Cryptographic algorithms generally rely for their security on having a "difficult problem". Most modern algorithms seem to use the factoring of very large numbers as their difficult problem - if you multiply two large numbers together, computing their factors is "difficult" (i.e. time-consuming). If those two numbers are prime numbers, then there is only one answer, which makes it even more difficult, and also guarantees that when you find the answer, it's the right one, not some other answer that just happens to give the same result.

我认为在密码学中重要的不是质数本身，而是质因数分解问题的难点

假设你有一个非常非常大的整数，已知是两个质数m和n的乘积，要找出m和n是什么并不容易。RSA等算法就是基于这个事实。

顺便说一下，有一篇发表的算法论文可以在可接受的时间内使用量子计算机“解决”这个质因数分解问题。因此，当量子计算机问世时，新的密码学算法可能不再依赖质因数分解的“困难”:)

最基本和一般的解释:密码学都是关于数论的，所有的整数(除了0和1)都是由质数组成的，所以你在数论中要处理很多质数。

More specifically, some important cryptographic algorithms such as RSA critically depend on the fact that prime factorization of large numbers takes a long time. Basically you have a "public key" consisting of a product of two large primes used to encrypt a message, and a "secret key" consisting of those two primes used to decrypt the message. You can make the public key public, and everyone can use it to encrypt messages to you, but only you know the prime factors and can decrypt the messages. Everyone else would have to factor the number, which takes too long to be practical, given the current state of the art of number theory.

Cryptographic algorithms generally rely for their security on having a "difficult problem". Most modern algorithms seem to use the factoring of very large numbers as their difficult problem - if you multiply two large numbers together, computing their factors is "difficult" (i.e. time-consuming). If those two numbers are prime numbers, then there is only one answer, which makes it even more difficult, and also guarantees that when you find the answer, it's the right one, not some other answer that just happens to give the same result.

为了更具体地说明RSA如何使用素数的性质，RSA算法主要依赖于欧拉定理，该定理指出，对于相对素数“a”和“N”，a^e等于1模N，其中e是N的欧拉totient函数。

质数是怎么进来的?为了有效地计算N的欧拉totient函数，需要知道N的质因数分解。在RSA算法中，对于一些质数“p”和“q”，N = pq，那么e = (p - 1)(q - 1) = N - p - q + 1。但是如果不知道p和q, e的计算是非常困难的。

更抽象地说，许多密码学协议使用各种活板门函数，这些函数易于计算但难以反演。数论是这类活板门函数的丰富来源(例如大素数的乘法)，而素数是数论的绝对中心。

有一些很好的资源可以加强加密。这里有一个:

http://research.microsoft.com/en-us/groups/crypto/firstcrypto.aspx

从那一页开始:

In the most commonly used public-key cryptography system, invented by Ron Rivest, Adi Shamir, and Len Adleman in 1977, both the public and the private keys are derived from a pair of large prime numbers according to a relatively simple mathematical formula. In theory, it might be possible to derive the private key from the public key by working the formula backwards. But only the product of the large prime numbers is public, and factoring numbers of that size into primes is so hard that even the most powerful supercomputers in the world cant break an ordinary public key.

布鲁斯·施奈尔的《应用密码学》是另一本。我强烈推荐这本书;读起来很有趣。