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Archive for the ‘mathematics’ Category

back ashore

every thursday, after two days and nights of being asea, struggling to keep afloat on the waves of number fields and integer rings that threaten to engulf me, i steer my tiny paper boat to shore with just a pencil and eraser, and return to the world of men.

it is comforting to know that life goes on as usual, despite 2 O_K resisting all efforts at factorization in \mathbb{Q}[x]/(x^3 + x^2 - 2x + 8).

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First of all, 5OSME is here in Singapore! (Registration is over, but I think there are some portions of this convention for Origami in Science, Math and Education which are open to the public.)

Which explains why Erik Demaine and Robert Lang are here in Singapore!

Erik Demaine gave a talk today at NUS, and among the many amazing things he displayed, he demonstrated a “trick” that makes use of something called monotonous Boolean functions.

Before proceeding, I shall state the problem.

First, suppose we have a picture frame hanging from a nail by a piece of string like so:

If we remove the nail, the frame will fall.

Now suppose we have two nails instead of one. We can hang the frame from both nails in a few ways:

In the leftmost example, the frame will still remain suspended if we remove either nail; only by removing both nails can we cause it to fall. In the middle example, removing the red peg will cause the frame to remain suspended, while removing the green one will cause it to fall. The problem is this:

Is it possible to loop our string around the two pegs in such a way that removing either nail will cause the frame to fall?

This should be relatively simple to solve. (A version of the answer is here. There is no picture frame in the answer, so you have to imagine it hanging from the bottom of the loop, and convince yourself that removing either nail will cause the frame to fall)

Erik demonstrated the above in his talk today. He then asked: can we generalize this to more nails? Can we hang a picture from n nails such that removing any nail causes the picture to fall?

I’ll post about the solution, and its relation to monotone Boolean functions, in subsequent posts.

.

(Picture Credits:

Pictures of curved origami sculptures in glass

Picture frame)

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The story of SAGE

Couldn’t stop reading it. Fascinating. William Stein’s account of the development of SAGE.

To Sage and it’s developers, and a bright, free future for mathematical software ahead!

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Cryptographers have a sense of humour.

Or maybe they’re just nuts.

Read the following excerpts from peer reviewed papers and judge for yourself.

Inspired by Carter and Wegman, we use simple primitives which we call NUT (for “n-Universal Transformation”) since they are so cheap to implement. We propose construction methods for block ciphers that we call COCONUT (for “Cipher Organized with Cute Operations and NUT”), PEANUT (for “Pretty Encryption Algorithm with NUT”), and WALNUT (for “Wonderful Algorithm with Light NUT”).

Decorrelation: a theory for block cipher security, S. Vaudenay

Then this other bunch of guys comes up with a new cipher:

In this paper we will suggest a new block cipher called DONUT (Double Operations with NUT) which is made by two pairwise perfect decorrelation modules. DONUT is secure against boomerang attack.

New Block Cipher DONUT Using Pairwise Perfect Decorrelation, Dong Hyeon Cheon et al

(Don’t ask about the boomerang…)

I’m not surprised if, when cryptanalyst come up with a new, powerful attack that breaks all known ciphers, they call it Cryptonite.

And it pays to watch movies after all… Who knows? One day you might get to cite them in your papers! See citation 28:

citation1

The title of this paper is… <drumroll>… “Dial C for Cipher”. And in case you didn’t catch the allusion, the authors are kind enough to add a footnote:

Refering to the famous movie by Alfred Hitchcock Dial M for Murder[28]…

Dial C for Cipher, Thomas Baignères and Matthieu Finiasz

The same authors are responsible for another cipher, the Krazy Feistel Cipher. Why “Krazy” and not “Crazy”? There might be other reasons (remember what I said about them being nuts?), but take a look at the initials…

(do I hear clucking?)

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mmddyy

Happy “mm * dd = yy” day!

Oh, and a belated “mm ^ dd = yy” day, too!

Edit: Thanks GGY for the poem! Happened to learn LaTeX today, so here’s the equation for the poem:

ze

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proof on the screen

I had no idea that David Auburn’s play, “Proof”, which I wrote about a while back, was made into a movie in 2005.

The movie script apparently follows the play quite closely. Which is good, because this means some of the best lines in the play are available on IMDB. 🙂

Gotta find out where I can get this movie… don’t think it opened in cinemas here… seems kinda non-mainstream.

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The bird has taken to the air!!!

So many things I wanna do! Better list them out (O.O Did i just say “list them out”?? Ah! I mean, “list them down”… better start revising my phrasal verbs):

Learn

  1. C++
  2. LaTex
  3. Cryptography
  4. Number Theory
  5. Economics
  6. Mathematics of Finance
  7. Tai Chi

Program

  1. Cryptogram solver
  2. Sudoku solver, implementing my method
  3. Sudoku solver, implementing Knuth’s Algorithm X
  4. Generalised sudoku solver (any M x N grid)
  5. C++ Implementation of the RSA Algorithm

Other targets

  1. Touch my toes (you know what I mean… don’t tell me, “But you can touch your toes what… if you bend your knees”)
  2. Stop slouching/ hunching
  3. Finish reading “Bleak House”and “Foucault’s Pendulum”
  4. Attempt “Ulysses” (both Homer’s and Joyce’s)
  5. Reread Lewis, Chesterton, H.A. Williams, Herbert, Hopkins and Eliot, and find more of their like.

Will add more to the list as time goes by. Hope to start striking out some soon 🙂

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