1 / 34

Secret Codes , Unforgeable Signatures , and Coin Flipping on the Phone Martin Tompa

Secret Codes , Unforgeable Signatures , and Coin Flipping on the Phone Martin Tompa Computer Science & Engineering University of Washington. Secret Codes , Unforgeable Signatures, and Coin Flipping on the Phone. What Is a Cryptosystem?. K AB. K AB. A Sender. B Receiver. M.

strohl
Download Presentation

Secret Codes , Unforgeable Signatures , and Coin Flipping on the Phone Martin Tompa

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. Secret Codes, Unforgeable Signatures, and Coin Flipping on the Phone Martin Tompa Computer Science & Engineering University of Washington

  2. Secret Codes, Unforgeable Signatures, and Coin Flipping on the Phone

  3. What Is a Cryptosystem? KAB KAB A Sender B Receiver M C = EAB(M) M = DAB(C)

  4. What Is a Cryptosystem? KAB KAB A Sender B Receiver M C = EAB(M) M = DAB(C) Cryptanalyst (bad guy)

  5. What Is a Cryptosystem? KAB KAB A Sender B Receiver M C = EAB(M) M = DAB(C) Cryptanalyst (bad guy) M C KAB Message Encryption Key Plaintext Cyphertext Cleartext

  6. What Is a Public Key Cryptosystem? KAB KAB A Sender B Receiver M C = EAB(M) M = DAB(C) Cryptanalyst (bad guy) M C KBEB Message Encryption Key Public Key Plaintext Cyphertext Private Key Cleartext

  7. The RSA Public Key Cryptosystem • Invented by Rivest, Shamir, and Adleman in 1977. • Has proven resistant to all cryptanalytic attacks.

  8. Receiver’s Set-Up • Choose 500-digit primes p and q, with p 2 (mod 3) and q 2 (mod 3) p = 5, q = 11

  9. Receiver’s Set-Up • Choose 500-digit primes p and q, with p 2 (mod 3) and q 2 (mod 3) p = 5, q = 11 • Let n = pq. n = 55

  10. Receiver’s Set-Up • Choose 500-digit primes p and q, with p 2 (mod 3) and q 2 (mod 3) p = 5, q = 11 • Let n = pq. n = 55 • Let s = (1/3) (2(p - 1)(q - 1) + 1). s = (1/3) (2  4  10 + 1) = 27

  11. Receiver’s Set-Up • Choose 500-digit primes p and q, with p 2 (mod 3) and q 2 (mod 3) p = 5, q = 11 • Let n = pq. n = 55 • Let s = (1/3) (2(p - 1)(q - 1) + 1). s = (1/3) (2  4  10 + 1) = 27 • Publish n. Keep p, q, and s secret.

  12. Encrypting a Message • Break the message into chunks. H I C H R I S …

  13. Encrypting a Message • Break the message into chunks. H I C H R I S …

  14. Encrypting a Message • Break the message into chunks. H I C H R I S … • Translate each chunk into an integer M (0 < M < n) by any convenient method. 8 9 3 8 18 9 19 …

  15. Encrypting a Message • Break the message into chunks. H I C H R I S … • Translate each chunk into an integer M (0 < M < n) by any convenient method. 8 9 3 8 18 9 19 … • Let E(M) = M3 mod n. M = 8, n = 55 83 = 512 = 9×55 + 17 E(8) = 17

  16. Decrypting a Cyphertext C • Let D(C) = Cs mod n. C = 17, n = 55, s = 27 1727 = 1,667,711,322,168,688,287,513,535,727,415,473 = 30,322,024,039,430,696,136,609,740,498,463 × 55 + 8 D(17) = 8

  17. Decrypting a Cyphertext C • Let D(C) = Cs mod n. C = 17, n = 55, s = 27 1727 = 1,667,711,322,168,688,287,513,535,727,415,473 = 30,322,024,039,430,696,136,609,740,498,463 × 55 + 8 D(17) = 8 • Translate D(C) into letters. H

  18. Decrypting a Cyphertext C Efficiently • C = 17, n = 55, s = 27 172 289  14 (mod 55) 174 172  172 14 14 196  31 (mod 55) 178 174  174 31 31 961  26 (mod 55) 1716 178  178 26 26 676  16 (mod 55) 1727 1716  178  172  171 16  26  14  17 416  14  17  31  14  17  434  17  (-6)  17  -102  8 (mod 55) D(17) = 8

  19. Why Does It Work? Euler’s Theorem (1736): Suppose • p and q are distinct primes, • n = pq, • 0 <M < n, and • k > 0. Then Mk(p-1)(q-1)+1mod n = M.

  20. Why Does It Work? Euler’s Theorem (1736): Suppose • p and q are distinct primes, • n = pq, • 0 <M < n, and • k > 0. Then Mk(p-1)(q-1)+1mod n = M. (M3)s = (M3) (1/3)(2(p-1)(q-1)+1) = M 2(p-1)(q-1)+1  M (mod n)

  21. Leonhard Euler 1707-1783

  22. Why Is It Secure? • To find M = D(C), you seem to need s.

  23. Why Is It Secure? • To find M = D(C), you seem to need s. • To find s, you seem to need p and q.

  24. Why Is It Secure? • To find M = D(C), you seem to need s. • To find s, you seem to need p and q. • All the cryptanalyst has is n = pq.

  25. Why Is It Secure? • To find M = D(C), you seem to need s. • To find s, you seem to need p and q. • All the cryptanalyst has is n = pq. • How hard is it to factor a 1000-digit number n? With the grade school method, doing 1,000,000,000 steps per second it would take …

  26. Why Is It Secure? • To find M = D(C), you seem to need s. • To find s, you seem to need p and q. • All the cryptanalyst has is n = pq. • How hard is it to factor a 1000-digit number n? With the grade school method, doing 1,000,000,000 steps per second it would take … 10483 years.

  27. State of the Art in Factoring • 1977: Inventors encrypt a challenge using “RSA129,” a 129-digit number n= pq. • 1981: Pomerance invents Quadratic Sieve factoring method. • 1994: Using Quadratic Sieve, RSA129 is factored over 8 months using 1000 computers on the Internet around the world. • 1999: Using a new method, RSA140 is factored.

  28. State of the Art in Factoring • 1977: Inventors encrypt a challenge using “RSA129,” a 129-digit number n= pq. • 1981: Pomerance invents Quadratic Sieve factoring method. • 1994: Using Quadratic Sieve, RSA129 is factored over 8 months using 1000 computers on the Internet around the world. • 1999: Using a new method, RSA140 is factored. • Using Quadratic Sieve, a 250-digit number would take 800,000,000 months instead of 8.

  29. Secret Codes, Unforgeable Signatures, and Coin Flipping on the Phone

  30. Signed Messages • How A sends a secret message to B A B C = EB(M) M = DB(C) C

  31. Signed Messages • How A sends a secret message to B A B C = EB(M) M = DB(C) • How A sends a signed message to B A B C = DA(M) M = EA(C) C C

  32. Signed and Secret Messages • How A sends a secret message to B ... A B C = EB(M) M = DB(C) • How A sends a signed secret message to B ... A B C = EB(DA(M)) M = EA (DB(C)) C C

  33. Secret Codes, Unforgeable Signatures, and Coin Flipping on the Phone

  34. Flipping a Coin Over the Phone A B Choose random x. y = EA(x) Guess ifxis even or odd. Checky = EA(x). • B wins if the guess about x was right, or y= EA(x). y “even” “odd” x

More Related