中国矿业大学：密码学_Public Key Cryptography

Public Key cryptography 曹天杰 Tianjie Cao ticao(cumt. edu. cn College of Computer science and echnology, China University of Mining and Technology Xuzhou, China 中国矿业大学计算机科学与技术学院 2003.6.2

1 曹天杰 Tianjie Cao tjcao@cumt.edu.cn College of Computer Science and Technology, China University of Mining and Technology, Xuzhou, China 中国矿业大学计算机科学与技术学院 2003.6.2 Public Key Cryptography

Diffie-Hellman Diffie-hellman is a public key distribution scheme First public-key type scheme, proposed in 1976 WDiffie Me Hellman. "New directions ir Cryptography", IEEE Trans. Information Theory, IT-22, pp644-654, Nov 1976 2

2 Diffie-Hellman • Diffie-Hellman is a public key distribution scheme • First public-key type scheme, proposed in 1976. – W Diffie, M E Hellman, "New directions in Cryptography", IEEE Trans. Information Theory, IT-22, pp644-654, Nov 1976

whitfield Diffie Martin Hellman

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Diffie-Hellman Public-key distribution scheme Cannot be used to exchange an arbitrary message Exchange only a key, whose value depends on the participants(and their private and public key information) The algorithm is based on exponentiation in a finite(Galois) field, either over integers modulo a prime, or a polynomial field

4 Diffie-Hellman • Public-key distribution scheme • Cannot be used to exchange an arbitrary message • Exchange only a key, whose value depends on the participants (and their private and public key information) • The algorithm is based on exponentiation in a finite (Galois) field, either over integers modulo a prime, or a polynomial field

Diffie-Hellman Security relies on the difficulty of computing logarithms in these fields discrete logarithms takes O(e log n log log n)operations The algorithm two people alice and bob who wish to exchange some key over an insecure communications channe They select a large prime p(200 digit), such as(p )2 should also be prime They also select g, a primitive root mod p g is a primitive if for each n from 0 to p-l, there exists some a where ga=n mod p

5 Diffie-Hellman • Security relies on the difficulty of computing logarithms in these fields – discrete logarithms takes O(e log n log log n) operations • The algorithm: – two people Alice and Bob who wish to exchange some key over an insecure communications channel. – They select a large prime p (~200 digit), such as (p- 1)/2 should also be prime – They also select g, a primitive root mod p • g is a primitive if for each n from 0 to p-1, there exists some a where ga = n mod p

Diffie-Hellman · The algorithn The values of g and p dont need to be secret Alice then chooses a secret number a Bob also chooses a secret number b Alice and Bob compute ya and yg respectively, which are then exchanged ya -ga mod p yB= moc Both alice and bob can calculate the key as KaR=gab mod p yab mod p( which B can compute) b mod p (which A can compute) The key may then be used in a private-key cipher to secure communications between a and B

6 Diffie-Hellman • The algorithm – The values of g and p don’t need to be secret – Alice then chooses a secret number a – Bob also chooses a secret number b – Alice and Bob compute yA and yB respectively, which are then exchanged • yA = ga mod p yB = gb mod p – Both Alice and Bob can calculate the key as • KAB = gab mod p = yA b mod p (which B can compute) = yB a mod p (which A can compute) – The key may then be used in a private-key cipher to secure communications between A and B

Diffie-Hellman alice a mod p Bob g mod The private key is: gab mod p where p is a prime and g is a generator of Z

7 Alice g a mod p Bob g b mod p The private key is: g ab mod p where p is a prime and g is a generator of Zp Diffie-Hellman

Diffie-Hellman Knows ga, gb, g, and p So we want to find the key k b This is believed to be hard If one knows how to compute discrete logs efficient ntly then one can break this scheme (and other schemes based on public key cryptography)

8 • Knows g a , gb , g, and p • So we want to find the key, k – k = gab – This is believed to be hard. • If one knows how to compute discrete logs efficiently, then one can break this scheme (and other schemes based on public key cryptography) Diffie-Hellman

Diffie-Hellman Can be expanded to be used with many parties · Can be extended to: Finite fields gFp Elliptic curves Galois field GF2 Computing discrete logarithms is closely related to factoring. If you can solve the discrete logarithm problem. the len you can factor. (The converse has never been proven to be true

9 Diffie-Hellman • Can be expanded to be used with many parties • Can be extended to: – Finite fields GFp – Elliptic curves – Galois field • Computing discrete logarithms is closely related to factoring. If you can solve the discrete logarithm problem, then you can factor. (The converse has never been proven to be true.) GF k 2

El Gamal a variant of the Diffie-Hellman key distribution scheme, allowing secure exchange of messages Published in 1985 by Elgamal in T ElGamal, "A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms IEEE Trans. Information Theory, vol IT-31(4), pp469 472,July1985 Like Diffie-Hellman its security depends on the difficulty of factoring logarithms

10 El Gamal • A variant of the Diffie-Hellman key distribution scheme, allowing secure exchange of messages • Published in 1985 by ElGamal in – T. ElGamal, "A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms", IEEE Trans. Information Theory, vol IT-31(4), pp469- 472, July 1985. • Like Diffie-Hellman its security depends on the difficulty of factoring logarithms