TUCoPS :: Crypto
:: crypfq06.txtCryptogrphy Faq Part 6 |

Archive-name: cryptography-faq/part06 This is the sixth of ten parts of the sci.crypt FAQ. The parts are mostly independent, but you should read the first part before the rest. We don't have the time to send out missing parts by mail, so don't ask. Notes such as ``[KAH67]'' refer to the reference list in the last part. Contents: 6.1. What is public-key cryptography? 6.2. What's RSA? 6.3. Is RSA secure? 6.4. How fast can people factor numbers? 6.5. What about other public-key cryptosystems? 6.1. What is public-key cryptography? In a classic cryptosystem, we have encryption functions E_K and decryption functions D_K such that D_K(E_K(P)) = P for any plaintext P. In a public-key cryptosystem, E_K can be easily computed from some ``public key'' X which in turn is computed from K. X is published, so that anyone can encrypt messages. If D_K cannot be easily computed from X, then only the person who generated K can decrypt messages. That's the essence of public-key cryptography, published by Diffie and Hellman in 1976. In a classic cryptosystem, if you want your friends to be able to send secret messages to you, you have to make sure nobody other than them sees the key K. In a public-key cryptosystem, you just publish X, and you don't have to worry about spies. This is only the beginning of public-key cryptography. There is an extensive literature on security models for public-key cryptography, applications of public-key cryptography, other applications of the mathematical technology behind public-key cryptography, and so on. 6.2. What's RSA? RSA is a public-key cryptosystem defined by Rivest, Shamir, and Adleman. Here's a small example. See also [FTPDQ]. Plaintexts are positive integers up to 2^{512}. Keys are quadruples (p,q,e,d), with p a 256-bit prime number, q a 258-bit prime number, and d and e large numbers with (de - 1) divisible by (p-1)(q-1). We define E_K(P) = P^e mod pq, D_K(C) = C^d mod pq. Now E_K is easily computed from the pair (pq,e)---but, as far as anyone knows, there is no easy way to compute D_K from the pair (pq,e). So whoever generates K can publish (pq,e). Anyone can send a secret message to him; he is the only one who can read the messages. 6.3. Is RSA secure? Nobody knows. An obvious attack on RSA is to factor pq into p and q. See below for comments on how fast state-of-the-art factorization algorithms run. Unfortunately nobody has the slightest idea how to prove that factorization---or any realistic problem at all, for that matter---is inherently slow. It is easy to formalize what we mean by ``RSA is/isn't strong''; but, as Hendrik W. Lenstra, Jr., says, ``Exact definitions appear to be necessary only when one wishes to prove that algorithms with certain properties do _not_ exist, and theoretical computer science is notoriously lacking in such negative results.'' 6.4. How fast can people factor numbers? It depends on the size of the numbers. In October 1992 Arjen Lenstra and Dan Bernstein factored 2^523 - 1 into primes, using about three weeks of MasPar time. (The MasPar is a 16384-processor SIMD machine; each processor can add about 200000 integers per second.) The algorithm there is called the ``number field sieve''; it is quite a bit faster for special numbers like 2^523 - 1 than for general numbers n, but it takes time only exp(O(log^{1/3} n log^{2/3} log n)) in any case. An older and more popular method for smaller numbers is the ``multiple polynomial quadratic sieve'', which takes time exp(O(log^{1/2} n log^{1/2} log n))---faster than the number field sieve for small n, but slower for large n. The breakeven point is somewhere between 100 and 150 digits, depending on the implementations. Factorization is a fast-moving field---the state of the art just a few years ago was nowhere near as good as it is now. If no new methods are developed, then 2048-bit RSA keys will always be safe from factorization, but one can't predict the future. (Before the number field sieve was found, many people conjectured that the quadratic sieve was asymptotically as fast as any factoring method could be.) 6.5. What about other public-key cryptosystems? We've talked about RSA because it's well known and easy to describe. But there are lots of other public-key systems around, many of which are faster than RSA or depend on problems more widely believed to be difficult. This has been just a brief introduction; if you really want to learn about the many facets of public-key cryptography, consult the books and journal articles listed in part 10.