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15-251

15-251. Some. Great Theoretical Ideas in Computer Science. for. Modular Arithmetic and the RSA Cryptosystem. Lecture 17 (March 18, 2008). p-1. 1.  p. Public Key Cryptography. -----BEGIN PGP PRIVATE KEY BLOCK----- Version: PGP 8.1 - not licensed for commercial use: www.pgp.com

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15-251

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  1. 15-251 Some Great Theoretical Ideas in Computer Science for

  2. Modular Arithmetic and the RSA Cryptosystem Lecture 17 (March 18, 2008) p-1 1 p

  3. Public Key Cryptography -----BEGIN PGP PRIVATE KEY BLOCK----- Version: PGP 8.1 - not licensed for commercial use: www.pgp.com lQHNBEOedR4RBAC6bfed3ULzOwVF/BouyO8kfs8wkOmk3vaMF6+6JyeEJqyImaVh pVhU7lst6QtXTyAF734FtClM/9Dq9Dn7GDoO3E9+nGVO7wJ1OT+X4lgkoiM68WWG eioT958Hg0zq0KquHUBFM+Kldc+nr0e0Q6uCgwIYM7oH60/WX8e2WnvycwCgzba0 kRzxNtmw9w9IEQUk9pa2CK8D/2FjRxtEDN6nY7/l1wUrkMjI/uXYnsNWwrIAbwHp qhUZQYst27XpwNplAmI6YuS+3O+L4vgURj1hVcnNG+2bZXjbt4Fg3RLhrTV/jiuL ohBayAAdZZ4+72Cja1+18xp700GVTF96jYc8dIoxNx1AgHzQUffj3GnscAzo7ud3 HyYHA/9sI6Gijh/ubr1qTzHwZPdilDjfnEyQwR6+forUUegwCO0YawsC2lG6F7MG q3RHnJSwI3DiH/gY5bYk3XxhinkKxqk54DiL6vrHIw/E9J6RazY+ocicLRZ+6XjO JPXpK8s2v64pH5gyrsfANwSTTyECPx/hp2G+0BW/mtMU+JqFPP8DAwKvgRN08aTW 1WAW5/ak/URD4OAOT6OXlyg4YwhaJodb9vfwck4V8bnNLVNhbXBsZSBTZWN1cmVC bGFja2JveCBQR1Aga2V5PGluZm9AZWxkb3MuY29tPp0DJgRDnnWkEAgAxIwIsmEC mLGfQMH6GfNqn8XG9/bvT/nL/m7TjUVv1JGrKDASbASsPlYLnDel4opyp24Azu5x nQ0/QlsOifmXjINcWzNtWcJkW3rBamP1Vw6ZhxN8e1ARRrxwOn42V/yn/HoMFH0O Hhm2M5r8W/rOxo4dPdKPmRPdMaPVMndgM8WcOGPJ6TbMb4g9hphb+E4o3glI6fhP 41GZy57wWdKY9DnQ3W2PGoFaFAlQdyAtekzOmixp2/jXpq4+br9Yd088Unp2sw4Z 56j8sAbTk7NFpcTUSxsnFXpiGr0WkV+qfDyYTBvEnFXIteiThAWYzJgge6Tb6qzL 1XjD5uR1z2FgTwACAgf/fGBkVe8yETYwARrcn0xXv991AEvA4wlNI0/gygJKXxTV 0wTImdsOnsKsxtTCKdchSCFRuGYsBPzc7lUsPKk4jNkAv1EZELhqHkZTC51tEPxA m3/i8iAYFz2rTPfZkrjBJ2xP6ChRqmj0BTcZjo3bzRwjlUMTejGaR9BQDqB9A59z 2rCBn6nz9f6F+1oZ62gThjS6WUbUbhBKCwvfntFsXFrpXWHr5Apv+5zbxteTdHr0 1x3NRl0RzvzJ4wQ4WsVFiZFI3l7HykRd37XE+hSvRSr2iWqE/PdR7alxAswc5LET GCH//3xzP1/7Au3XLUaD4LaFoXY1bIBui2Srmu1kcP8DAwIYqvVK6L5Df2CejmTt hiA1DBnNck4dF7gPOaYku6Rfw27EOvhWmdZ1pp13uw2Tm6SEBoG7rkq1a01UWEjs PhUPkfxhVT6qHd4Bs3EOGSh7sNFsv8IbbAyP3rPOtbt3m9t02xEzKl5ZOqD85EZC HYK/l6lLD8pUX2dJQqZwTN4lkdl99HOf7XYPxHvCmbh1S1CgTM3H2wc5M7QROMhr -----END PGP PRIVATE KEY BLOCK----- -----BEGIN PGP PUBLIC KEY BLOCK----- Version: PGP 8.1 - not licensed for commercial use: www.pgp.com mQGiBEOedR4RBAC6bfed3ULzOwVF/BouyO8kfs8wkOmk3vaMF6+6JyeEJqyImaVh pVhU7lst6QtXTyAF734FtClM/9Dq9Dn7GDoO3E9+nGVO7wJ1OT+X4lgkoiM68WWG eioT958Hg0zq0KquHUBFM+Kldc+nr0e0Q6uCgwIYM7oH60/WX8e2WnvycwCgzba0 kRzxNtmw9w9IEQUk9pa2CK8D/2FjRxtEDN6nY7/l1wUrkMjI/uXYnsNWwrIAbwHp qhUZQYst27XpwNplAmI6YuS+3O+L4vgURj1hVcnNG+2bZXjbt4Fg3RLhrTV/jiuL ohBayAAdZZ4+72Cja1+18xp700GVTF96jYc8dIoxNx1AgHzQUffj3GnscAzo7ud3 HyYHA/9sI6Gijh/ubr1qTzHwZPdilDjfnEyQwR6+forUUegwCO0YawsC2lG6F7MG q3RHnJSwI3DiH/gY5bYk3XxhinkKxqk54DiL6vrHIw/E9J6RazY+ocicLRZ+6XjO JPXpK8s2v64pH5gyrsfANwSTTyECPx/hp2G+0BW/mtMU+JqFPM0tU2FtcGxlIFNl Y3VyZUJsYWNrYm94IFBHUCBrZXk8aW5mb0BlbGRvcy5jb20+wkkEEBECABMCmQEF AkOedaQJEKl84ZsqNet0AAAROgCggFcJOrmvNvpdmADv0iEzVUVci+gAoJDD9wbm WOq7M06k4rSOZSj2me0GuQINBEOedaQQCADEjAiyYQKYsZ9AwfoZ82qfxcb39u9P +cv+btONRW/UkasoMBJsBKw+VgucN6XiinKnbgDO7nGdDT9CWw6J+ZeMg1xbM21Z wmRbesFqY/VXDpmHE3x7UBFGvHA6fjZX/Kf8egwUfQ4eGbYzmvxb+s7Gjh090o+Z E90xo9Uyd2AzxZw4Y8npNsxviD2GmFv4TijeCUjp+E/jUZnLnvBZ0pj0OdDdbY8a gVoUCVB3IC16TM6aLGnb+Nemrj5uv1h3TzxSenazDhnnqPywBtOTs0WlxNRLGycV emIavRaRX6p8PJhMG8ScVci16JOEBZjMmCB7pNvqrMvVeMPm5HXPYWBPAAICB/98 YGRV7zIRNjABGtyfTFe/33UAS8DjCU0jT+DKAkpfFNXTBMiZ2w6ewqzG1MIp1yFI IVG4ZiwE/NzuVSw8qTiM2QC/URkQuGoeRlMLnW0Q/ECbf+LyIBgXPatM99mSuMEn bE/oKFGqaPQFNxmOjdvNHCOVQxN6MZpH0FAOoH0Dn3PasIGfqfP1/oX7WhnraBOG NLpZRtRuEEoLC9+e0WxcWuldYevkCm/7nNvG15N0evTXHc1GXRHO/MnjBDhaxUWJ -----END PGP PUBLIC KEY BLOCK-----

  4. Public Key Cryptography Secret M Private Key Public Key P EP[M]

  5. The RSA Cryptosystem Rivest, Shamir, and Adelman (1978) RSA is one of the most used cryptographic protocols on the net. Your browser uses it to establish a secure session with a site.

  6. Pick secret, random large primes: p,q “Publish”: n = p*q (n) = (p) (q) = (p-1)*(q-1) Pick random e  Z*(n) “Publish”: e Compute d = inverse of e in Z*(n) Hence, e*d = 1 [ mod (n) ] “Private Key”: d Mumbo jumbo… More Mumbo jumbo…

  7. But how does it all work? What is φ(n)? What is Zφ(n)*? … Why do all the steps work? To understand this, we need a little number theory...

  8. MAX(a,b) + MIN(a,b) = a+b n|m means that: m is an integer multiple of n. (We say that “n dividesm”.)

  9. Greatest Common Divisor: GCD(x,y) = greatest k ≥ 1 such that k|x and k|y Least Common Multiple: LCM(x,y) = smallest k ≥ 1 such that x|k and y|k Fact: GCD(x,y) × LCM(x,y) = x × y You can use MAX(a,b) + MIN(a,b) = a+b to prove the above fact.

  10. Modulus (a mod n) means: the remainder when a is divided by n If a = dn + r with 0 ≤ r < n Then r = (a mod n)and d = (a div n)

  11. Modular Equivalence a  b [mod n]  (a mod n) = (b mod n)  n|(a-b) Written as a n b, and spoken “a and b are equivalent modulo n” Example: 31  81 [mod 2] 31 2 81

  12. n is an Equivalence Relation In other words, it is: Reflexive: a n a Symmetric: (a n b)  (b n a) Transitive: (a n b and b n c)  (a n c) n induces a partition of Z into n classes a and b are said to be in the same “residue class” or “congruence class” when a n b

  13. a n b  n|(a-b) “a and b are equivalent modulo n” Define Residue class [i] = the set of all integers that are congruent to i modulo n Residue Classes Mod 3: [0] = { …, -6, -3, 0, 3, 6, ..} [1] = { …, -5, -2, 1, 4, 7, ..} [2] = { …, -4, -1, 2, 5, 8, ..} [-6] = { …, -6, -3, 0, 3, 6, ..} [7] = { …, -5, -2, 1, 4, 7, ..} [-1] = { …, -4, -1, 2, 5, 8, ..}

  14. Fact: equivalence mod n implies equivalence mod any divisor of n. If (x n y) and (k|n) then: x k y Example: 10 6 16  10 3 16 Proof: x n y  n|(x-y)  k|(x-y)  x k y

  15. Fundamental lemma of plus, minus, and times mod n: If (x n y) and (a n b), then: 1. x + a n y + b 2. x - a n y – b 3. x * a n y * b When doing plus, minus, and times modulo n, you can at any time replace a number with a number in the same residue class modulo n What is (249)(504) mod 251? When working mod 251: (-2)(2) = -4 = 247

  16. A Unique Representation System Modulo n: We pick exactly one representative from each residue class. We do all our calculations using these representatives.

  17. Unique Representation System Modulo 3 Finite set S = {0, 1, 2} + and * defined on S:

  18. Unique Representation System Modulo 3 Finite set S = {0, 1, -1} + and * defined on S:

  19. Perhaps The Most Convenient Set of Representatives The reduced system modulo n: Zn = {0, 1, 2, …, n-1} Define operations +n and *n: a +n b = (a+b mod n) a *n b = (a*b mod n)

  20. The Reduced System Modulo 2 Z2 = {0, 1} Two binary, associative operators on Z2:

  21. The Reduced System Modulo 2 Z2 = {0, 1} Two binary, associative operators on Z2:

  22. The Reduced System Z4 = {0,1,2,3}

  23. The Reduced System Z6 = {0,1,2,3,4,5} An operator has the permutation property if each row and each column has a permutation of the elements. For every n, +n on Zn has the permutation property

  24. What about multiplication?Does *6 on Z6 have the permutation property? NO!

  25. What about *8 on Z8? 0 3 6 1 4 7 2 5 0 4 0 4 0 4 0 4 Which rows have the permutation property?

  26. A visual way to understand multiplication and the “permutation property”.

  27. 0 7 1 2 6 5 3 4 The multiples of c modulo n is the set: {0, c, c +n c, c +n c +n c, ….} = {kc mod n | 0 ≤ k ≤ n-1} There are exactly 8 distinct multiples of 3 modulo 8: Hit all numbers  row 3 has the “permutation property”

  28. 0 7 1 2 6 5 3 4 There are exactly 2 distinct multiples of 4 modulo 8

  29. 0 7 1 2 6 5 3 4 There is exactly 1 distinct multiple of 8 modulo 8

  30. 0 7 1 2 6 5 3 4 There are exactly 4 distinct multiples of 6 modulo 8

  31. There number of distinct multiples of c modulo n is: LCM(n,c)/c = n/GCD(c,n) Hence only those values of c with GCD(c,n) = 1 have the permutation property for *n on Zn

  32. Theorem: There are exactly k = n/GCD(c,n) = LCM(c,n)/c distinct multiples of c modulo n, and these are: { c*i mod n | 0 ≤ i < k } Proof: Clearly, c/GCD(c,n) ≥ 1 is a whole number n 0 ck = cn/GCD(c,n) = n(c/GCD(c,n)) So there are at most k multiples of c mod n: c*0, c*1, c*2, …, c*(k-1) Also, k = all the factors of n missing from c  cx n cy  n|c(x-y)  k|(x-y)  x-y ≥ k Hence, there are exactlyk multiples of c.

  33. Fundamental lemma of plus, minus, and times mod n: If (x n y) and (a n b), then: 1. x + a n y + b 2. x - a n y – b 3. x * a n y * b

  34. Is there a fundamental lemma of division modulo n? cx n cy  x n y ? Of course not! If c=0[mod n], cx n cy for all x and y. Canceling the c is like dividing by zero.

  35. Let’s Fix That! Repaired fundamental lemma of division modulo n? if c  0 [mod n], thencx n cy  x n y ? 6*3 10 6*8, but not 3 10 8 2*2 6 2*5, but not 2 6 5 This also doesn’t work!

  36. When Can’t I Divide By c? Theorem: There are exactly n/GCD(c.n) distinct multiples of c modulo n Corollary: If GCD(c,n) > 1, then the number of multiples of c is less than n Corollary: If GCD(c,n) > 1 then you can’t always divide by c. Proof: There must exist distinct x,y < n such that c*x=c*y (but xy). Hence can’t divide.

  37. Fundamental lemma of division modulo n: if GCD(c,n)=1, then ca n cb  a n b ca n cb  n|(cb-ca) Proof:  n|c(b-a)  n|(b-a) (because GCD(c,n)=1)  a n b

  38. Fundamental lemma of division modulo n: if GCD(c,n)=1, then ca n cb  a n b Consider the set: Zn* = {x  Zn | GCD(x,n) =1} Multiplication over this set Zn*will have the cancellation property

  39. Z6 = {0, 1,2,3,4,5} Z6* = {1,5}

  40. Z12* = {0 ≤ x < 12 | gcd(x,12) = 1} = {1,5,7,11}

  41. Z5* = {1,2,3,4} = Z5 \ {0} For all primes p, Zp* = Zp \ {0}, since all 0 < x < p satisfy gcd(x,p) = 1

  42. Euler Phi Function (n) Define (n) = size of Zn* (number of 1 ≤ k < n that are relatively prime to n) If p is prime, then (p) = p-1

  43. Z12* = {0 ≤ x < 12 | gcd(x,12) = 1} = {1,5,7,11} (12) = 4

  44. Theorem: if p,q distinct primes then: f(pq) = (p-1)(q-1) Proof: # of integers from 1 to pq: pq # of multiples of q up to pq: p # of multiples of p up to pq: q # of multiples of both p and q up to pq: 1 f(pq) = pq – p – q + 1 = (p-1)(q-1)

  45. Additive Inverse The additive inverse of a  Zn is the unique b  Zn such that a +n b n 0. We denote this inverse by “–a” It is trivial to calculate: “-a” = n-a

  46. Multiplicative Inverse The multiplicative inverse of a  Zn* is the unique b  Zn*such that a *n b n 1 We denote this inverse by “a-1” or “1/a” The unique inverse of “a”must exist because the “a” row contains a permutation of the elements and hence contains a unique 1.

  47. Efficient Algorithm To Compute a-1 From a and n Run Extended Euclidean Algorithm on the numbers a and n It will give two integers r and s such that ra + sn = gcd(a,n) = 1 Taking both sides modulo n, we obtain: ra n 1 Output r, which is the inverse of a

  48. Fundamental Lemmas Until Now If (x n y) and (a n b), then: 1. x + a n y + b 2. x - a n y – b 3. x * a n y * b For a,b,c in Zn* then ca n cb  a n b

  49. NO! If (a n b) then xan xb ? (2 3 5) , but it is not the case that: 223 25

  50. Euler’s Theorem: a  Zn*, a(n)n 1 Fermat’s Little Theorem: p prime, a  Zp* ap-1p 1

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