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## Cryptography FAQ (06/10: Public Key Cryptography)

```Archive-name: cryptography-faq/part06
Last-modified: 94/06/07

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
FAQ is posted to the newsgroups sci.crypt, talk.politics.crypto,

Contents:

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
algorithms.

6.5. What's RSA?

RSA is a public-key cryptosystem defined by Rivest, Shamir, and

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
results.''

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
system reduces communications overhead with the ability to have
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
else'.

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

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
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