Short Overview of Cryptography

(Lecture II)

John C. Mitchell Stanford 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) Secret vs. public key Public key: publishing key does not reveal key Secret key: more efficient; can have key -1 = key -1 Hash function map long text to short hash key; ideally, no collision Signature scheme public key -1 and private key provide “authentication”

Cryptosystem

 A cryptosystem consists of five parts     A set P A set C A set K of plaintexts of ciphertexts of keys A pair of functions encrypt : K  P  C decrypt : K  C  P such that for every key k  K and plaintext decrypt(k, encrypt(k, p)) = p 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 Set C Set K of plaintexts {a, b, c, … , x, y, z} of ciphertexts {a, b, c, … , x, y, z} 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 ~ 0.04

CUMWFGYPB 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 Set C of plaintexts: all n-bit sequences 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

Plaintext 0 1 0 1 0 1 0  Key 1 1 0 1 0 0 1 Ciphertext = 1 0 0 0 0 1 1 Ciphertext 1 0 0 0 0 1 1  Key 1 1 0 1 0 0 1 = Plaintext 0 1 0 1 0 1 0

Evaluation of one-time pad

  Advantages   Easy to compute encrypt , decrypt from As hard to break as possible  This is an information-theoretically secure cipher  key, text 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

L i-1 L i R i-1 f  R i K i   Function f(R i-1 ,K i )   Expand R i-1 and XOR w/ K i Divide into 8 6-bit blocks   Apply “S-box” table-lookup functions to each block Permute resulting bits K i  is permutation of key K Invertible if K known See Biham and Shamir for analysis

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

hard

PSpace NP

easy

BPP P 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

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) = 3 x f(x) = x 3 mod p, where p is prime 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 {f a   :D f  R f | a  A} is one-way if Efficient algorithm for f a (x), given a, x No efficient alg computes x, given a, f a (x) where we assume D f , R f 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) = x 3 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 e  x = x  e = x inverse ( ) -1 with x  x -1 = x -1  x = e Cyclic group  Group G =  G,  , e, ( ) -1   with G = { g 0 , g 1 , g 2 , ... , g k = g 0 }   element g is called a generator of G number of distinct elements if called the order of group

Number theory for RSA

  Group Z n * 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

  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 Z n * 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  Encrypt(x) = x a mod n   1 mod  (n) Decrypt(y) = y b mod n Since ab   1 mod  (n), have x ab  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 = k e * m e = 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(k e *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) = x a mod n Verify(y) = y b mod n Since ab  1 mod  (n), have x ab  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 : 2 2k -> 2 k reduces bits by 2 Repeat h 0 = some seed h i+1 = f(h i , x i ) Some final function g completes calculation f(x i-1 Pad to x=x ) f x x 1 i x 2 …x k g

General Basis for Cryptography

  Cyclic group with one-way properties   multiplication, inverse easy to compute discrete log   a, a n   n not in O(log 2 Note: randomized algorithm in O(sqrt |G|) |G|) Examples  Integers modulo prime p  Elliptic curve groups Important: complexity depends on group presentation

Public-Key Cryptography

[ElGamal]     Public encryption key:  Decryption  Compute g -ab = ((g b ) a ) -1  Decrypt g -ab  g ab  msg g, g a Private decryption key: a Encryption function   Choose random b  [2, |G|-1] Send encrypt(msg) =  g b , g ab   msg  This is classical algorithm; better security with hash(g ab )  msg