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

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.

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

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

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

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

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

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

challenge-rsa-honor-roll@rsa.com

challenge-partition-honor-roll@rsa.com