1 / 19

Public Key Cryptography

Public Key Cryptography. Bryan Pearsaul. Outline. What is Cryptology? Symmetric Ciphers Asymmetric Ciphers Diffie-Hellman RSA (Rivest/Shamir/Adleman) Moral Issues. Outline. Summary References. What is Cryptology?. The science of keeping data secure. Two transformation algorithms:

shada
Download Presentation

Public Key Cryptography

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Public Key Cryptography Bryan Pearsaul

  2. Outline • What is Cryptology? • Symmetric Ciphers • Asymmetric Ciphers • Diffie-Hellman • RSA (Rivest/Shamir/Adleman) • Moral Issues

  3. Outline • Summary • References

  4. What is Cryptology? • The science of keeping data secure • Two transformation algorithms: Enciphering and Deciphering • Symmetric ciphers • Asymmetric ciphers

  5. Symmetric Ciphers • Also known as private key • Both parties must agree on the key in advance • D_K(E_K(P)) = P • Not very computationally intensive • Key must be securely sent to both parties

  6. Symmetric Cipher Example D E E_K(X) Deciphering Enciphering D_K(E_K(X)) = X X K • k = 4 • Turn plaintext SECRET into ciphertext • S+4=W, E+4=I, C+4=G, R+4=V, E+4=I, T+4=X

  7. Symmetric Cipher Example • Much more elaborate transformations are available • Some that are so complicated that even if the transformation was public a key would still be needed • Still require a distributed key

  8. Asymmetric cipher • Also known as public key • Two keys: public k, private k’ • Private key not required for both parties • More computationally intensive D E E_K(X) Deciphering Enciphering D_K’(E_K(X)) = X X K K’

  9. Diffie-Hellman • One of the first public key cryptographic systems • Developed by Martin Hellman, Ralph Merkle, and Whitfield Diffie at Stanford University in 1976

  10. Diffie-Hellman • Based on a special case of the subset-sum, or knapsack, problem 20 11 8 6 5 4 Subset-sum Problem

  11. Diffie-Hellman Example • Block cipher • Block size of 7 bits. Possible 27 combinations • Private key (a’1, a’2, … , a’n) of 7 integers: (1, 2, 5, 11, 32, 87, 141) • Chose two special integers, w and m,such that w and m are relatively prime, • meaning gcd(w,m) = 1: w = 901, m = 1234 • Public key (a1, a2, … , an)of 7 integers using the equation: ai = w * a’i mod m: • (901, 568, 803, 39, 450, 645, 1173) • Partition SECRET into 7 bit blocks each block consisting of xn bits (x1, x2, …, xn) S 1010011 E 1000101 C 1000011 R 1010010 E 1000101 T 1010100 n • Bx = ∑ xiai i=1 • S = 1 X (901) + 0 X (568) + 1 X (803) + 0 X (39) + 0 X (450) + 1 X (645) + 1 X (1173) • S = 3522

  12. Diffie-Hellman Example • Encrypted blocks Bx received. Special version of subset-sum problem • Which subset of (a’1, a’2, … , a’n) sums to B’x where B’x = Bx*w-1 mod m • w-1 is the modular inverse of w for m, w*w-1 mod m = 1 • B’x = 3522 X (901)-1 mod 1234 • B’x = 3522 X 1171 mod 1234 • B’x = 234 • 1. sum← 0 • 2. for i = n step -1 until 1 do • if ai + sum <= B’x • then sum←sum + ai; • subset(i) ← 1 • else subset(i) ← 0 • 3. if sum = B’x then exit with subset • else exit with “failure” • Private key (1, 2, 5, 11, 32, 87, 141), B’x = 234, find subset (1, 0, 1, 0, 0, 1, 1) = S

  13. Diffie-Hellman • An algorithm that solves the particular problem on which a cryptographic system is based. • Two possible points of vulnerability • An algorithm which solves NP-complete problems quickly

  14. RSA • Factorization so far is unsolvable in polynomial-time • Developed by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT in 1977. • Based on the difficulty of factoring large numbers

  15. RSA Example • Find two large prime integers, p and q, and form product n = pq • Find a random integer, e, that is relatively prime to Ф(n) = (p-1)(q-1) • p and q are kept private, (n,e) are the public key • Message is partitioned into blocks, b, such that b < n • Each block is encrypted using the equation: c = be mod n • For the private key, calculate integer d which is the modular inverse of e • for Ф(n), or e * d mod Ф(n) = 1 • Once d is calculated it becomes your private key and all records of • p and q should be destroyed • Each encrypted block, c, is decrypted using the equation: b = cd mod n • p = 61, q = 53, n = 3233, Ф(n) = 3120, e = 17, d = 2753 • encrypt(123) = 12317 mod 3233 = 855 • decrypt(855) = 8552753 mod 3233 = 123

  16. RSA • Factorization cannot be done in polynomial-time • Security of RSA relies on two assumptions • Factoring is required to break the system

  17. Moral Issues • Information Theft • Privacy • Who does the data belong to?

  18. Summary • Diffie-Hellman and RSA • Cryptology • Symmetric and Asymmetric ciphers • Pros and Cons • Moral Issues

  19. References • A.K. Dewdney, The New Turning Omnibus, pp. 250-257, Henry Holt and Company, 2001. • RSA Cryptosystem, http://primes.utm.edu/glossary/page.php?sort=RSA. • Cryptology FAQ, http://www.faqs.org/faqs/cryptography-faq/part06/. • The Extended Euclidian Algorithm, http://www.grc.nasa.gov/WWW/price000/pfc/htc/zz_xeuclidalg.html. • A. Shamir, “A Polynomial-Time Algorithm for Breaking the Basic Merkle-Hellman Cryptosystem", Advances in Cryptology - CRYPTO '82 Proceedings, pp. 279-288, Plenum Press, 1983. IEEE Transactions on Information Theory, Vol. IT-30, pp. 699-704, 1984.

More Related