pattern

• encrypt(key,e) is same for all occurrences of letter e• can use letter-frequency tables, etc

Letter frequency in English

Five frequency groups [Beker and Piper]

E has probability 0.12 TAOINSHR have probability 0.06 - 0.09 DL have probability ~ 0.04CUMWFGYPB have probability 0.015 - 0.028 VKJXQZ have probability < 0.01

Possible to break many letter-to-letter substitution ciphers.

One-time Pad

Secret-key encryption scheme (symmetric) Encrypt plaintext by xor with sequence of bits Decrypt ciphertext by xor with same bit sequence

Scheme for pad of length n Set P of plaintexts: all n-bit sequences Set C of ciphertexts: all n-bit sequences Set K of keys: all n-bit sequences Encryption and decryption functions encrypt(key, text) = key text (bit-by-bit)

decrypt(key, text) = key text (bit-by-bit)

Example one-time pad

10

01010

PlaintextKey Ciphertext KeyPlaintext

11

01001

01

00011

11

01001

10

01010

= =

01

00011

Ciphertext

Evaluation of one-time pad

Advantages Easy to compute encrypt, decrypt from key, text As hard to break as possible

This is an information-theoretically secure cipher Given ciphertext, all possible plaintexts are equally likely,

assuming that key is chosen randomly

Disadvantage Key is as long as the plaintext

How does sender get key to receiver securely?

Idea can be combined with pseudo-random generators ...

What is a “secure” cryptosystem?

Idea If an enemy intercepts your ciphertext, cannot

recover plaintext Issues in making this precise

What else might your enemy know? The kind of encryption function you are using Some plaintext-ciphertext pairs from last year Some information about how you choose keys

What do we mean by “cannot recover plaintext” ? Ciphertext contains no information about plaintext No efficient computation could make a reasonable guess

Information-theoretic Security

Remember conditional probability... Random variables X, Y, … Conditional probability P(X=x|Y=y)

Probability that X takes value x, given that Y=y

Apply to plaintext, ciphertext Cryptosystem is info-theoretically secure if

P(Plaintext=p | Ciphertext=c) = P(Plaintext=p)

Ciphertext gives no advantage in guessing the plaintext.

Data Encryption Standard

Developed at IBM, widely used Regular structure

Permute input bits Repeat application of a certain function Apply inverse permutation to produce output

Appears to work well in practice Efficient to encrypt, decrypt Not provably secure

One round of DES

Function f(Ri-1 ,Ki) Expand Ri-1 and XOR w/ Ki

Divide into 8 6-bit blocks Apply “S-box” table-lookup

functions to each block Permute resulting bits

Ki is permutation of key K Invertible if K known

See Biham and Shamir for analysis

L i-1 R i-1

R iL i

f

K i

Properties of DES

Not a simple mathematical function Difficult to analyze All operations are linear except “S-boxes”

Security depends on “magic” S-box functions These were designed secretly by NSA

• No S-box is a linear function• Changing one input bit changes two output bits

Efficient to compute Combination of bit operations and table lookup

Differential cryptanalysis of DES Can break 8-round DES, but not 16-round DES (yet)

Complexity-based Cryptography

Some computational problems provably hard Undecidability of halting problem Presburger arithmetic is non-elementary Commutative semi-groups require exponential space

Some problems are believed intractable NP-complete optimization problems

Traveling salesman as hard as any problem in NP No known polynomial time algorithm, in spite of effort

Factoring is not believed to be poly-time Not NP-complete, but many years of effort

Still, useful to relate crypto to standard problems

Review: Complexity Classes

Answer in polynomial space may need exhaustive search

If yes, can guess and check in polynomial time

Answer in polynomial time, with high probability

Answer in polynomial time compute answer directly

P

BPP

NP

PSpace

easy

hard

One-way functions

A function f is one-way if it is Easy to compute f(x), given x Hard to compute x, given f(x), for most x

Examples (we believe) f(x) = divide bits, x = yz, and multiply f(x)=y*z f(x) = 3x mod p, where p is prime f(x) = x3 mod pq, where p,q are primes with |p|

=|q|

Easy and hard (more precisely)

For any finite f, can build a table and invert f Measure “hardness” using classes of functions

Want this to be hard as a function of choice of f

A class {fa :Df Rf | aA} is one-way if Efficient algorithm for fa (x), given a, x

No efficient alg computes x, given a, fa (x)

where we assume Df , Rf finite and measure running time as a function of |a|

One-way trapdoor

A function f is one-way trapdoor if Easy to compute f(x), given x Hard to compute x, given f(x), for most x There is extra “trapdoor” information making it

easy to compute x from f(x)

Example (we believe) f(x) = x3 mod pq, where p,q are primes with |p|

=|q| Compute cube root using (p-1)*(q-1)

Group theory for RSA

Group G = G, , e, ( )-1 Set of elements with

associative “multiplication” identity e with ex = xe = x inverse ( )-1 with xx-1 = x-1 x = e

Cyclic group Group G = G, , e, ( )-1 with

G = { g0, g1 , g2 , ... , gk = g0} element g is called a generator of G number of distinct elements if called the order of group

Number theory for RSA

Group Zn* of integers relatively prime to n

multiplication mod n is associative operation 1 is identity x-1 computed by Euclidean algorithm for gcd order of group is (n) = | { k<n | gcd(k,n) =1 } |

What if x not relatively prime to n? Can have zero divisors, no multiplicative inverse

If y divides x and n, then yi=x, yj=n and therefore xj = yij 0 mod n

Only numbers relatively prime to n form group

RSA Encryption

Let p, q be two distinct primes and let n=p*q Encryption, decryption based on group Zn

* For n=p*q product of primes, (n) = (p-1)*(q-1)

Proof: (p-1)*(q-1) = p*q - p - q + 1

Key pair: a, b with ab 1 mod (n) Encrypt(x) = xa mod n Decrypt(y) = yb mod n Since ab 1 mod (n), have xab x mod n

Proof: if gcd(x,n) = 1, then by general group theory, otherwise use “Chinese remainder theorem”.

How well does this work?

Can generate modulus, keys fairly efficiently Efficient rand algorithms for generating primes p,q

May fail, but with low probability

Given primes p,q easy to compute n=p*q and (n) Choose a randomly with gcd(a, (n))=1 Compute b = a-1 mod (n) by Euclidean algorithm

Public key n, a does not reveal b This is not proven, but believed

But if n can be factored, all is lost ...

Message integrity

Theoretically, a weak point encrypt(k*m) = (k*m)e = ke * me

= encrypt(k)*encrypt(m) This leads to “chosen ciphertext” form of attack

If someone will decrypt new messages, then can trick them into decrypting m by asking for decrypt(ke *m)

Implementations reflect this problem “The PKCS#1 … RSA encryption is intended

primarily to provide confidentiality. … It is not intended to provide integrity.” RSA Lab. Bulletin

Recall security objectives

Secrecy Info not revealed

Authentication Know identity of

individual or site Data integrity

Msg not altered Message

Authentication Know source of msg

Receipt Know msg received

Access control Revocation Anonymity Non-repudiation

Digital Signatures

Public-key encryption Alice publishes encryption key Anyone can send encrypted message Only Alice can decrypt messages with this key

Digital signature scheme Alice publishes key for verifying signatures Anyone can check a message signed by Alice Only Alice can send signed messages

RSA Signature Scheme

Publish decryption instead of encryption key Alice publishes decryption key Anyone can decrypt a message encrypted by

Alice Only Alice can send encrypt messages

In more detail, Alice generates primes p, q and key pair a, b Sign(x) = xa mod n Verify(y) = yb mod n Since ab 1 mod (n), have xab x mod n

Cryptographic hash functions

Function h with two main properties Map arbitrary strings to strings of fixed length Given h(x), impractical to find y with h(y)=h(x)

Variety of uses More efficient digital signatures

Sign hash of message instead of entire message

Data integrity Compute and store hash of some data Check later by recomputing hash and comparing

Keyed hash fctns provide message authentication ???

Iterated hash functions

Repeat use of block cipher (like DES, …) Pad input to some multiple of block length Iterate a length-reducing function f

f : 22k -> 2k reduces bits by 2 Repeat h0= some seed

hi+1 = f(hi, xi)

Some final function g completes calculation

Pad to x=x1x2 …xk

f

g

xi

f(xi-1)

x

General Basis for Cryptography

Cyclic group with one-way properties multiplication, inverse easy to compute discrete log a, an n not in O(log2 |G|)

Note: randomized algorithm in O(sqrt |G|)

Examples Integers modulo prime p Elliptic curve groups

Important: complexity depends on group presentation

Public-Key Cryptography [ElGamal]

Public encryption key: g, ga Private decryption key: a Encryption function

Choose random b [2, |G|-1] Send encrypt(msg) = gb , gab msg

Decryption Compute g-ab = ((gb)a) -1

Decrypt g-ab gab msg

This is classical algorithm; better security with hash(gab) msg