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Cryptography and Network Security Chapter 10

Cryptography and Network Security Chapter 10. Fifth Edition by William Stallings . Chapter 10 – Other Public Key Cryptosystems.

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Cryptography and Network Security Chapter 10

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  1. Cryptography and Network SecurityChapter 10 Fifth Edition by William Stallings

  2. Chapter 10 – Other Public Key Cryptosystems No Singhalese, whether man or woman, would venture out of the house without a bunch of keys in his hand, for without such a talisman he would fear that some devil might take advantage of his weak state to slip into his body. —The Golden Bough, Sir James George Frazer

  3. Diffie-Hellman Key Exchange • first public-key type scheme proposed • by Diffie & Hellman in 1976 along with the exposition of public key concepts • is a practical method for public exchange of a secret key • used in a number of commercial products

  4. Diffie-Hellman Key Exchange • a public-key distribution scheme • cannot be used to exchange an arbitrary message • rather it can establish a common key • known only to the two participants • value of key depends on the participants (and their private and public key information) • based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) - easy • security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard

  5. Discrete Logs Given Find x We denote this as Why is this hard?

  6. Diffie-Hellman Setup • all users agree on global parameters: • large prime integer or polynomial q • a being a primitive root mod q • each user (eg. A) generates their key • chooses a secret key (number): xA < q • compute their public key: yA = axA mod q • each user makes public that key yA

  7. Diffie-Hellman Key Exchange • shared session key for users A & B is KAB: KAB = axA.xB mod q = yAxB mod q (which B can compute) = yBxA mod q (which A can compute) • KAB is used as session key in private-key encryption scheme between Alice and Bob • if Alice and Bob subsequently communicate, they will have the same key as before, unless they choose new public-keys • attacker needs an x, must solve discrete log

  8. Diffie-Hellman Example • users Alice & Bob who wish to swap keys: • agree on prime q=353 and a=3 • select random secret keys: • A chooses xA=97, B chooses xB=233 • compute respective public keys: • yA=397 mod 353 = 40 (Alice) • yB=3233 mod 353 = 248 (Bob) • compute shared session key as: • KAB= yBxA mod 353 = 24897 = 160 (Alice) • KAB= yAxB mod 353 = 40233 = 160 (Bob)

  9. Key Exchange Protocols • users could create random private/public D-H keys each time they communicate • users could create a known private/public D-H key and publish in a directory, then consulted and used to securely communicate with them • both of these are vulnerable to a meet-in-the-Middle Attack • authentication of the keys is needed

  10. Man-i-the-middle attack on Diffie-Hellman Darth prepares for the attack by generating two random private keys XD1 and XD2 and then computing the corresponding public keys YD1=a XD1 mod q and YD2 =a XD2 mod q Alice transmitsYA to Bob. Darth intercepts YA but transmits YD1 to Bob. Darth also calculates K2 = (YA) XD2 mod q. Bob receives YD1 and calculates K1 = (YD1)XB mod q. Bob transmits YB to Alice. Darth intercepts YB but transmits YD2 to Alice. Darth calculates K1 = (YB) XD1 mod q. Alice receives YD2 and calculates K2 = (YD2)XA mod q. Allice and Bob think they share a secret key, but actually Bob and Darth share K1, and Alice and Darth share K2.

  11. ElGamal Cryptosystem • Another public-key cryptosystem like RSA. • Bob publishes (a, p, b), where 1 < m < p and b=aa • Alice chooses secret k, computes and sends to Bob the pair (r,t) where • r=ak (mod p) • t = bkm (mod p) • Bob calculates: tr-a=m (mod p) • Why does this decrypt?

  12. Bob publishes (a, p, b), where 1 < m < p and b=aa Alice chooses secret k, computes and sends to Bob the pair (r,t) where r=ak (mod p) t = bkm (mod p) Bob finds: tr-a=m (mod p) Why does this work? Multiplying m by bk scrambles it. Eve sees a, p, b, r, t. If she only knew a or k! Knowing a allows decryption. Knowing k also allows decryption. Why? Can’t find k from r or t.Why? ElGamal Cryptosystem

  13. Elliptic Curve Cryptography • majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials • imposes a significant load in storing and processing keys and messages • an alternative is to use elliptic curves • offers same security with smaller bit sizes • newer, but not as well analysed

  14. Elliptic Curve CryptographyAbelian Group • A set of elements G and operation * among elements (G,*) with some • Axioms: • (A1) Closure:  a,b  G, a*b  G • (A2) associative law:(a*b)*c = a*(b*c) • (A3) has identity e: e*a = a*e = a • (A4) has inverses a-1: a*a-1 = e • (A5) commutative law a*b = b*a

  15. Elliptic Curve CryptographyOperations • If the operation * is x, and we perform all operations mod q, a x a x ... x a = ak mod q • If the operation * is +, a + a + ... + a = k a k k

  16. Real Elliptic Curves • an elliptic curve is defined by an equation in two variables x & y, with coefficients • consider a cubic elliptic curve of form • y2 = x3 + ax + b • where x,y,a,b are all real numbers • also define zero point O • Note: More general form of the elliptical curve (Weierstrass equation): y2 + axy + by = x3 + cx2 + dx + e can be transformed to the form: y2 = x3 + ax + b • have addition operation for elliptic curve • geometrically sum of Q+R is reflection of intersection R

  17. Real Elliptic Curve Example - - - R= - -

  18. Elliptic Curve Cryptography Slope of the line between P and Q

  19. Elliptic Curve Cryptography • ECC addition is analog of modulo multiply • ECC repeated addition is analog of modulo exponentiation • need “hard” problem equiv to discrete log • Q= kP = P+P+…+P, where Q,P belong to a prime curve • is “easy” to compute Q given k, P • but “hard” to find k given Q, P • known as the elliptic curve logarithm problem

  20. Finite Elliptic Curves • Elliptic curve cryptography uses curves whose variables & coefficients are finite • have two families commonly used: • prime curves Ep(a,b) defined over Zp • use integers modulo a prime • best in software • binary curves E2m(a,b) defined over GF(2n) • use polynomials with binary coefficients • best in hardware

  21. Certicom example: E23(1,1) All operations work mod 23. Not all values for x and y satisfy y2 = x3 + x + 1 For x=9, only y=7 and y=16

  22. ECC Diffie-Hellman • can do key exchange analogous to D-H • users select a suitable curve Ep(a,b) • select base point G=(x1,y1) • with large order n s.t. nG=O • A & B select private keys nA<n, nB<n • compute public keys: PA=nAG, PB=nBG • compute shared key: K=nAPB,K=nBPA • same since K=nAnBG

  23. ECC Encryption/Decryption • several alternatives, will consider simplest • must first encode any message M as a point on the elliptic curve Pm • select suitable curve & point G as in D-H • each user chooses private key nA<n • and computes public key PA=nAG • to encrypt Pm : Cm={kG, Pm+kPb}, k random • decrypt Cm compute: Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm

  24. ECC Security • relies on elliptic curve logarithm problem • fastest method is “Pollard rho method” • compared to factoring, can use much smaller key sizes than with RSA etc • for equivalent key lengths computations are roughly equivalent • hence for similar security ECC offers significant computational advantages

  25. Comparable Key Sizes for Equivalent Security

  26. Legal use of ECC A lot of ECC techniques are patented by the company Certicom NSA bought some patents from them and made the royalty free via NIST standardisation BUT …

  27. Summary • have considered: • distribution of public keys • public-key distribution of secret keys • Diffie-Hellman key exchange • EllGamal encryption • Elliptic Curve cryptography

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