An Interesting Insight into the Notion of Calculus

Yesterday, I had an interesting insight! An important notion of "calculus" is the notion of symbol -- and that the "calculus" of identity management would benefit from a "symbolic" approach. Before I explain why this is an interesting insight, I will have to go through a little digression.

Several months ago, I came across a math book in a used bookstore, that had an interesting claim in its introduction: that mathematicians over the years have distanced themselves from the notion of "variable" -- that the notion is rather confusing, and they have been thinking about things in different ways. My reaction at the time was "well, that is weird! I'm a mathematician, and I have never known about this controversy". I put the book back on the shelf, even though I really wanted to get it, just for the introduction alone, but I couldn't bring myself to justify, at the time, even that tiny expense, especially since I have so little space for yet another book.

Bah! I should have bought that book!

The notion wormed its way into my head as I wondered "what would I use, if not variables?" -- and then it hit me: "variables" imply too much -- the word implies that what you are working with can be expected to change -- so the notion you really need is symbols.

A "symbol" can be anything -- it can be a word, a letter, a wedge or dot or dash above a letter, a picture, a collection of lines and wedges pressed into clay, even a physical object sitting in front of you. Anything can be a symbol, and any given symbol can represent anything.
All you need to do, when you want to talk about a particular idea, is to choose a symbol, describe what you want that symbol to mean, and then use the symbol for that purpose.
Symbols are the foundation of civilization. When sequences of sounds are assigned "meaning", they make spoken language possible. When they are used as hieroglyphics for words or letters for sounds, they make writing possible. Those same words can capture abstract ideas, and whether they are manipulated by sentences or by letters, or combined with dots, circles, and lines, they become the foundations for exploring mathematical ideas. And they are one of the features that makes Common Lisp the powerful computer language it is -- you can create whatever symbol you need for the task at hand, and even create "hidden" symbols, used once for a very specific purpose, so that no one can "clobber" that symbol by accidentally using it.

If you think Algebra is challenging, try doing it without letters! There is a common notion that the devil introduced letters into math, but compare this: "consider a function that takes a number, multiplies it by itself, adds three of itself to that, and adds five" to this: "f(x) = x^2 + 3x + 5". In the latter statement, "f" is commonly recognized as a function, "x" as a number, "^2" as "multiply the number by itself", etc -- and because of its compact form, it's far easier to work with! (Indeed, when one considers the "Cubic Formula" -- the equivalent of the "Quadratic Formula" used to find zeros of something like "f(x)" above -- and is a big complicated jumble of cube roots and square roots, requiring the use of "imaginary" numbers to get to "real" number solutions, which may very well be plain vanilla integers disguised as a mix of square and cubic roots that non-obviously can be simplified -- when one considers all this, what is amazing is that it was all done in words, well before letters and other symbols were introduced to mathematics!)

So, what does all this have to do with Identity Management? It has occurred to me that perhaps discussions of the various components of Identity Management can be simplified by treating their applications symbolically -- making it a lot easier to see what's going on -- than is typically handled by Cryptography texts.

Consider an example (putting aside that I haven't really discussed what each of these things mean!): Let M be a message, H be a hash function, and (Af, Ap) be a Alice's Public-Private Key Pair -- I call the key "A" for "Alice", but the "f" here means free, and "p" means "private" (it kindof drives me nuts that "public" and "private" start with the same letter; however, I may want to come up with better ways to designate the parts of the key) -- with these things in mind, how would I sign a message, and send it to Bob, who has a Public-Private key pair (Bf, Bp)?

First, Alice would need to sign the message, by obtaining the hash H(M) of the message, signing the hash with the private key, Ap(H(M)), and then appending the result to the message itself: [M|Ap(H(M))]. (Since this is an "exploration" on my part, I'm going to leave the parentheses in for now, but it might be easier to see what's going on by shortening all this to [M|ApHM].) Now, to send this to Bob, she'll want to encrypt it with Bob's public key: Bf([M|Ap(H(M))]). She then sends this message to Bob.

So, how does Bob read all this? He applies his private key to the message: Bp(Bf([M|Ap(H(M))])) -- and since Bp(Bf) cancel each other out, he has the message [M|Ap(H(M))]. To confirm that the message really came from Alice, he then takes the hash of the message H(M), and "decrypts" the message from Alice with her public key: Af(Ap(H(M)) -- the Af(Ap) cancel each other out, so if the H(M) that Bob calculated matches the H(M) that Alice "signed", the message indeed came from Alice!

Ah, so the approach has promise! (And I'll definitely want to drop the parentheses!) Perhaps this approach has already been done -- I may be duplicating other people's efforts while creating an equivalent notation that no one else uses -- but if that's indeed the case, it means I'm on the right track! Having said that, while I'm not an expert in this field, I have read a few documents describing these principles over the years, and while I remember a lot of pretty pictures, I don't recall anyone doing this kind of thing.

And this approach will go a long way to justifying the use of the term "calculus" to describe the field!

And this may also mean I'll want to figure out how to display mathematical notation in Blogger ....