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Cry ptographic Concepts

CS 99j. Cry ptographic Concepts . John C. Mitchell Stanford University. Basic Concepts. Encryption scheme: functions to encrypt, decrypt data key generation algorithm Secret vs. public key Public key: publishing key does not reveal key -1

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Cry ptographic Concepts

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  1. CS 99j Cryptographic Concepts John C. Mitchell Stanford University

  2. Basic Concepts • Encryption scheme: • functions to encrypt, decrypt data • key generation algorithm • Secret vs. public key • Public key: publishing key does not reveal key-1 • Secret key: more efficient; can have key = key-1 • Hash function • map text to short hash key; ideally, no collisions • Signature scheme • functions to sign data and confirm signature

  3. 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: K  P  C decrypt: K  C  P such that for every key kK and plaintextpP decrypt(k, encrypt(k, p)) = p Good def’n for now, but doesn’t include key generation or prob encryption.

  4. 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, stanford) = vwdqirug

  5. 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,x) is same for all occurrences of letter x • can use letter-frequency tables, etc Brutus the cat from Rachel Levengood, www.yerf.com/leverach/

  6. 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 letter-to-letter substitution ciphers.

  7. 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)

  8. 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 Example one-time pad Plaintext Key Ciphertext Ciphertext Key Plaintext  =  =

  9. Evaluation of one-time pad • Advantages • Easy to compute encrypt, decryptfromkey, 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 ...

  10. 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

  11. 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 2 1 P(even| red) = 2/3 3 P(even) = 1/2 4 5 6

  12. H Sea 2 T T Sea Land 2 H Key Plaintext Land Ciphertext 1 1 Information-theoretic security • Cryptosystem is info-theoretically secure if P(Plaintext=p | Ciphertext=c) = P(Plaintext=p) • Ciphertext gives no info about plaintext Prob(1 is for Land) = Prob(1 is for Sea) assuming that all keys are equally likely

  13. In practice ... • Information-theoretic security is possible • Shift cipher, one-time pad are info-secure • But often not practical • Keys would be long • No public-key system • Therefore • Cryptosystems in use are either • Just found to be hard to crack, or • Based on computational notion of security

  14. Example cryptosystems • Feistel constructions • Iterate a “scrambling function” • Example: DES, … • Complexity-based cryptography • Multiplication, exponentiation are “one-way” functions • Examples: RSA, El Gamal, elliptic curve systems, ...

  15. Feistel networks • Many block algorithms are Feistel networks • Examples • DES, Lucifer, FREAL, Khufu, Khafre, LOKI, GOST, CAST, Blowfish, … • Standard form for • Iterating a function f on parts of a message • Producing invertible transformation

  16. L i-1 R i-1 L i R i Feistel network Divide n-bit input in half and repeat • Scheme requires • Function f(Ri-1 ,Ki) • Computation for Ki • e.g., permutation of key K • Advantage • Systematic calculation • Easy if f is table, etc. • Invertible if Ki known • Get Ri-1 from Li • Compute f(R i-1,Ki) • Compute Li-1 by  f K i 

  17. Data Encryption Standard • Developed at IBM, widely used • Feistel structure • Permute input bits • Repeat application of a S-box function • Apply inverse permutation to produce output • Appears to work well in practice • Efficient to encrypt, decrypt • Not provably secure

  18. 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 hard PSpace NP BPP P easy

  19. 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 = y@z 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|

  20. 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 • Extra “trapdoor” information makes 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)

  21. Public-key Cryptosystem • Trapdoor function to encrypt and decrypt • encrypt(key, message) • decrypt(key -1, encrypt(key, message)) = message • Resists attack • Cannot compute m from encrypt(key, m) and key (without key-1) key pair

  22. Example: RSA • Arithmetic modulo pq • Generate secret primes p, q • Generate secret numbers a, b with xab  x mod pq • Public encryption key n, a • Encrypt(n, a, x) = xa mod n • Private decryption key n, b • Decrypt(n, b, y) = yb mod n • Main properties • This works • Cannot compute b from n,a • Apparently, need to factor n = pq n

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

  24. 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

  25. 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”.

  26. 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 ...

  27. 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

  28. 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

  29. Properties of signatures • Functions to sign and verify • Sign(Key-1, message) • Verify(Key, x, m) = • Resists forgery • Cannot compute Sign(Key-1, m) from m and Key • Resists existential forgery: given Key, cannot produce Sign(Key-1, m) for any random or otherwise arbitrary m Look for where this is used! • true if x = Sign(Key-1, m) • false otherwise

  30. 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

  31. One-way hash functions • Length-reducing function h • Map arbitrary strings to strings of fixed length • One way • Given y, hard to find x with h(x)=y • Given m, hard to find m’ with h(m) = h(m’) • Collision resistant • Hard to find any distinct m, m’ with h(m)=h(m’)

  32. Applications of one-way hash • Password files (one way) • Digital signatures (collision resistant) • 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 for message authentication

  33. Summary • Encryption scheme: encrypt(key, plaintext) decrypt(key ciphertext) • Secret vs. public key • Public 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 collisions • Signature scheme • private key and public key provide “authentication” -1 -1 -1 -1

  34. Extra slides • Not used in WICS course

  35. Dan’s info on crypto history • Substitution cipher • 1400’s: Arabs did careful analysis of words in Koran • 1500’s: realized that with letter-frequency information, could break substitution ciphers • Vigenere cipher • Permutation cipher • Rotor machines

  36. 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

  37. 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)

  38. 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

  39. 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 | aA} 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|

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

  41. 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

  42. 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”.

  43. 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 ...

  44. 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

  45. Pad to x=x1x2 …xk xi f(xi-1) f g 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 x

  46. 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

  47. 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 withhash(gab)  msg

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