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Chapter 10 Key Management; Other Public Key Cryptosystems

Chapter 10 Key Management; Other Public Key Cryptosystems. Key Management. public-key encryption helps address key distribution problems have two aspects of this: distribution of public keys use of public-key encryption to distribute secret keys. Distribution of Public Keys.

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Chapter 10 Key Management; Other Public Key Cryptosystems

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  1. Chapter 10Key Management; Other Public Key Cryptosystems

  2. Key Management • public-key encryption helps address key distribution problems • have two aspects of this: • distribution of public keys • use of public-key encryption to distribute secret keys

  3. Distribution of Public Keys • can be considered as using one of: • Public announcement of public keys • Publicly available directory • Public-key authority • Public-key certificates

  4. Public Announcement • users distribute public keys to recipients or broadcast to community at large • eg. append PGP keys to email messages or post to news groups or email list • major weakness is forgery • anyone can create a key claiming to be someone else and broadcast it • until forgery is discovered can masquerade as claimed user

  5. Publicly Available Directory • can obtain greater security by registering keys with a public directory • directory must be trusted with properties: • contains {name, public-key} entries • participants register securely with directory • participants can replace key at any time • directory is periodically published • directory can be accessed electronically • still vulnerable to tampering or forgery

  6. Public-Key Authority • improve security by tightening control over distribution of keys from directory • has properties of directory • and requires users to know public key for the directory • then users interact with directory to obtain any desired public key securely • does require real-time access to directory when keys are needed

  7. Public-Key Authority

  8. Public-Key Certificates • certificates allow key exchange without real-time access to public-key authority • a certificate binds identity to public key • usually with other info such as period of validity, rights of use etc • with all contents signed by a trusted Public-Key or Certificate Authority (CA) • can be verified by anyone who knows the public-key authorities public-key

  9. Public-Key Certificates

  10. Public-Key Distribution of Secret Keys • use previous methods to obtain public-key • can use for secrecy or authentication • but public-key algorithms are slow • so usually want to use private-key encryption to protect message contents • hence need a session key • have several alternatives for negotiating a suitable session

  11. Simple Secret Key Distribution • proposed by Merkle in 1979 • A generates a new temporary public key pair • A sends B the public key and their identity • B generates a session key K sends it to A encrypted using the supplied public key • A decrypts the session key and both use • problem is that an opponent can intercept and impersonate both halves of protocol

  12. Simple Secret Key Distribution

  13. Public-Key Distribution of Secret Keys • if have securely exchanged public-keys:

  14. Diffie-Hellman Key Exchange • first public-key type scheme proposed • by Diffie & Hellman in 1976 along with the exposition of public key concepts • note: now know that James Ellis (UK CESG) secretly proposed the concept in 1970 • is a practical method for public exchange of a secret key • used in a number of commercial products

  15. 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

  16. Diffie-Hellman Example • users Alice & Bob who wish to swap keys: • agree on prime q=353 and α=3 • select random secret keys: • A chooses xA=97, B chooses xB=233 • compute 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)

  17. ElGamal Public Key Cryptosystem • security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard • User Alice wants to send a message m to Bob • Bob: q - prime number  - a primitive root of q X - private key  = X mod q – public key • Alice: • Downloads (, q, ) • Chooses a secret random integer k • Encryption: r  k mod q; t  km mod q • Send (r, t)=(k, km)to Alice • Bob: • Decryption: t/rX = m

  18. 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 • Certain conventional systems with a 4096-bit key size can be replaced by 313-bit elliptic curve system.

  19. Elliptic Curves Over Real Number • an elliptic curve is defined by an equation in two variables x & y, with coefficients • consider a cubic elliptic curve E(a, b) of • y2 = x3 + ax + b • 4a3 + 27b2 0 to form a group • where x,y,a,b are all real numbers • O: the point at infinity (or be called as zero point) • have addition operation for elliptic curve • geometrically sum of P+Q is reflection of the line L through P and Q intersecting E • If P=Q then L is the tangent line of P

  20. Elliptic Curves Over Real Number • O: point at infinity (or be called as zero point) • O: additive identity • O = -O • P+O=P • P=(x, y), -P=(x, -y); P+(-P)=O • P=(x, 0), 2P=O, 3P=P, 4P=O, … • P+Q+R=O  P, Q, R are collinear • Associative: (P+Q)+R=P+(Q+R) • Commutative: P+Q=Q+P

  21. Elliptic Curves Over Real Number E(a, b)= y2 = x3 + ax + b P1=(x1, y1), P2=(x2, y2), P1+P2=P3=(x3, y3) x3=m2-x1-x2 y3=m(x1-x3)-y1 if P1P2 then m=(y2-y1)/(x2-x1) if P1=P2 then m=(3x12+ a)/2y1 If m= then P3=O

  22. 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

  23. Elliptic Curves over Zp prime curves Ep(a,b) defined over Zp • y2 mod p = (x3 + ax + b) mod p • a,b: nonnegative integer and less than p • (4a3 + 27b2) mod p  0 to form a group • P(x, y): x and y are nonnegative integers and less than p • (x, y)  Ep(a,b)(x, -y)  Ep(a,b)

  24. Elliptic Curves over Zp prime curves Ep(a,b): y2 = (x3 + ax + b) • For each x (0x<p), calculate (x3 + ax + b) mod p • if (x3 + ax + b) is a square mod p then there are two square roots: y and –y. i.e. (x, y)Ep&(x, -y)Ep Ex: E23(1,1): y2 = (x3 + x + 1) mod 23 The number of points NEp is bounded by For large p, N is approximately equal to p+1 points.

  25. Elliptic Curves Over Zp Ep(a, b): y2 = x3 + ax + b (mod p), p5 is prime P1=(x1, y1), P2=(x2, y2), P1+P2=P3=(x3, y3) x3=(m2-x1-x2 ) mod p y3=(m(x1-x3)-y1) mod p if P1P2 then m=((y2-y1)/(x2-x1)) mod p if P1=P2 then m=((3x12+ a)/2y1) mod p If m= then P3=O

  26. Elliptic Curves Over Zp E23(1, 1): y2 = x3 + x + 1 (mod 23) P1=(3, 10), P2=(9, 7), P1+P2=(m2-x1-x2 , m(x1-x3)-y1)=(17, 20) m=(y2-y1)/(x2-x1)=7-10/9-3=-1/2=11 P1=(3, 10), P2=(3, 10), P1+P2=(m2-x1-x2 , m(x1-x3)-y1)=(7, 12) m=(3x12+ a)/2y1=(3*32+ 1)/2*10=1/4=6

  27. Elliptic Curve Cryptography • ECC addition is analog of modulo multiply • ECC multiplication is analog of modulo exponentiation

  28. Elliptic Curve Cryptography • ECC • given k, P, find Q=kP EASY • given Q, P, find k HARD • known as the elliptic curve logarithm problem • RSA • Given two primes p, q, find N=p*q EASY • Given N, find p, q HARD • known as factoring problem • Diffle-Hellman • Given p, k, find q=pk EASY • Given q, p find k HARD • known as discrete logarithm problem

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

  30. ECC Diffie-Hellman E211(0, -4): y2 = x3 - 4(mod 211) G=(2, 2) 240G=O nA=121, nB=203 compute public keys: PA=nA×G=(115, 48) PB=nB×G=(130, 203) compute shared key: K=nA×PB =(161, 69) K=nB×PA =(161, 69)

  31. 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=nA*G • to encrypt Pm : Cm={kG, Pm+k*Pb}, k random • to decrypt Cm : Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm

  32. ElGamal using ECC Encryption/Decryption E11(1, 6): y2 = x3 + x + 6(mod 11) G=(2, 7) M=(10, 9) nA=7,PA=nA*G=(7,2) to encrypt M : k=3 Cm={kG,M+k*PA}={(8,3), (10,2)} To decrypt Cm : M+kPA–nA(kG)=(10,2)–(3,5)=(10,2)+(3,6)=(10,9)

  33. Menezes-Vanstone ECC Encryption/Decryption • M =(m1, m2) • select suitable curve & point G as in D-H • each user chooses private key nA<n • and computes public key PA=nA*G • to encrypt M : Cm={kG, c1, c2}, k random kPA=(x, y)=k*nA*G c1 =x*m1 mod p c2 =y*m2 mod p • to decrypt Cm : kG*nA =(x, y)  (x-1, y-1) (c1*x-1, c2*y-1)= (m1, m2) = M

  34. Menezes-Vanstone ECC Encryption/Decryption E11(1, 6): y2 = x3 + x + 6(mod 11); G=(2, 7); M=(9, 1) nA=7,PA=nA*G=(7,2) to encrypt M : k=6 Cm={kG, c1, c2}= {(7, 9), 6, 3} kG=6(2, 7)=(7, 9) kPA=6(7, 2)=(8, 3) c1 =x*m1 mod p = 8*9 mod 11=6 c2 =y*m2 mod p = 3*1 mod 11=3 to decrypt Cm : kG*nA =7(7,9)=(8,3)  (x-1,y-1)=(7,4) (c1*x-1, c2*y-1)=(6*7, 4*3)=(9, 1)

  35. 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

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