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Cryptography FAQ (06/10: Public Key Cryptography)

( Part1 - Part2 - Part3 - Part4 - Part5 - Part6 - Part7 - Part8 - Part9 - Part10 )
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Archive-name: cryptography-faq/part06
Last-modified: 94/06/07

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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 
as /pub/usenet/news.answers/cryptography-faq/part[xx]. The Cryptography 
FAQ is posted to the newsgroups sci.crypt, talk.politics.crypto, 
sci.answers, and news.answers every 21 days.


6.1. What is public-key cryptography?
6.2. How does public-key cryptography solve cryptography's Catch-22?
6.3. What is the role of the `trapdoor function' in public key schemes?
6.4. What is the role of the `session key' in public key schemes?
6.5. What's RSA?
6.6. Is RSA secure?
6.7. What's the difference between the RSA and Diffie-Hellman schemes?
6.8. What is `authentication' and the `key distribution problem'?
6.9. How fast can people factor numbers?
6.10. What about other public-key cryptosystems?
6.11. What is the `RSA Factoring Challenge?'

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 decryption D_K cannot be easily 
  computed from public key X without knowledge of private key K, but 
  readily with knowledge of K, then only the person who generated K can 
  decrypt messages. That's the essence of public-key cryptography, 
  introduced by Diffie and Hellman in 1976. 
  This document describes only the rudiments 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; consult the references at the end for more 
  refined and thorough presentations.

6.2. How does public-key cryptography solve cryptography's Catch-22?

  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 the most vexing problems of all prior 
  cryptography: the necessity of establishing a secure channel for the 
  exchange of the key. To establish a secure channel one uses 
  cryptography, but private key cryptography requires a secure channel! 
  In resolving the dilemma, public key cryptography has been considered 
  by many to be a `revolutionary technology,' representing a 
  breakthrough that makes routine communication encryption practical 
  and potentially ubiquitous.

6.3. What is the role of the `trapdoor function' in public key schemes?
  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 tractable if the private key K is known.

6.4. What is the role of the `session key' in public key schemes?

  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 the public key algorithm. The message 
  is actually encrypted using a faster private key algorithm with the 
  session key. At the receiver side, the session key is decrypted using 
  the public-key algorithms and the recovered `plaintext' key is used 
  to decrypt the message.
  The session key approach blurs the distinction between `keys' and 
  `messages' -- in the scheme, the message includes the key, and the 
  key itself is treated as an encryptable `message'. Under this 
  dual-encryption approach, the overall cryptographic strength is 
  related to the security of either the public- and private-key 

6.5. 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. All quantities are
  readily computed from classic and modern number theoretic algorithms 
  (Euclid's algorithm for computing the greatest common divisor yields
  an algorithm for the former, and historically newly explored
  computational approaches to finding large `probable' primes, such as 
  the Fermat test, provide the latter.)

  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.

6.6. 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

  Note that there may even be a `shortcut' to breaking RSA other than
  factoring. It is obviously sufficient but so far not provably 
  necessary. That is, the security of the system depends on two 
  critical assumptions: (1) factoring is required to break the system,
  and (2) factoring is `inherently computationally intractable',
  or, alternatively, `factoring is hard' and `any approach that can 
  be used to break the system is at least as hard as factoring'.

  Historically even professional cryptographers have made mistakes
  in estimating and depending on the intractability of various 
  computational problems for secure cryptographic properties. For 
  example, a system called a `Knapsack cipher' was in vogue in the
  literature for years until it was demonstrated that the instances
  typically generated could be efficiently broken, and the whole
  area of research fell out of favor.

6.7. What's the difference between the RSA and Diffie-Hellman schemes?

  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 have 
  static, unchanging keys for each receiver that are `advertised' by 
  a formal `trusted authority' (the hierarchical model) or distributed 
  in an informal `web of trust'.

6.8. What is `authentication' and the `key-exchange problem'?

  The ``key exchange problem'' involves (1) ensuring that keys are
  exchanged so that the sender and receiver can perform encryption and
  decryption, and (2) doing so in such a way that ensures an
  eavesdropper or outside party cannot break the code. `Authentication'
  adds the requirement that (3) there is some assurance to the receiver
  that a message was encrypted by `a given entity' and not `someone 

  The simplest but least available method to ensure all constraints 
  above are satisfied (successful key exchange and valid authentication)
  is employed by private key cryptography: exchanging the key secretly.
  Note that under this scheme, the problem of authentication is 
  implicitly resolved. The assumption under the scheme is that only the
  sender will have the key capable of encrypting sensible messages
  delivered to the receiver. 

  While public-key cryptographic methods solve a critical aspect of the 
  `key-exchange problem', specifically their resistance to analysis
  even with the presence a passive eavesdropper during exchange of keys, 
  they do not solve all problems associated with key exchange. In
  particular, since the keys are considered `public knowledge,'
  (particularly with RSA) some other mechanism must be
  developed to testify to authenticity, because possession of keys 
  alone (sufficient to encrypt intelligible messages) is no evidence
  of a particular unique identity of the sender.

  One solution is to develop a key distribution mechanism that assures
  that listed keys are actually those of the given entities, sometimes
  called a `trusted authority'. The authority typically does not actually
  generate keys, but does ensure via some mechanism that the lists of 
  keys and associated identities kept and advertised for reference
  by senders and receivers are `correct'. Another method relies on users
  to distribute and track each other's keys and trust in an informal,
  distributed fashion. This has been popularized as a viable alternative
  by the PGP software which calls the model the `web of trust'.

  Under RSA, if a person wishes to send evidence of their identity in
  addition to an encrypted message, they simply encrypt some information
  with their private key called the `signature', additionally included in
  the message sent under the public-key encryption to the receiver. 
  The receiver can use the RSA algorithm `in reverse' to verify that the
  information decrypts sensibly, such that only the given entity could
  have encrypted the plaintext by use of the secret key. Typically the
  encrypted `signature' is a `message digest' that comprises a unique
  mathematical `summary' of the secret message (if the signature were
  static across multiple messages, once known previous receivers could 
  use it falsely). In this way, theoretically only the sender of the
  message could generate their valid signature for that message, thereby
  authenticating it for the receiver. `Digital signatures' have many 
  other design properties as described in Section 7.

6.9. How fast can people factor numbers?

  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 value 
  or relevance to particular instances (and those generated for use
  in cryptographic systems are specifically `filtered' to resist such
  approaches.) The most important observation about factoring is that
  all known algorithms require an exponential amount of time in the
  _size_ of the number (measured in bits, log2(n) where `n' is the 
  number). Cryptgraphic algorithms built on the difficulty of factoring
  generally depend on this exponential-time property. (The distinction
  of `exponential' vs. `polynomial time' algorithms, or NP vs. P, is a 
  major area of active computational research, with insights very 
  closely intertwined with cryptographic security.)
  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.)

6.10. 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.

6.11. What is the ``RSA Factoring Challenge''?

  [Note: The e-mail addresses below have been reported as invalid.]
  In ~1992 the RSA Data Securities Inc., owner and licensor of multiple
  patents on the RSA hardware and public key cryptographic techniques in
  general, and maker of various software encryption packages and 
  libraries, announced on sci.math and elsewhere the creation of an 
  ongoing Factoring Challenge contest to gauge the state of the art in
  factoring technology. Every month a series of numbers are posted and
  monetary awards are given to the first respondent to break them into
  factors. Very significant hardware resources are required to succeed 
  by beating other participants. Information can be obtained via 
  automated reply from

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