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Cryptography Concepts: Prime Testing and Factoring Techniques

Learn about prime testing techniques like Fermat's Theorem and Miller-Rabin test in the context of RSA encryption. Understand how compositeness testing plays a crucial role in cryptographic systems. Discover how to apply these techniques for secure communication.

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Cryptography Concepts: Prime Testing and Factoring Techniques

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  1. DTTF/NB479: Dszquphsbqiz Day 21 • Announcements: • Congrats on reaching the halfway point once again! • DES graded soon • Short “pop” quiz on Ch 3. (Thursday at earliest) • Reminder: pass in worksheet on using RSA today or tomorrow. • Term project groups and topics due by Friday. • Questions? • This week: • Primality testing, factoring • Discrete Logs

  2. Plus-delta • 5-min

  3. Term projects • Use Ch 10 – 19 as inspiration. • Elliptic curves? • Quantum crypto? • Security protocols? • Deliverables: • A paper demonstrating your understanding of the topic • A 20-min in-class presentation 9th/10th week • Groups of ~3 to bound presentation time.

  4. Pulling 479 back into cache • RSA: public-key system: n, e known • Easy to encrypt • But need factorization of n (pq) to find d to decrypt. • Factorization is a “one-way” function • Builds on lots of ch 3 number theory, like Euclid, Fermat, and Euler. • You used Maple to send messages • You looked at some “implementation mistakes” (for example, using small values for e)

  5. Compositeness testing Oops, did I say primality testing? Today, we discuss three techniques that can guarantee a number is composite, and guess when one is prime. • Square Root Compositeness Theorem + • Fermat’s Theorem = • Miller-Rabin Compositeness Test

  6. Square Root Compositeness Theorem Given integers n, x, and y: Then n is composite, and gcd(x-y, n) is a non-trivial factor Proof: live in class Toy example showing 35 is composite using x=2 and y=12.

  7. Review • Fermat’s little theorem: • If n is prime and doesn’t divide a, then • Contrapositive: • If then n is composite • In practice, • If then n is probably prime • Rare counterexamples (15k our of first 10B ints) called pseudoprimes • Notes • Never gives factors • Compute using powermod

  8. Miller-Rabin Compositeness Test To test whether n is prime or composite 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.

  9. 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

  10. Examples of Miller-Rabin • n=189 • n=561 (Fermat says prob prime) 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. 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 Those composites that even get by M-R are called strong pseudoprimes (~20% of pseudoprimes < 10B). Why does it work?

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

  12. 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 Odd? no div by other small primes? no Pass M-R? yes Prime by Factoring/advanced techn.? yes prime

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