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  A typical one-way hash function takes a variable-length message and
  produces a fixed-length hash. Given the hash it is computationally
  impossible to find a message with that hash; in fact one can’t
  determine any usable information about a message with that hash, not
  even a single bit. For some one-way hash functions it’s also
  [...]

 It depends on the size of the numbers, and their form. Numbers
  in special forms, such as a^n - b for `small’ b, are more readily
  factored through specialized techniques and not necessarily related
  to the difficulty of factoring in general. Hence a specific factoring
  `breakthrough’ for a special number form may have no practical [...]

  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 [...]

  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
  [...]

  In virtually all public key systems, the encryption and decryption
  times are very lengthy compared to other block-oriented
  algorithms such as DES for equivalent data sizes. Therefore in most
  implementations of public-key systems, a temporary, random `session
  key’ of much smaller length than the message is generated for each
  message and alone encrypted by [...]

Intrinsic to public key cryptography is a `trapdoor function’ D_K
  with the properties that computation in one direction (encryption,
  E_K) is easy and in the other is virtually impossible (attack,
  determining P from encryption E_K(P) and public key X). Furthermore,
  it has the special property that the reversal of the computation
  (decryption, D_K) is again [...]

  A standard cryptanalytic attack is to know some plaintext matching a
  given piece of ciphertext and try to determine the key which maps one
  to the other.  This plaintext can be known because it is standard (a
  standard greeting, a known header or trailer, …) or because it is
  guessed.  If text is guessed [...]

  Diffie and Hellman proposed a system that requires the dynamic
  exchange of keys for every sender-receiver pair (and in practice,
  usually every communications session, hence the term `session key’). 
  This two-way key negotiation is useful in further complicating
  attacks, but requires additional communications overhead. The RSA
  system reduces communications overhead with the ability to [...]

  Some don’t know any better. Often amateurs think they can design
  secure systems, and are not aware of what an expert cryptanalyst
  could do. And sometimes there is insufficient motivation for anybody
  to invest the work needed to crack a system.

  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. Hence public key
  cryptography `solves’ one of [...]