Source: TIPTON, et. al, Official (ISC)2 Guide to the CISSP CBK, 2007 edition, page 254.
And from the RSA web site, http://www.rsa.com/rsalabs/node.asp?id=2214 :
The RSA cryptosystem is a public-key cryptosystem that offers both encryption and digital
signatures (authentication). Ronald Rivest, Adi Shamir, and Leonard Adleman developed
the RSA system in 1977 [RSA78]; RSA stands for the first letter in each of its inventors' last
names.
The RSA algorithm works as follows: take two large primes, p and q, and compute their
product n = pq; n is called the modulus. Choose a number, e, less than n and relatively
prime to (p-1)(q-1), which means e and (p-1)(q-1) have no common factors except 1. Find
another number d such that (ed - 1) is divisible by (p-1)(q-1). The values e and d are called
the public and private exponents, respectively. The public key is the pair (n, e); the private
key is (n, d). The factors p and q may be destroyed or kept with the private key.
It is currently difficult to obtain the private key d from the public key (n, e). However if one
could factor n into p and q, then one could obtain the private key d. Thus the security of the
RSA system is based on the assumption that factoring is difficult. The discovery of an easy
method of factoring would "break" RSA (see
Question 3
.1.3 and Question 2.3.3).
Here is how the RSA system can be used for encryption and digital signatures (in practice,
the actual use is slightly different; see Questions 3.1.7 and 3.1.8):
Encryption
Suppose Alice wants to send a message m to Bob. Alice creates the ciphertext c by
exponentiating: c = me mod n, where e and n are Bob's public key. She sends c to Bob. To
decrypt, Bob also exponentiates: m = cd mod n; the relationship between e and d ensures
that Bob correctly recovers m. Since only Bob knows d, only Bob can decrypt this
message.
Digital Signature
Suppose Alice wants to send a message m to Bob in such a way that Bob is assured the
message is both authentic, has not been tampered with, and from Alice. Alice creates a
digital signature s by exponentiating: s = md mod n, where d and n are Alice's private key.
She sends m and s to Bob. To verify the signature, Bob exponentiates and checks that the
message m is recovered: m = se mod n, where e and n are Alice's public key.
Thus encryption and authentication take place without any sharing of private keys: each
person uses only another's public key or their own private key. Anyone can send an
encrypted message or verify a signed message, but only someone in possession of the
correct private key can decrypt or sign a message.