1. LOGO Public Key Encryption Kyle Schmidt

2. A Brief History of Cryptography  Ancient Greeks  Scytale Cipher  Julius Caesar  Caesar Cipher  “Enigma”  Automated Cipher

3. What is Cryptography?  Secure and private communication  Encryption  Rendering a message unintelligible WEDNESDAY THE SIXTEENTH JRQARFQNL GUR FVKGRRAGU

4. Symmetric vs. Asymmetric  Symmetric  Single key  Asymmetric (Public Key)  Two keys Public key & Private key  Mailbox Concept  Digital Signature

5. Branches of Cryptology Cryptology CryptographyCryptanalysis SymmetricAsymmetric Encryption Message Authentication Encryption

6. Advantages of Asymmetric  Secure Exchange of Keys  Can’t trust the middleman  Nonrepudiation  Keep track of your own key  More Uses  Encryption  Message Authentication  Digital Signatures

7. Modular Arithmetic  Most cryptosystems based on finite, discrete sets modulus = 12

8. Modulus Operation  Formal Definition: Given integers a, r, and m, we say a ≡ r mod m if (r – a) is divisible by m a = r mod m  Note that there are infinitely many remainders  Not to be confused with:

9. The Ring Z m  Ring of integers with properties:  Arithmetic operations always yield result in Z m e.g. ∀ a, b ε Z m then (a + b) ε Z m  Neutral elements 0 for addition, 1 for multiplication e.g. ∀ a ε Z m, a + 0 ≡ a mod m  Additive inverse always exists i.e. ∀ a ε Z m, ∃ b = -a such that a + b ≡ 0 mod m  Multiplicative inverse only exists for some elements

10. Euclidean Algorithm  Calculates Greatest Common Divisor (GCD)  Simplify the problem  GCD(a, b) = GCD(a – b, b)

11. Euclidean Algorithm a = bq + r a = su b = tu r = a – bq r = (su) – (qt)u r = (s – qt)u a = bq + r a = (s’v)q + (t’v) a = (s’q + t)v b = s’v r = t’v

12. Euclidean Algorithm 1q 1 = a / ba = bq 1 + r 1 r 1 = a – b q 1 2q 2 = b / r 1 b = q 2 r 1 + r 2 r 2 = b – q 2 r 1 3q 3 = r 1 / r 2 r 1 = q 3 r 2 + r 3 r 3 = r 1 – q 3 r 2 nq n = r n-2 / r n-1 r n-2 = q n r n-1 + r n r n = r n-2 – q n r n-1 n+1q n+1 = r n-1 / r n r n-1 = q n+1 r n + 0--- Procedure of Euclidean Algorithm

13. Extended Euclidean Algorithm  Modular Division  Multiplication by multiplicative inverse  ba -1 instead of b/a  Multiplicative Inverse:  aa -1 ≡ 1 mod m  Extended Euclidean Algorithm:  Fast, efficient way to find multiplicative inverse

14. Extended Euclidean Algorithm  Perform regular Euclidean Algorithm  GCD(a, b) must be 1  Then for ax + by = 1,  x is the multiplicative inverse of a, and  y is the multiplicative inverse of b

15. Extended Euclidean Algorithm a = bq 1 + r 1 b = q 2 r 1 + r 2 r 1 = q 3 r 2 + r 3 r n-2 = q n r n-1 + 1 r 1 = a – bq 1 r 2 = b – q 2 r 1 r 3 = r 1 – q 3 r 2 1 = r n-2 – q n r n-1 1 = r n-2 – q n (r 1 – q 3 r 2 ) 1 = r n-2 – q n (r 1 – q 3 (b – q 2 r 1 )) 1 = r n-2 – q n (r 1 – q 3 (b – q 2 (a – b q 1 ))) 1 = ax + by

16. Extended Euclidean Algorithm  Proof ax + by = 1 ax + by  1 mod a by  1 mod a aa -1  1 mod a

17. Euler’s Totient Function  Essential for RSA Scheme  and most likely others  Totient  (n)  Number of totatives of an integer n  Totative: An integer m, 0 < m < n, GCD(m, n) = 1  Prime factorization of n must be known

18. {1, 2, 3, …, 30} Example:  (30) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (5) (2) (3) S = C = A = = B

19. Example:  (30)  Calculate totients from frequency  De Morgan’s Theorem:  Probability a number is in a subset is equal to Probability a number is not in all other subsets  Probability a number is NOT in a set is equal to 1 – (Probability of being IN the set)  Probability = (1 – 1/2) * (1 – 1/3) * (1 – 1/5)  Frequency = (1 – 1/2) * (1 – 1/3) * (1 – 1/5) * 30

20. Euler’s Totient Function  Formula:  (n) = n(1 – 1/p 1 )(1 – 1/p 2 )…(1 – 1/p m ) =  (n) = (p 1 – 1)p 1 k1–1 (p 2 – 1)p 2 k2–1 …(p m – 1)p m km–1

21. RSA  Ronald Rivest, Adi Shamir, Leonard Adleman  1977  Most widely used asymmetric scheme today  Two main uses:  Secure exchange of keys  Digital signatures

22. How RSA Works  Keys are pairs of integers  Encrypting key: (e, n)  Decrypting key: (d, n)  Encryption/Decryption: Exponentiation within Z n  Encrypt message: C = M e  Decrypt cyphertext: M = C d  Before encrypting:  Convert plaintext to integer with hash function

23. RSA: Key Generation 1.Choose two arbitrary prime numbers p and q 2.Calculate n = pq 3.Calculate  (n)  = (p – 1)(q – 1) 4.Choose arbitrary integer e <  (n) – 1 such that GCD(e,  (n) ) = 1 5.Calculate d = multiplicative inverse of e mod  (n) using Extended Euclidean Algorithm

24. RSA: Key Generation  Basic requirement:  After choosing p, q, choose e, d, k satisfying: ed – 1 = k(p – 1)(q – 1)  Extended Euclidean Algorithm requires two integers that are relatively prime  Thus, requiring e and  (n) to be relatively prime ensures that there will be a matching private key

25. How RSA Works  M e = C; C d = M  Prove C d ≡ (M e ) d ≡ M ed ≡ M mod n  Fermat’s Little Theorem  M  (n) ≡ 1 mod n if M and n are relatively prime  M k  (n) ≡ 1 mod n  M*M k  (n) ≡ M mod n  M k  (n)+1 ≡ M mod n M ed ≡ M mod n  ed – 1 = k(p – 1)(q – 1)  ed = k(p – 1)(q – 1) + 1  ed = k  (n) + 1

26. How RSA Works  M = M ed  = M 1+  (n)k  = (M)M  (n)k  = (M)(M  (n) ) k  = (M)(1) k  = M  M = M

27. RSA: Faster Encryption  “Square-and-Multiply” Algorithm  Quick and efficient, even with large numbers  Based on binary representation of exponent  Iterative through bits, left to right  Consider y = x h mod n  Starting with 2 nd bit from left: 1.Calculate y = x 2.Calculate y = y 2 mod n 3.If current bit of h is 1, calculate y = yx mod n 4.Repeat steps 2 and 3 for each bit in exponent

28. RSA: Faster Encryption  Example: y = 2 26 mod 5 IterationCurrent BitCalculationValue of y y = x2 11 [1] 0 1 0y = y 2 mod n4 mod 5 = 4 11 [1] 0 1 0y = y * x mod n8 mod 5 = 3 21 1 [0] 1 0y = y 2 mod n9 mod 5 = 4 31 1 0 [1] 0y = y 2 mod n16 mod 5 = 1 31 1 0 [1] 0y = y * x mod n2 mod 5 = 2 41 1 0 1 [0]y = y 2 mod n4 mod 5 = 4

29. RSA: Faster Encryption  Square-and-Multiply has complexity O(log n), where n is the number of bits in the exponent  Relatively efficient  Although still intensive for small devices  Speed up encryption more: smaller public key  No significant loss of security

30. RSA: Faster Decryption  Can’t use smaller private key  Major security loss  Chinese Remainder Theorem  Allows computation of y = x mod (pq) given: y 1 = x mod p and y 2 = x mod q  Break down C d mod n into smaller computations  More computations, but less intensive  Requires knowledge of p and q, thus cannot be used to speed up encryption

31. RSA: Faster Decryption  Variation of Fermat’s Little Theorem:  x p-1 ≡ 1 mod p  Using this, break down exponent d into d 1 = d mod (p – 1) and d 2 = d mod (q – 1)  Decryption now requires two exponentiations:  Using Chinese Remainder Theorem, compute: y ≡ y 1 q(q –1 mod p) + y 2 p(p –1 mod q) mod n  On average, four times faster

32. Practical Uses of RSA  Even with these methods to speed up RSA, it is still much slower than symmetric systems  Not typically used for large-scale encryption  Encrypt smaller messages  Passwords  Symmetric keys  Digital Signatures  Used together with symmetric systems  Secure key exchange + fast, efficient encryption

33. Problem  Modern computers becoming more efficient  Factoring large numbers is becoming easier Larger keys required for RSA to remain secure –RSA becoming slower and slower

34. Alternative  Elliptic Curve Cryptography (ECC)  1985  Neal Koblitz, Victor S. Miller  Estimated to be widespread within next decade

35. Elliptic Curve Cryptography: Premise  Point “Addition” (addition of ordered pairs)  Given a set E of points, and an operator “+”:  Compute “sum” of two points as another point  P + Q = R; P, Q, R ɛ E  NOT actual arithmetic addition  Point “Multiplication”  G = P + P + … + P k = kP; G, P ɛ E, k ɛ R

36. Elliptic Curve Cryptography: Premise  The set E is drawn from points of an elliptic curve  y 2 = x 3 + ax + b  Security comes from difficulty of finding k if given G and P  Elliptic Curve Discrete Logarithm Problem  Can’t just divide G by P Not arithmetic multiplication!  More similar to finding k in a = b k  No efficient algorithm exists to solve this problem

37. Computing P + Q  Since elliptic curves are cubic, there are generally three points a line intersects the curve  Use this fact to calculate P + Q 1.Draw line from P to Q 2.Define the third point of intersection to be –R 3.Thus R is the mirror reflection of –R

38. Computing P + Q  If there is no third point (the line is vertical), P + Q is said to be “infinity”, denoted as O  O is an additive identity (P + O = P)  To compute P + P, use P’s tangent line instead

39. Elliptic Curve Algebra  Algebraic Formulae:  P + Q x P+Q = β 2 – x P – x Q y P+Q = β(x P – x R ) – y P –β is the slope of the line  P + P (or 2P) x 2P = ([3x 2 P + a] / 2y P ) 2 – 2x P y 2 P = ([3x 2 P + a] / 2y P ) * (x P – x R ) – y P –a is the same parameter from the cubic equation

40. How it is Applied to Cryptography  To ensure security, some restrictions:  Curve must be smooth (no cusps, intersections, etc)  Can’t use all real numbers – must be discrete In particular, prime numbers or binary numbers  No longer a “curve,” but algebra still holds  Why ECC is harder to crack than RSA:  Algebra is more complex than factoring numbers

41. Secure Key Exchange  Variation of Diffie-Hellman Scheme 1.Alice and Bob agree on parameters for curve  a, b in y 2 = x 3 + ax + b and a point G ɛ E 2.Alice chooses a private integer X A and calculates a point Y A = X A G 3.Bob does similar, calculating Y B from integer X B 4.Alice and Bob publicly exchange Y A and Y B 5.The secret key K is computed by:  For Alice, K = X A Y B  For Bob, K = X B Y A

42. Secure Key Exchange  Alice and Bob get the same private key, because:  K = X A Y B  = X A (X B G)  = X B X A G  = X B Y A  = K

43. The Bigger Picture  ECC found to be 10x faster than RSA  Requires less memory and computational power  Equal security as RSA  Ideal for use on:  Smart cards  Wireless devices  Other constrained devices RSA is unsuitable for

44. The Bigger Picture  Security of RSA  Increasingly more vulnerable  Security of ECC  No significant increase in vulnerability over 25 years Symmetric Key SizeRSA Key SizeECC Key Size 801024160 1122048224 1283072256 1927680384 25615360521 NIST Recommended Key Sizes for Equal Security

45. References [1] Alayont, Feryâl. (2005). “ RSA: A Public Key Cryptosystem ”. [2] Kak, Avi. (2011). “ Elliptic Curve Cryptography and Digital Rights Management ”. Lecture Notes on Computer and Network Security. [3] Kotas, William A. (2000). “ A Brief History of Cryptography ”. University of Tennessee Honors Thesis Projects. [4] National Security Agency. (2009). “ The Case for Elliptic Curve Cryptography ”. [5] Paar, Christof and Pelzl, Jan. (2010). “ Introduction to Cryptography ”. Understanding Cryptography – A Textbook for Students and Practitioners (online slides). [6] Paar, Christof and Pelzl, Jan. (2010). “ The RSA Cryptosystem ”. Understanding Cryptography – A Textbook for Students and Practitioners (online slides). [7] RSA Laboratories. (2000). “ RSA Laboratories ’ Frequently Asked Questions About Today ’ s Cryptography, Version 4.1 ”. [8] Turner, Clay S. (2008). “ Euler ’ s Totient Function and Public Key Cryptography ”. [9] Vinck, A.J. Han. (2011). “ Introduction to Public Key Cryptography ”. [10] Wagner, Neal R. (2003). “ The RSA Public Key Cryptosystem ”. The Laws of Cryptography with Java Code. [11] Weisstein, Eric W. “ Euclidean Algorithm ”. MathWorld – A Wolfram Web Resource.

46. References  Additional images for this presentation retrieved from: http://en.wikipedia.org/wiki/Enigma_machine http://en.wikipedia.org/wiki/Public-key_cryptography http://www.usc.edu/dept/molecular-science/RSA-2003.htm http://en.wikipedia.org/wiki/Leonhard_Euler http://physicsworld.com/cws/article/news/47723 http://en.wikipedia.org/wiki/Credit_card