Cosmology

[desiresjab]

Those articles like a good starting point. I haven't read read all, but I will start with the one on Carmichael numbers.

I will be going over them again and again, lad, combing for details I can uptake. The only thing I can see which unifies all Carmichael numbers is the criterion that the p-1 of all primes dividing n also divides n-1. That seems to be it. It would be wonderful to find more connections unifying them. There might be some other unifying principle which would significantly lessen the work involved in computer searches for them.

* * * * *

Several more things to mention. The first is the difficulty of the math now facing us. How difficult for you I do not know, but for me quite difficult are the great abstract fields ahead. There can hardly be enough preparation. I am like a wolf that has been picking off theorems from the outskirts of the herd for years. I rather know what to expect and still find the slogging torturous. Those articles created far more questions than they answered for me. The basic idea of an ideal seems simple enough at first presentation. The idea of a field consisting of multiples of single number such as 2 or 3, is easy to grasp. For instance the even numbers. Every multiplication or addition of any two elements in the field produces another element of the field, a basic requirement of groups. Easy, right?

But then one learns that Carmichael numbers also form an ideal field (my language is not yet straight). And one says, What? I thought ideals were simple multiples of a single number. Carmichael numbers are not multiples of one another that I can see.

So I have a long ways to go to get the basics of this higher mathematics under control. If Category Theory is not higher mathematics, then I do not know what is. If Category Theory is still elementary mathematics, I am, sir, a moslem's uncle. From what I have seen so far it offers the highest and most abstract view available in mathematics. I have been toiling in the grease of the gears of raw numbers for too long. I now must attempt to take the step to the lofty views which dash away whole categories with a few slashes of chalk.

[desiresjab]

The other topic today is the prevelance of mathematical genius. I hope some who normally only read here will pitch in with some ideas.

Are there talents equivalent to Gauss, Euler, Newton and Ramanujan in the world today? If not, why not?

There are more people in the world than ever and that means more researchers than ever. Statistically, we should have math men as great as those aforementioned in the world today. But one gets the feeling a Gauss or Newton might have already completely realized the mathematics of string theory, tying it to physics and forcing the paradigm shift in thinking. There are great mathematical minds in every age. Of course we have them now, too.

Fifteen and seventeen year old thinkers do not make major discoveries anymore. They might make a small one here and there. Perhaps the period of preparation needed to bring oneself up to specs on contemporary research topics is so long it precludes that happening anymore. At eighteen one could make a reasonable argument that Gauss and Newton were already the world's best mathematicians. I doubt if this is still possible, but I hope it still is. I would love to see some fifteen year old force the world to shift its paradigm of reality.

[desiresjab]

One limiting factor of mathematics is the lack of poetry. Yes, I said lack of poetry. For poetry consists always of employing one word in a context where it is superior to all others. Mathematicians are decent at choosing appropriate symbols, but are often worse than mediocre when it comes to choosing terms in a spoken language, and downright awful for accepting as standard some of the terms they have.

Either the term order refers to the power (mod p) to which one must raise the constant a until the value wraps back around to one, or it refers to the number of elements in a Finite Galois Field. Which is it, math boys? You boys do not even care that these two branches are closely aligned, and expect newcomers to put up with your perfectly avoidable ambiguities as if they did not exist. Ignore them, you teach.

Let the student instruct the master, the master whose purpose is to eliminate ambiguity. In how many ways in how many branches must you idiots continue to use the words Order, Class, Congruent, et al, to mean different things? I can find such examples all over mathematics.

* * * * *

While we are at it, I do not think the symbol for pi had ought to be used for anything in math other than the ratio of a circumference to a diameter. Admittedly, there is a genuine paucity of available symbols not found in the actual language where these symbols combine to form words, many of which are synonyms, so there is more excuse for bad notation in math than bad poetry where a multiplicity words are available.

The notion will make more sense to me (in Group Theory) if it turns out that it equates to the power we must raise a to (mod p) before it becomes a again. This would be just another angle on Fermat's Little Theorem, the unreduced version where a^p≡a (mod p). Is that what is going on with the confusion of words mathematicians have chosen? I could accept that linguistic disparity because both notions of order are at least based on the same theorem in two different forms, those two forms being:

a^p≡a (mod p), and a^p-1≡1 (mod p).

[YesNo]

I don't know much about category theory.

We have been traveling to avoid the cold in Chicago, although it was rather warm there last week. I just got the computer connected to the internet again.

There are a lot of overused words in mathematics as you mentioned like "order". Whatever they refer to in some context should be unambiguous or one won't be able to use logic to show something is false or true.

[desiresjab]

And speaking of ugly and unimaginative...the slash /, should never be used in mathematics to mean anything other than simple division. In the group theory I now need to look at they seem to use it all the time for something else. I hate this. I detest it. It makes mathematicians look stupid in their own way.

[desiresjab]

While we are waiting for formal higher math to reveal the best port of entry for my weaknesses, let us take a look at the Euler phi function, with the eventual purpose of understanding precisely a modrern encrytion system such as RSA through this function. We looked at this function before but it always deserves more. I do not remember what we said before.

The function is always even. For an even number it is easy to see why by inspection of this formula for ф(n):

ф(n)=nΠ_p|n(1-1/p).

Notice the right side is multiplied by n, which we already know is an even number. Case closed.

When n is odd, that is where there are no powers of 2 in its prime factorization, the following formula for a power of a prime makes the answer transparently obvious:

ф(p^k)=p^k(1-1/p).

That subtraction on the right must result in an odd number minus an odd number, which we know is always even. Case closed. The ф function is always even.

* * * * *

I am having a difficult time penetrating very far into ideal numbers. I can see it will take more than a few new tools. One thing about the ф function is that it is always even. I may be a dummy, but I believe that allows me to say the function could be stated in terms of the ideal of the even numbers. The function does play a significant role in ideal theory, if I am not mistaken. No surprise there. That is what great functions do--show up everywhere.

The elementary functions are so interrelated one could probably state any of them in terms of the others. The Euler ф function would be no more than a special and quite extended exercise or circumstance of the divisor function. That would be quite a mess. Thankfully, there is a division of labor among a bevy, a cluster, of closely related elementary functions, parcelling out the various applications and implications. This is much nicer.

I can already tell anyone riding along and reading who wants to follow the mechanical details of the encrytion process, that it will boil down to some tricks of exponents on ф, almost magic-like tricks, where you have to follow the bouncing ball. Even after you see it, it can get away from you again in the next moment. But once you do see it, you will be able to retrace your steps and see it again, even when you lose sight of it.

Does this mean I see it? Actually, friends, I do not. I forgot. I used to know more about than I do now. I followed the reasoning in detail once and saw that it was a game of exponents and how they worked in a round robin of substitution until you arrived back where you started. The ф function was never mentioned way back then in the article I read. It is not the only function that encryption systems have been based on, but I believe it is used in RSA which is the most widely used security system.

If we can make the mathematical mechanics of RSA perfectly and intuitively transparent to ourselves, we have the right to toy around with ideas for an encryption system of our own devising, but not before, certainly not before!

[desiresjab]

Eventually, Yes/No will drag back to the big shoulders of frigid Chicago and give us some good direction and pointed tips. He is a mathematician and computers are his gig. He may have made the encryption process completely transparent to himself already. It is hard telling what a mathematician will get himself up to and involved with. Myself, I am retired so allowed the luxury of going only where I think it matches my desires--one cannot always tell in advanved foreign territory--and my desire is to understand numbers (from natural to complex) on the deepest levels I can.

Instead of prodigy, Paul Erdos called one who remained intellectually active in their dotage a dotagy. I am a dotagy who feels that not only the universe but we ourselves and our consciousness are underlain by deeper layers of unrevealed structure, just like mere numbers are continually rediscovered to be. To us, these deep layers are necessarily more complex, or we would have discovered them first. To me, they possibly hint of what men have correctly or erroneously attributed to objects like spirits and souls in times past. I am one who feels it is probable there are structures of increasing complexity within us that explain age old mysteries and legends, myths, dreams, ESP, prescience, et al. I am trying to edge closer to it before I die. Maybe there is some advantage to be had from pursuing the deeper structures as a pure amateur who will never publish a math paper. I wrote novels, too, that I believe no one will ever read. But in writing them I felt that perhaps I could edge closer to Yeats' golden Byzantium of artifice and creation. What I mean is, I felt that they might have counted in some deeper way as part of another structure, whether any person ever read them or not. Byzantium would recognize me at my death. Okay. Pretty weird.

Back to numbers.

[desiresjab]

The general idea is this: Multiply two gigantic (and I mean gigantic!) primes together, and no one can tell from looking at the result which the hell two numbers you multiplied to get this result.

There are a couple of reasons:

(1) You are in Gigantic territory.
(2) Factoring numbers is inherently hard.

That explains how at the heart of modern e-encryption systems lies the problem of factoring a large number. You--the person trying to break the code--even have some advantages, it would seem at first sight. First, you know it is a composite and not a prime, though it looks exactly like the kind of number computational number theorists would put in great work upon to determine its primality or compositness. Factoring is hard, especailly when your two prime number factors are in the neighborhood of four hundred digits long apiece. You know there are only two and you also know they are relatively close together, i.e. not all that far from the square root of the giant product--and you still cannot do it, even with your computer. If you had a quantum computer, yes, you could do it.

* * * * *

Okay, Jabby boy, you say, how does the ф function come into the picture?

Well, you see, being able to hide what those big factors are is only part of the job, in fact it is the easy part. For anyone can multiply two gigantic primes together and you will not be able to factor them, whether you are the world's best mathematician or the Cray super computer. The clever part was turning the ability to hide those factors into a way to transmit a secure message. This took some footwork worthy of Jersey Joe Walcott. It is like mathematical sleight of hand.

The inventors of RSA truly invented their system, but independently, as they say in the sciences. The work of the guy who got their first, decades before RSA, was immediately marked classified, or his grandchildren might now be vastly rich.

It usually turns out that some smart people got there early in the game, so to speak, before their time.

Anyway, what is needed after multiplying together two extremely large primes, is a way to carry a secure message with the even larger product.

Ask yourself this as you ponder the things above and I go to my slates:

If given the ф of an enormous number but not the number itself, could you determine the number?

If given the ф of an enormous number and given the number as well, could you determine its factors, since ф(pq)=ф(p)·ф(q)?

[desiresjab]

Okay, I think I have a good handle on the basic mechanics of RSA. What I want to do is pare the explanation down to a simple minimum. It will not be the full presentation of how the code is implemented into ASCI, etc., but a bare revealing of the mechanics.

All this is for a later post, once I figure out the best presentation and clear up any lingering questions I myself have.

RSA is indeed clever, but easily understandable with the precise tools we have been using in this thread for months. It is only commutative modular arithmetic in a finite field or ring, to throw some fancy words around. It depends on being able to find the modular inverse of a number with respect to ф(n) when ф(n)=ф(pq)=ф(p)·ф(q) and you only know n but not p or q.

I will try to set it up so it is easy to understand.

[YesNo]

If I remember right, the slash / originated with Fibonacci to represent rational numbers. It took some time to get a decimal representation of them.

Since I am too old to be a prodigy, I'll have to try for a dotagy.

Robert Prechtner uses the ratios of consecutive integers of the Fibonacci sequence to show how this might affect our herding ability as well as other things in nature. I think there is something to this. It is called Elliott Wave theory.

[desiresjab]

If I remember right, the slash / originated with Fibonacci to represent rational numbers. It took some time to get a decimal representation of them.

Since I am too old to be a prodigy, I'll have to try for a dotagy.

Robert Prechtner uses the ratios of consecutive integers of the Fibonacci sequence to show how this might affect our herding ability as well as other things in nature. I think there is something to this. It is called Elliott Wave theory.

That sounds like a likely place for the slash to have begun.

[desiresjab]

Here is my take on RSA encryption. If I am wrong Yes/No or someone else will set me straight.

First, here is the idea in a nutshell. When you raise m to the d power and then to the inverse of d, you end up back at m, the actual message being sent. In this case the inverse is very hard to find because you do not have the right information.

Dusty would like to receive secure messages from Garret, and Mandy would like to know what is in them. So Dusty finds two really large primes p and q, and multiplies them together to get n. Then he calculates ф(n), which he can easily do because he knows what p and q are, and their ф is merely (p-1)(q-1). He then chooses a number d relatively prime to and smaller than ф(n) to use as a power. Then he declares to the world that anyone wanting to send him a message must only raise that message to the power of d first. When Dusty receives a message from Garret it is raised to the power of d, so no one else is able to make sense of it. Dusty can decrypt it because he knows the inverse of d which we call e.

Let the message be m. Garret only wants to send the number 7 to Dusty. He is telling him that afternoon's horse race will be fixed and where to put his money.

Garret makes this number: 7^d, and sends it to Dusty. 7^d is some other number, very large, certainly it is not 7. All Dusty has to do to know which horse to bet on is this: 7^de=7, and we are back at 7, the original message, because d and e are inverses (with respect to ф(n)), and therefore must equal 1 when multipled together as exponents in the term 7^de, and of course 7¹=7. Dusty deciphers the message by simply applying the inverse of what is called the “public exponent,” for that is the power everyone knows they must raise their messages for Dusty to.

Mandy, who is watching all this, cannot figure out what the message is. All she knows is that Dusty raised “some number,” to a very large power, and she even knows the power, which itself is an extraordinarily large number, and she even knows the number n. If she only knew what p and q were she could calculate the value of ф(n), and then she would know how to get the inverse of that huge number d in (7^d) with respect to ф(n) that she sees Dusty received. 7^de would put her right back at 7, the content of Garret's message to Dusty.

* * * * *

The numbers involved in reality are beyond huge. Depending on the sensitivity of the application, some might have upwards of a thousand digits. To get the ball rolling, encryption folk like the public exponent d to be 65537, but only when that value is relatively prime to ф(n), otherwise they have to choose the next prime number. When Garret raises his message to the 65537^th power, the result is a fairly large number, to use an understatement. Mandy cannot raise this number to the inverse power of d with respect to ф(n) because she would need to know the values of p and q to calculate ф(n), which would then make her task easy. Without knowing ф(n), she is reduced to brute force approaches. When the numbers are truly gigantic, brute force can take a long, long time—like the age of the universe or longer—to bring about the correct solution.

We could all agree that 7⁶⁵⁵³⁷ is a pretty large number. It is especially mysterious for Mandy because she does not know the base is 7. So just which number did Garret raise to this large power? You can see where that might take a long time to figure out without the right knowledge which consists of the values of p and q.

That is it. Pretty simple, but it definitely took genius to conceive of.

[desiresjab]

The above post is the bare bones approach. It does not say anything about all the techniques of implementation that require computer experts, or the padding schemes necessary to make encryption more secure, which require even more computer experts. I think one could devise an encryption system without being a computer expert, just from number theoretic knowledge. Let the computer guys figure out how to implement it.

My feeling is that none of the encryption systems are much different from the others in basic technique. They are all based on some common number theoretic function. We have already discovered that elementary number theoretic formulas could all be expressed in terms of one another, if we worked hard enough at it. So, at heart, these systems are not much different from one another, I suspect.

Just remember that the message we sent would be but one alphabetic character or number in a longer message, if we were sending a longer one. We chose a one character message to make the process more transparent.

[desiresjab]

I have finally filled the missing link (for me) in Eisenstein's lattice point proof of Quadratic Reciprocity. It is now clear that the lattice points in the lower left triangle (labeled WAXY in the Wiki-peja article) really do represent quadratic residues. I do not know why it took so long for me to put the last piece in place, but it is now clear. There are so many angles to understand QR from. I understand only one proof of the theorem, but I do fully undrstand that one at least at last. How many ways can you pick pairs from sets of 5 elements and 3 elements with no pair ever being from the same set? Of course the two sets are (p-1)/2 and (q-1)/2.

[Dreamwoven]

All this advanced maths is way above my head, like a new language.