Joining the RSA Cryptosystem, Lecture notes of Algorithms and Programming

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Cryptography: Joining the RSA Cryptosystem

Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin

Joining the RSA Cryptosystem: Overview

• First, Bob randomly chooses two large (e.g., 512-bit) primes p and q
• Then, Bob computes n = pq, φ(n) = (p − 1)(q − 1), and a positive integer d < n such that d and φ(n) are relatively prime - For example, any prime exceeding max(p, q) (and less than n) is a valid choice for d
• Then, Bob computes e such that de is congruent to 1 modulo φ(n)
  • Thus e and φ(n) are also relatively prime
• Bob’s public key is (e, n) and Bob’s private key is (d, n)
  • Remark: The scheme will also work if we use (d, n) as the public key and (e, n) as the private key
• We will discuss each of these steps in greater detail in the slides that follow

Prime Number Theorem

• Prime number theorem: As n tends to infinity, the fraction of the first n positive integers that are prime tends to (^) ln^1 n
• For example, about a. 002818 fraction of the numbers less than 2512 are prime, and about a. 002823 fraction of the numbers less than 2511 are prime - Thus, about a. 002813 fraction of 512-bit numbers (i.e., with leading 1 bit in bit position 511, indexing from 0), or about one in 355.5, is prime - So if we pick a set S of a few thousand 512-bit numbers independently and uniformly at random, we are overwhlemingly likely to pick at least one prime - We can use an efficient primality test to find a prime in S - We can make this approach more efficient by, e.g., picking only odd 512-bit numbers

Joining the RSA Cryptosystem: Implementation

• Given that we’ve seen how to generate random large primes, we now know how to compute p, q, and d - Of course, p and q also give us n = pq and φ(n) = (p − 1)(q − 1)
• All that is left is to compute e such that de is congruent to 1 modulo φ(n) - We will compute e using the extended Euclid algorithm, as discussed on the next slide

Joining the RSA Cryptosystem: Example

• Suppose we choose primes p = 47 and q = 59 (in practice, we would choose much larger primes)
• Then n = 47 · 59 = 2773 and φ(n) = 46 · 58 = 2668
• Now suppose we choose d = 157
  • Note that 157 is a prime greater than p and q and hence is relatively prime to φ(n)
• Running the extended Euclid algorithm with x = 157 and y = 2668 yields e = 17
• Bob’s public key is (17, 2773) and his private key is (157, 2773)

RSA Encryption and Decryption

• Next time we’ll see how to use Bob’s public key to encrypt messages sent to Bob
• We’ll also see how Bob uses his private key to decrypt these messages