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## Elliptic Curve Cryptography and Finite Fields

Back around 2000, I found a copy of Neal Koblitz’s text A Course in Number Theory and Cryptography at the Borders bookstore in Bangor, Maine.  I only worked my way through the first chapter, but was fascinated with these ideas.  I found Professor Koblitz’s website which, at the time, had a tutorial section on finite fields and elliptic curve cryptography (this may have been on the Certicom website, I can’t remember now).  I moved on to other forms of digital cryptography, like the Diffie-Hellman Key Exchange and RSA Cryptosystem, but always appreciated Prof. Koblitz’s work.  Recently, we dressed up for Halloween as a number and I chose to be the number 4.  As part of my costume, I drew the addition table for the Galois Field of order 4$GF(4)=^{GF(2)[x]}/_{x^2+x+1}$, and did a lot of thinking that week about the element a, which was defined as the root of the equation $0=x^2+x+1$ in $GF(2)$

This past week, I decided to look at the mathematics behind Bitcoin and blockchain, and lo and behold, it is Finite Fields and Elliptic Curve Cryptography – I don’t know why it took me so long to find this out, but now I’m excited about these topics.  I am a little skeptical about the current “Bitcoin bubble.”  I’m not sure that these valuations are sustainable, but from everything I’ve read, the blockchain algorithm behind Bitcoin is revolutionary and the mathematics is “supercool.”

Here’s a graph of the equation $y=x+1$ in $GF(2)$.

## National Debt

While we’re on the topic of economics, here’s an interesting graphic I found at The Big Picture blog –

What we see here is that, although China and Japan do have substantial holdings of U.S. government debt, they are far from being majority stakeholders.

## The Problem with the Euro

For those who are economically minded, Paul Krugman has a great article in the upcoming Times Sunday Magazine (Jan. 16) on the creation of the euro and the benefits and difficulties involved in this process.

At its heart, his comparison of Ireland and Nevada highlights the issues very clearly,

Climate, scenery and history aside, the nation of Ireland and the state of Nevada have much in common. Both are small economies of a few million people highly dependent on selling goods and services to their neighbors. (Nevada’s neighbors are other U.S. states, Ireland’s other European nations, but the economic implications are much the same.) Both were boom economies for most of the past decade. Both had huge housing bubbles, which burst painfully. Both are now suffering roughly 14 percent unemployment. And both are members of larger currency unions: Ireland is part of the euro zone, Nevada part of the dollar zone, otherwise known as the United States of America.

But Nevada’s situation is much less desperate than Ireland’s.

First of all, the fiscal side of the crisis is less serious in Nevada. It’s true that budgets in both Ireland and Nevada have been hit extremely hard by the slump. But much of the spending Nevada residents depend on comes from federal, not state, programs. In particular, retirees who moved to Nevada for the sunshine don’t have to worry that the state’s reduced tax take will endanger their Social Security checks or their Medicare coverage. In Ireland, by contrast, both pensions and health spending are on the cutting block.

Also, Nevada, unlike Ireland, doesn’t have to worry about the cost of bank bailouts, not because the state has avoided large loan losses but because those losses, for the most part, aren’t Nevada’s problem. Thus Nevada accounts for a disproportionate share of the losses incurred by Fannie Mae and Freddie Mac, the government-sponsored mortgage companies — losses that, like Social Security and Medicare payments, will be covered by Washington, not Carson City.

And there’s one more advantage to being a U.S. state: it’s likely that Nevada’s unemployment problem will be greatly alleviated over the next few years by out-migration, so that even if the lost jobs don’t come back, there will be fewer workers chasing the jobs that remain. Ireland will, to some extent, avail itself of the same safety valve, as Irish citizens leave in search of work elsewhere and workers who came to Ireland during the boom years depart. But Americans are extremely mobile; if historical patterns are any guide, emigration will bring Nevada’s unemployment rate back in line with the U.S. average within a few years, even if job growth in Nevada continues to lag behind growth in the nation as a whole.

Over all, then, even as both Ireland and Nevada have been especially hard-luck cases within their respective currency zones, Nevada’s medium-term prospects look much better.

What does this have to do with the case for or against the euro? Well, when the single European currency was first proposed, an obvious question was whether it would work as well as the dollar does here in America. And the answer, clearly, was no — for exactly the reasons the Ireland-Nevada comparison illustrates. Europe isn’t fiscally integrated: German taxpayers don’t automatically pick up part of the tab for Greek pensions or Irish bank bailouts. And while Europeans have the legal right to move freely in search of jobs, in practice imperfect cultural integration — above all, the lack of a common language — makes workers less geographically mobile than their American counterparts.

The issue here is that the difficulty of creating a monetary union without a corresponding political fiscal union creates huge problems for the member states and their citizens.

## The Black Swan

I’ve finally gotten around to reading Nassim Nicholas Taleb‘s The Black Swan.  I’m about halfway through and I’m enjoying it very much.

It’s part math, part economics, part philosophy, and part quirky anecdotes.

I’m not sure what the rest of the book will be like and in the spirit of the book itself I am OK with that!

What it has made me think about so far is mainly that the real issue is not predicting the future because human beings are notoriously and demonstrably bad at predicting the future.

Rather, I think that our energies are better spent trying to develop plans and methods for continuity and adaptability in the face of the inherent and disruptive unpredictable nature of future events.

## The Quants

Last February, I wrote about the current financial crisis and the role that mathematical trading strategies played in the meltdown.

I recently finished reading The Quants by Wall Street Journal reporter Scott Patterson.  In The Quants Patterson tells the story of computer trading strategies at the big NY investment banks and various hedge funds.  He covers some of the history of quantitative analysis of financial markets as well as the meltdown that started in August of 2007.

One paragraph in particular caught my eye as I was reading:

The market moves PDT and other quant funds started to see early that week defied logic.  The fine-tuned models, the bell curves and random walks, the calibrated correlations – all the math and science that had propelled the quants to the pinnacle of Wall Street – couldn’t capture what was happening.  It was utter chaos driven by pure human fear, the kind that can’t be captured in a computer model or complex algorithm.  The wild fat-tailed moves discovered by Benoit Mandelbrot in the 1950s seemed to be happening on an hourly basis.  Nothing like it had ever been seen before.  This wasn’t supposed to happen. (Italics in original)

Things that the mathematical trading models had predicted to be so unlikely as to happen only once in 10,000 years happened three days in a row in August 2007.  This is not just like flipping a coin and getting 10 heads in a row, it’s like flipping a coin and having it land on edge 10 times in a row.

[By the way, if your model predicts that something will happen once in 10,000 years and then it happens three days in row – NEWSFLASH – there is something deeply flawed in your model!]

Part of the problem was laid out by Nassim Nicholas Taleb, author of the book The Black Swan.  The mathematics used in the world of physics and other hard sciences uses standard bell curves.  If you measure the height of 1,000 people off the street, even if you include a few NBA players, the average won’t change all that much.

The problem in finance is that the scale of the financial world is radically different from the scale of the physical world.  If, instead of measuring the height of 1,000 people, you are measuring the financial net worth of 1,000 people, having someone like Bill Gates in your sample of 1,000 can have extreme effects on the average.

This plays out in the financial world in the effect that very large pools of money inevitably have on markets as they move in and out of various trading positions.  Standard statistical models that were developed to deal with the natural world don’t account for the effect these outsized moves have on prices and markets.

Another idea that I found interesting in The Quants was that, as quantitative trading strategies caught on through the 90s and 00s, more and more people began using similar strategies, which made these strategies less profitable.

There are only so many trades to go around, so if, instead of \$1 billion chasing these slight pricing inefficiencies, you suddenly have \$100 billion and more chasing these same slight inefficiencies, there is less and less profit in each trade.

The solution – LEVERAGE, LEVERAGE, LEVERAGE.  In other words, once the profits on these strategies started becoming smaller and smaller (because they were effectively being split between more and more traders using similar strategies), they needed to use bigger and bigger trades to get the same amount of profit out.

Many of the hedge funds and investment banks were leveraged 10, 20 or even 30 to 1.  For example, in The Quants, Patterson describes the hedge fund Citadel Group as having \$140 billion in assets on only \$15 billion in actual capital.  This is about a 9 to 1 leverage ratio.

What makes this so important is that if these trades go bad on the hedge funds, they stand to lose A LOT more money than they can conceivably pay back, especially if what they’re invested in turns out to be worthless.

So, when things turned bad in August of 2007, all of a sudden, all the hedge funds that were making these trades with vast amounts of borrowed money were scrambling for the exits all at once.  They were all highly leveraged and they were all in almost the same trading strategies which meant they were all trying to get out of the same doorway at the same time.

In a 60 minutes interview with Steve Kroft last year, Frank Partnoy pointed out that “You can’t model human behavior with math.”

Another way of saying this is that finance and economics are social sciences – not hard sciences.

## The Financial Crisis – One Year Later

This isn’t really mathematical, but there has been a flurry of articles recently about the financial crisis – both its lingering effects and the day to day issues it raised last September in the Federal Government and the large, money center banks and invesment banks.

Andrew Ross Sorkin has an article in the new Vanity Fair that is an excerpt from his book Too Big to Fail, which comes out later this month.  In it, he details the events of the third week of September 2008.

On September 7th, Fannie Mae and Freddie Mac were taken over by the government.  Bear Stearns had been taken over by JPMorgan Chase in May 2008 (with a \$30 billion government guarantee serving as a spoonful of sugar to make the deal go down easier) and Lehman Brothers went into bankruptcy on Monday September 15th.

Sorkin’s story opens on Wednesday September 17, 2008.

The panic was already palpable in John Mack’s office at Morgan Stanley’s Times Square headquarters. Sitting on his sofa with his lieutenants, Walid Chammah, 54, and James Gorman, 50, drinking coffee from paper cups, Mack was railing: the major news on Wednesday morning, he thought, should have been the strength of Morgan Stanley’s earnings report, which he had released the afternoon before, a day early, to stem any fears of the firm’s following in Lehman’s footsteps….

Apart from the general nervousness about investment banks, he was facing a more serious problem than anyone on the outside realized: at the beginning of the week, Morgan Stanley had had \$178 billion in the tank—money available to fund operations and lend to its hedge-fund clients. But in the past 24 hours, more than \$20 billion of it had been withdrawn by anxious hedge-fund clients, in some cases closing their prime-brokerage accounts entirely.

“The money’s walking out of the door,” Chammah told Mack….

‘What’s wrong?” Mack asked in alarm as Colm Kelleher walked into his office later in the day, his face ashen. “John, we’re going to be out of money on Friday,” Kelleher said with his staccato British inflection. He had been nervously watching the firm’s tank—its liquid assets—shrink, the way an airline pilot might stare at the fuel gauge while circling an airport, waiting for landing clearance.

“That can’t be,” Mack said anxiously. “Do me a favor: go back to the financing desk—go through it again.”

Kelleher returned to Mack’s office 30 minutes later, less shaken, but only slightly. After finding some additional money trapped in the system between trades that hadn’t yet settled, he revised his prognosis: “Maybe we’ll make it through early next week.”

The article details the machinations of the major economic players in last September’s meltdown – Former Treasury Secretary Henry Paulson and then-Head of the New York Federal Reserve Bank and current Treasury Secretary Timothy Geithner, as well Morgan Stanley’s John Mack, Goldman Sach’s Lloyd Blankfein, JPMorgan Chase’s Jamie Dimon, Citibank’s Vikram Pandit, Bank of America’s Ken Lewis and many others.

Another article, this one a blog post from Chris Whalen at The Big Picture economics blog goes over some of the problems that came out of the wave of takeovers late last year as Bank of America took over Countrywide Mortgage and Merrill Lynch, Wells Fargo took over Wachovia Bank, and Bear Stearns was taken over by JPMorgan Chase.  Whalen is a professional bank analyst (who also worked as a republican congressional staffer in the 1990s) and his description is pretty heavy on banking jargon and details.

## Oil Trading Denominated in Dollars

Also today, there is an article in the British newspaper The Independent written by Robert Fisk about a plan to sell oil using a “basket” of currencies instead of the U.S. dollar.

Ever since Richard Nixon removed the gold standard in the early 1970s, the U.S. dollar has remained a reserve currency for the world mainly due its status as the currency in which oil trading is denominated.  This is set to change.

In the most profound financial change in recent Middle East history, Gulf Arabs are planning – along with China, Russia, Japan and France – to end dollar dealings for oil, moving instead to a basket of currencies including the Japanese yen and Chinese yuan, the euro, gold and a new, unified currency planned for nations in the Gulf Co-operation Council, including Saudi Arabia, Abu Dhabi, Kuwait and Qatar.

Secret meetings have already been held by finance ministers and central bank governors in Russia, China, Japan and Brazil to work on the scheme, which will mean that oil will no longer be priced in dollars.

The plans, confirmed to The Independent by both Gulf Arab and Chinese banking sources in Hong Kong, may help to explain the sudden rise in gold prices, but it also augurs an extraordinary transition from dollar markets within nine years….

Ever since the Bretton Woods agreements – the accords after the Second World War which bequeathed the architecture for the modern international financial system – America’s trading partners have been left to cope with the impact of Washington’s control and, in more recent years, the hegemony of the dollar as the dominant global reserve currency.

This could devalue the dollar and make imports (like oil itself) more expensive.

In addition to the article, The Independent also included an editorial about the proposed change:

Last autumn’s global financial crisis set off an economic earthquake. And we are still feeling the tremors. The latest sign of the ground shifting beneath our feet is our report today of plans by Gulf states, China, Russia, France and Japan to end their practice of conducting oil deals in US dollars, switching instead to a diverse basket of currencies.

It is not hard to see the motivation for oil exporters to move away from the dollar. The value of the US currency has fallen sharply since last year’s meltdown. And fears are growing, in the light of a spiralling US government deficit, that a further depreciation is likely. They do not want to sell their wares in return for a currency with an uncertain future.

## Corruption

Last on the list is a story about the New York State Pension Fund and a guilty plea in a case of corruption reportedly involving the New York State comptroller Alan Hevesi

ALBANY — Raymond B. Harding, one of the last of New York’s political bosses, admitted on Tuesday that he had accepted more than \$800,000 in exchange for doing favors for Alan G. Hevesi, the former state comptroller; among the favors was a scheme to secure an Assembly seat for Mr. Hevesi’s son….

The case has focused on the state’s \$116.5 billion pension fund, and how people close to Mr. Hevesi exploited their relationship with the former comptroller to enrich themselves. The comptroller serves as the fund’s sole trustee, a relatively unusual arrangement that gives him ultimate authority over what firms are allowed lucrative contracts to manage the fund’s money…

Mr. Cuomo’s office also said on Tuesday that Saul Meyer of Aldus Equity, a Dallas firm that consulted with the pension fund, had pleaded guilty to a similar charge that had been sealed since Friday. Mr. Meyer admitted to violating his fiduciary duty to pensioners in both New York and New Mexico and taking part in schemes allegedly orchestrated by Mr. Morris. He also faces four years in jail, but is cooperating with the investigation.

“These guilty pleas vividly depict the depth and breadth of corruption involving the New York State pension fund,” Mr. Cuomo said. “In one case, we see New York’s state pension fund looted to reward a political boss with hundreds of thousands of dollars in improper payments.”

“In the other, we see a pension fund adviser — the outside ‘gatekeeper’ who is supposed to safeguard the integrity of the pension fund process — recommending deals based on pressure from pension officials and politically connected people.”

This type of corruption is at the root of the problems in our financial system.  The recently released inspector general’s report on the investigation (or lack of investigation) into Bernard Madoff’s massive ponzi scheme details the failure of the Securities and Exchange Commission to look into repeated reports of fraud.

Cleaning up corrupt practices in both business and government must precede any meaningful resurgence of the U.S. economy.

## ORMATYC

I went to Lincoln City this past weekend for the ORMATYC Conference.  ORMATYC is the Oregon Mathematical Association of Two Year Colleges and is a part of the larger group AMATYC, the American Mathematical Association of Two Year Colleges.

Just about every year, we have a conference in Lincoln City – this is the fifth one I’ve attended.  It’s a great experience to get together with other community college math instructors from Oregon to talk about math and math education.  It’s a valuable forum to find out what other schools are doing in their math classes and how we compare with those other schools.

In attending this conference every year, I have learned more about both math and math education.  In some cases it was about how to present a particular topic in class, while in other cases it was something that provided additional background information about the topics I teach so that I can share these ideas with my students.

In addition to being exposed to new ideas about math and math education, another positive aspect of this conference is being part of a community.  After 5 years, I have gotten to know some of the other instructors and have a better idea about which talks to attend while I’m there.

This year, I saw Jim Ballard of OIT Klamath Falls give a talk on the mathematics of finance.  There was a lot of discussion about the current economic situation and he didn’t really get a chance to talk about the Black-Scholes equations which are somewhat controversial, but have been used in mathematical finance for over 20 years.  I wrote about math and finance in an earlier post.

I also attended a session with Ron Wallace of Blue Mountain CC about deciding which topics to teach and which to leave out in the math curriculum.  He asked if anyone there had used the quadratic formula in their lives outside of teaching in the past five years.  I was the only one to raise my hand.

I did use the quadratic formula a few years ago when my Mom asked me about the cost of Medicare Part D programs.  The formula for pricing in Medicare Part D is quadratic in that it initially becomes more expensive the longer you wait to enroll, but the money you save by not enrolling right away can offset the higher premiums you end up paying.

On Friday afternoon I went to a presentation by Art Peck of Lane CC.  I had seen his talk last year about the connection between the Fibonacci Sequence and the Mandelbrot Set, which was excellent.  This year, he talked about applications of mathematics to environmental problems, including alternative energy.  There is a lot of mathematics involved in scientific research that is focused on the environment.  For particular examples, he mentioned a textbook and companion website that have been developed and have some great application questions.

This is an important time for alternative energy generation and research directed at the environment in general.  I wrote an earlier post about the Solar Tres project.  The development of electric car motors and batteries has reached a point that production of “all-electric” car models is happening now.  One of my calculus textbooks has a cover page addressed to the instructor saying “The first person to invent a car that runs on water may be sitting right in your classroom.”

On Saturday morning, I saw a presentation by Geza Laszlo called “Rational Approximations of Roots of Polynomials.”  This is a very interesting topic.  It has connections to some of the material we cover in MTH 111 about roots of polynomials, but it is more closely related to the ideas we discuss in MTH 116 about using Newton’s method to approximate a square root.

Newton’s method uses calculus, but the method itself was known to the Greeks, even though they did not have formal knowledge of the methods of calculus.  The idea of approximating irrational numbers with rational numbers is also of great importance in constructing a musical scale.  Attempting to approximate (log3)/(log2) with a rational number determines how an octave will be separated into notes and how accurate the scale will be.