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RSA Encryption and Primality Testing

Learn about RSA encryption, primality testing, and factoring techniques such as Miller-Rabin and Quadratic Sieve. Discover how to factor large numbers and find prime numbers using these methods.

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RSA Encryption and Primality Testing

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  1. DTTF/NB479: Dszquphsbqiz Day 22 • Announcements: • Pass in worksheet on using RSA now. • DES graded soon • Short “pop” quiz on Ch 3 (Thursday at earliest) • Term project groups and topics due by Friday • Can use discussion forum to find teammates • HW6 posted, due date bumped back to next week (and a few questions added), but doing what’s there now might help your quiz prep. • Questions? • This week: • Primality testing, factoring • Discrete Logs

  2. Miller-Rabin So: k Given odd n>1, write n-1=2km, where k >=1. Choose a base a randomly (or just pick a=2) Let b0=am(mod n) If b0=+/-1, stop. n is probably prime by Fermat For i = 1..k-1 Compute bi=bi-12. If bi=1(mod n), stop. n is composite by SRCT, and gcd(bi-1-1,n) is a factor. If bi=-1(mod n), stop. n is probably prime by Fermat. If bk=1 (mod n), stop. n is composite by SRCT Else n is composite by Fermat. b0 b1 bk Big picture: Fermat on steroids By doing a little extra work (finding k to change the order of the powermod), we can call some pseudoprimes composite and find some of their factors

  3. Using within a primality testing scheme n (From Day 11) Odd? no div by other small primes? no Fermat? yes Prime by Factoring/advanced techn.? yes prime

  4. Using within a primality testing scheme n • Finding large probable primes • #primes < x = Density of primes: ~1/ln(x) For 100-digit numbers, ~1/230. So ~1/115 of odd 100-digit numbers are prime Can start with a random large odd number and iterate, applying M-R to remove composites. We’ll soon find one that is a likely prime. Maple’s nextprime() appears to do this, but also runs the Lucas test: http://www.mathpages.com/home/kmath473.htm Alternatively, could repeat M-R to get high probability prime Odd? no div by other small primes? no Pass M-R? yes Prime by Factoring/advanced techn.? yes prime

  5. Factoring • If you are trying to factor n=pq and know that p~q, use Fermat factoring: • Compute n + 12, n + 22, n + 32, until you reach a perfect square, say r2 = n + k2 • Then n = r2 - k2 = (r+k)(r-k) • The moral of the story? • Choose p and q such that _____

  6. Example Factor n = 3837523 • Concepts we will learn also apply to factoring really big numbers. They are the basis of the best current methods • All you have to do to win $30,000 is factor a 212 digit number. • This is the RSA Challenge: http://www.rsa.com/rsalabs/node.asp?id=2093#RSA704

  7. Quadratic Sieve (1) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 1: Pick a factor base, just a set of small factors. • In our examples, we’ll use those < 20. • There are 8: 2, 3, 5, 7, 11, 13, 17, 19

  8. Quadratic Sieve (2a) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 2: We want squares that are congruent to products of factors in the factor base. Our hope: Reasonably small numbers are more likely to be products of factors in the factor base. • Then which is small as long as k isn’t too big • Loop over small e, lots of k. • A newer technique, the number field sieve, is somewhat faster

  9. Quadratic Sieve (2b) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 2: We want squares that are congruent to products of factors in the factor base. Our hope: Reasonably small numbers are more likely to be products of factors in the factor base. Examples:

  10. Quadratic Sieve (3) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 3: Want two non-congruent perfect squares Example: This is close, but all factors need to be paired Recall:

  11. Quadratic Sieve (3b) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 3: Want two non-congruent perfect squares Example: This is close, but all factors need to be paired Generate lots of # and experiment until all factors are paired. So what? gcd(1147907-17745, n)=1093 Other factor = n/1093=3511

  12. Quadratic Sieve (3b) Factor n = 3837523 Want x,y:  gcd(x-y, n) is a factor Step 4: Want to get 2 non-congruent perfect squares Example: This is close, but all factors need to be paired Generate lots of # and experiment until all factors are paired. To automate this search: Can write each example are a row in a matrix, where each column is a prime in number base Then search for dependencies among rows mod 2. May need extra rows, since sometimes we get x=+/-y.

  13. My code Factor n = 3837523 To automate this search: • Each row in the matrix is a square • Each column is a prime in the number base • Search for dependencies among rows mod 2. • For last one (green) So we can’t use the square root compositeness theorem Sum: 0 2 2 2 0 4 0 0 Sum: 8 4 6 0 2 4 0 2 Sum: 6 0 6 0 0 2 0 2

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