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Introduction to Cryptography

Introduction to Cryptography. Lecture 9. Public – Key Cryptosystems. Each participant has a public key and a private key . It should be infeasible to determine the private key from knowledge of the public key. message. Alice. Public – Key Cryptosystems. Bob.

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Introduction to Cryptography

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  1. Introduction to Cryptography Lecture 9

  2. Public – Key Cryptosystems • Each participant has a public key and a private key. • It should be infeasible to determine the private key from knowledge of the public key.

  3. message Alice Public – Key Cryptosystems Bob • Bob encrypts message using Alice’s public key • Alice decrypt message using her private key

  4. Prime Numbers Definition: A prime number is an integer number that has only two divisors: one and itself. Example: 1, 2,17, 31. • Prime numbers distributed irregularly among the integers • There are infinitely many prime numbers

  5. Factoring • The Fundamental Theorem of Arithmetic tells us that every positive integer can be written as a product of powers of primes in essentially one way. Example:

  6. The RSA Public – Key Cryptosystem • In 1978, Ronald Rives, Adi Shamir, and Leonard Adelman wrote a paper called “A Method for Obtaining Digital Signatures and Public Key Cryptosystem”. • They described a cipher system in which senders encrypt message using a method and a key that are publicly distributed.

  7. The RSA Public – Key Cryptosystem Alice: • Selects two prime numbers p and q. • Calculates m = pq and n = (p - 1)(q - 1). • Selects number e relatively prime to n • Finds inverse of e modulo n • Publishes e and m

  8. The RSA Public – Key Cryptosystem To encrypt the message x: • Bob computes: . • Bob sends y to Alice. To Decrypt the message y: • Alice computes: .

  9. The RSA Public – Key Cryptosystem Example: • p =127, q = 223. • Then m = 28321 and n = 27972 • Let e = 5623, check gcd(n,e) = 1. • Then using Extended Euclidean Algorithm d = 22495. • Public Key: (5623, 28321).

  10. The RSA Public – Key Cryptosystem Example: • Let the message be x = 3620. • Then • Alice gets one and decrypts it • Then

  11. The RSA Public – Key Cryptosystem • Why does this method work? • Last step is a little bit more complicate • How secure is RSA? • Can opponent deduce d and n from (m,e)? • The opponent can find n and d only if he can factor m.

  12. Factoring • Problem of factoring a number is very hard • Fermat’s factoring method sometimes can be used to find any large factors of a number fair quickly (pg.251) • Want to make sure Fermat’s factoring method does not work for your key • p and q should be at least 155 decimal digits each

  13. Homework • Read pg.286-293. • Exercises: 2(a), 4(c), 5(a) on pg.294. • Those questions will be a part of your collected homework.

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