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Lecture 3.2: Public Key Cryptography II

Lecture 3.2: Public Key Cryptography II. CS 436/636/736 Spring 2012 Nitesh Saxena. Today’s Informative/Fun Bit – Acoustic Emanations. http://www.google.com/search?source=ig&hl=en&rlz=&q=keyboard+acoustic+emanations&btnG=Google+Search http://tau.ac.il/~tromer/acoustic/.

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Lecture 3.2: Public Key Cryptography II

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  1. Lecture 3.2: Public Key Cryptography II CS 436/636/736 Spring 2012 Nitesh Saxena

  2. Today’s Informative/Fun Bit – Acoustic Emanations • http://www.google.com/search?source=ig&hl=en&rlz=&q=keyboard+acoustic+emanations&btnG=Google+Search • http://tau.ac.il/~tromer/acoustic/ Public Key Cryptography -- II

  3. Course Administration • HW2 – due at 11am on Feb 06 • Any questions, or help needed? Public Key Cryptography -- II

  4. Outline of Today’s Lecture • Number Theory • Modular Arithmetic Public Key Cryptography -- II

  5. Modular Arithmetic • Definition: x is congruent to y mod m, if m divides (x-y). Equivalently, x and y have the same remainder when divided by m. Notation: Example: • We work in Zm = {0, 1, 2, …, m-1}, the group of integers modulo m • Example: Z9 ={0,1,2,3,4,5,6,7,8} • We abuse notation and often write = instead of Public Key Cryptography -- II

  6. Addition in Zm : • Addition is well-defined: • 3 + 4 = 7 mod 9. • 3 + 8 = 2 mod 9. Public Key Cryptography -- II

  7. Additive inverses in Zm • 0 is the additive identity in Zm • Additive inverse of a is -a mod m = (m-a) • Every element has unique additive inverse. • 4 + 5= 0 mod 9. • 4 is additive inverse of 5. Public Key Cryptography -- II

  8. Multiplication in Zm : • Multiplication is well-defined: • 3 * 4 = 3 mod 9. • 3 * 8 = 6 mod 9. • 3 * 3 = 0 mod 9. Public Key Cryptography -- II

  9. Multiplicative inverses in Zm • 1 is the multiplicative identity in Zm • Multiplicative inverse (x*x-1=1 mod m) • SOME, but not ALL elements have unique multiplicative inverse. • In Z9 : 3*0=0, 3*1=3, 3*2=6, 3*3=0, 3*4=3, 3*5=6, …, so 3 does not have a multiplicative inverse (mod 9) • On the other hand, 4*2=8, 4*3=3, 4*4=7, 4*5=2, 4*6=6, 4*7=1, so 4-1=7, (mod 9) Public Key Cryptography -- II

  10. Which numbers have inverses? • In Zm, x has a multiplicative inverse if and only if x and m are relatively prime or gcd(x,m)=1 • E.g., 4 in Z9 Public Key Cryptography -- II

  11. Extended Euclidian: a-1 mod n • Main Idea: Looking for inverse of a mod n means looking for x such that x*a – y*n = 1. • To compute inverse of a mod n, do the following: • Compute gcd(a, n) using Euclidean algorithm. • Since a is relatively prime to m (else there will be no inverse) gcd(a, n) = 1. • So you can obtain linear combination of rm and rm-1 that yields 1. • Work backwards getting linear combination of ri and ri-1 that yields 1. • When you get to linear combination of r0 and r1 you are done as r0=n and r1= a. Public Key Cryptography -- II

  12. Example – 15-1 mod 37 • 37 = 2 * 15 + 7 • 15 = 2 * 7 + 1 • 7 = 7 * 1 + 0 Now, • 15 – 2 * 7 = 1 • 15 – 2 (37 – 2 * 15) = 1 • 5 * 15 – 2 * 37 = 1 So, 15-1 mod 37 is 5. Public Key Cryptography -- II

  13. Modular Exponentiation:Square and Multiply method • Usual approach to computing xc mod n is inefficient when c is large. • Instead, represent c as bit string bk-1 … b0 and use the following algorithm: z = 1 For i = k-1 downto 0 do z = z2 mod n if bi = 1 then z = z* x mod n Public Key Cryptography -- II

  14. Example: 3037 mod 77 z = z2 mod n if bi = 1 then z = z* x mod n Public Key Cryptography -- II

  15. Other Definitions • An element g in G is said to be a generator of a group if a = gi for every a in G, for a certain integer i • A group which has a generator is called a cyclic group • The number of elements in a group is called the order of the group • Order of an element a is the lowest i (>0) such that ai = e • A subgroup is a subset of a group that itself is a group Public Key Cryptography -- II

  16. Lagrange’s Theorem • Order of an element in a group divides the order of the group Public Key Cryptography -- II

  17. Euler’s totient function • Given positive integer n, Euler’s totient function is the number of positive numbers less than n that are relatively prime to n • Fact: If p is prime then • {1,2,3,…,p-1} are relatively prime to p. Public Key Cryptography -- II

  18. Euler’s totient function • Fact: If p and q are prime and n=pq then • Each number that is not divisible by p or by q is relatively prime to pq. • E.g. p=5, q=7: {1,2,3,4,-,6,-,8,9,-,11,12,13,-,-,16,17,18,19,-,-,22,23,24,-,26,27,-,29,-,31,32,33,34,-} • pq-p-(q-1) = (p-1)(q-1) Public Key Cryptography -- II

  19. Euler’s Theorem and Fermat’s Theorem • If a is relatively prime to n then • If a is relatively prime to p then ap-1 = 1 mod p Proof : follows from Lagrange’s Theorem Public Key Cryptography -- II

  20. Euler’s Theorem and Fermat’s Theorem EG: Compute 9100 mod 17: p =17, so p-1 = 16. 100 = 6·16+4. Therefore, 9100=96·16+4=(916)6(9)4 . So mod 17 we have 9100  (916)6(9)4 (mod 17)  (1)6(9)4 (mod 17)  (81)2 (mod 17)  16 Public Key Cryptography -- II

  21. Some questions • 2-1 mod 4 =? • What is the complexity of • (a+b) mod m • (a*b) mod m • a-1 mod (m) • xc mod (n) • Order of a group is 5. What can be the order of an element in this group? Public Key Cryptography -- II

  22. Further Reading • Chapter 4 of Stallings • Chapter 2.4 of HAC Public Key Cryptography -- II

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