Talk:Cryptography/Mathematical Background

Structure of this article as of 6/2011
As it currently stands, this article is a brief (and excellent) review of much of the finite math which pertians to cryptographic algorithm design. This is a good thing.

It is not the best thing for an article in this book however. Perhaps an appendix?

From a writing perspective (this is an introductory text on crypto), a better approach would be to forego so much mathematics and to explain instead (probably without the mathematical groundwork) the why of the importance cryptographically.

Thus:

the mathematics of modern cryptography is relatively understandable, but is rather involved. Understanding why will be quicker than understanding the details of how (see appendix for many of these). Public key / private key algorithms (the most important part of the wider asymmetric key algorithm group) depend on mathematical relationships between keys and the operation of the algorithm.

If there could be one-way connections between keys, it might be possible to build encryption algorithms which would allow one key to encrypt with only the other key being usable for decryption. In practice, it must be necessary to generate a key pair with this relationship relatively easily, while producing the other key from knowledge of one only with much greater effort. This is the one way property from a cryptographic viewpoint.

In mathematics, there exist several such problems. None have been proved to inherently ad inescapably one way in this sense, but in the case of at least two of problems, some thousands of years of effort have produced little reason to believe there exists a way round the one-wayness.

Prime factoring is the first of these problems which found an application in cryptography (in what is generally called the RSA algorithm). It is easy, if tedious, to multiply large numbers together. It is much much harder to determine what numbers were multiplied together, and much more so whether the result is a prime number (a number with 2000 or more digits is remarkably difficult to divide evenly, much less to be certain there are no divisors except 1). Another cryptographically useful mathematical problem is the discrete logarithm problem, which is the basis of the elGamal encryption algorithm.

--Some estimates of the difficulties of brute forcing either or both. Perhaps a table. --

So, by using these difficult mathematical problems (possibly actual one-way problems for everyone everywhere, though none have been proved to be so) as the basis of an encryption algorithm, it turned out that one can produce one or more algorithms (and a pair of keys) which allow publication of one of the keys, but not the deduction or calculation of the other key. Thus, anyone anywhere can encrypt a message that only the person who has retained a copy of the other key can decrypt. This possibility was first noticed by James Ellis of GCHQ in the early 1970s and first reduced to a practical algorithm by Clifford Cocks (also of GCHQ) a little later. His algorithm was essentially reinvented, and patented, by Rivest, Shamir, and Adelson (thus, RSA) at MIT in the mid-70s,

Various elaborations on this idea have produced several cryptosystems using these algorithms. There remain practical problems, the most notable of which is being sure that this key (purported to be one a pair generated by Alice) actually belongs to Alice. Much spoofing and scamming is possible, and the problem has found no fully satisfactory solution to date.

Two alternative one-way mathematical problems have also been widely implemented as part of cryptographic algorithms and indeed cryptosystems since then. The Discrete logarithms problem was first and is used as part of the elGamal algorithm, the DSA system (developed at NSA), and the SSL / TSL techniques widely used on the Internet in Web browser software. The second was a variant, called elliptic curve algorithms, which are recent enough to be still subject to one or more patents as of late in the first decade of 2000.

And so on.

This would be an improvement for the average user, since mathematical notation is difficult for a great many such people. And so would result in increased understanding, though a less than complete mathematical one, in more readers. Sufficient, in my view, to justify the change.

Response from author
Thank you for your review on this chapter. Please allow me the privilege of responding to your concerns.

When you state, "It is not the best thing for an article in this book," please bear in mind that while I may agree with you that this section is most appropriate as an appendix, I did not place this section in its current position, I merely populated it.

When you state, "a better approach would be to forego so much mathematics," please bear in mind that the title of this section is the Mathematical Background of Cryptography. As such, it seems to me that a mathematical discussion using the language of mathematics is warranted. It is confusing to me to consider a chapter on mathematics "without the mathematical groundwork." I fully understand that I am writing a chapter which most people will glance at (in horror), only to quickly skip to the next section. However, in order for the book to be complete in its presentation, this section as it stands is completely warranted. I have kept in mind that this is an introductory text by including a plethora of worked examples (which were the most time consuming to write), which a more advanced text would leave to the reader.

As to your comment, "the mathematics of modern cryptography is relatively understandable," if this were so, you should have had no reason to comment on the rigorousness this section. Since however, modulo arithmetic is officially classified as an advanced subject, this comment seems unwarranted.

When you state that public/private key cryptography is, "the most important part of the wider asymmetric key algorithm group," please give an example of an asymmetric key algorithm that is not part of the public/private key space, as I believe these two terms are interchangeable. Asymmetric cryptography means public/private key pairs, as opposed to symmetric cryptography, which contains only one key.

"If there could be one-way connections between keys." This is a bit confusing to me, as "one-way" functions in cryptography are usually limited in discussion to so-called "trap door" functions such as hashes.

Without spending too much time responding to your comments, most, if not all, of what you suggest for the mathematical section belongs in a section on history, which is not my area of expertise, and I cannot comment on it.

Just because math is hard doesn't mean it does not belong in a chapter on mathematics.

Thank you for taking the time to comment,

Kl-robertson (discuss • contribs) 14:24, 13 February 2012 (UTC)

Content of this article (08/23)
I'm attempting to learn some of the mathematical background behind cryptography.

This source seems very useful. I have discovered and fixed a few issues in this article and I'm not sure if there are other issues.

I'm relatively new to editing and commenting on Wikipedia, so I'm not sure how this all works. This article hasn't been edited since 2019 and may not be current or accurate.

This article should probably be reviewed by someone with enough background to ensure all the content is accurate. DrewMSmith (discuss • contribs) 03:02, 1 September 2023 (UTC)