Archive-name: cryptography-faq/part06 Last-modified: 1993/4/15   FAQ for sci.crypt, part 6: Public-Key Cryptography  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.  The sections of this FAQ are available via anonymous FTP to rtfm.mit.edu  as /pub/usenet/news.answers/cryptography-faq/part[xx].  The Cryptography  FAQ is posted to the newsgroups sci.crypt, sci.answers, and news.answers  every 21 days.   Contents:  * What is public-key cryptography? * What's RSA? * Is RSA secure? * How fast can people factor numbers? * What about other public-key cryptosystems?   * 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.  * What's RSA?    RSA is a public-key cryptosystem defined by Rivest, Shamir, and   Adleman.  For full details, there is a FAQ available by ftp at   RSA.COM. Here's a small example.    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.  * 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.''  * 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 about 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.)  * What about other public-key cryptosystems?