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[Phys-L] thin lenses, Feistel ciphers, symplectic integrators, et cetera



Question:
-- What does simple geometric optics have to do with modern cryptography?
-- What does either of those things have to do with good numerical methods
for integrating an equation of motion, such as the Keplerian orbit of a
particle?

Answer: Entropy-preserving updates. In particular: independent, orthogonal
updates.

A) For a system of thin lenses, the update rule has two parts:
-- Free propagation: X is updated by an amount that depends on dX/dZ,
_independent of X_
-- Refraction: dX/dZ is updated by an amount that depends on X
_independent of dX/dZ_

B) In a Feistel cipher, you break the block into two pieces, X and Y.
-- X is updated based on some function of Y _independent of X_
-- Y is updated based on some function of X _independent of Y_

C) At each step of the integration, if you're smart you'll break it
into two substeps:
-- the position X is updated based on the momentum _independent of X_
-- the momentum P is updated based on the position _independent of P_


In all cases, you can define a notion of phase space. The aforementioned
properties of the update rule suffice to ensure that area in phase space
is conserved. That means that the entropy (and information) are unchanged
at each step.

For more on all this, including diagrams, see
http://www.av8n.com/physics/symplectic-integrator.htm#sec-analogous

=====================================================
Pedagogical remarks:

I am fond of quoting what William James said about memory. More than
100 years ago he talked about the /associations/ between ideas:

Each of the associates is a hook to which it hangs, a means
to fish it up when sunk below the surface. Together they form
a network of attachments by which it is woven into the entire
tissue of our thought. The ’secret of a good memory’ is thus
the secret of forming diverse and multiple associations with
every fact we care to retain. But this forming of associations
with a fact, – what is it but thinking about the fact as much
as possible? Briefly, then, of two men with the same outward
experiences, the one who thinks over his experiences most, and
weaves them into the most systematic relations with each other,
will be the one with the best memory.

Obviously we are not talking about mere rote memory here. We are talking
about the highest form of memory, and indeed about learning and reasoning
in all generality.

I tell students -- repeatedly and emphatically --

Getting a good grade on the test does not depend primarily on how
hard you work during the test. To a much greater extent, it depends
on how you lived your life in the months and years leading up to the
test. Every time you hear a new idea you need to mull it over in your
mind, checking to see how it connects (or conflicts!) with other stuff
you know. This takes time and effort.

Nothing you do in physics class will make you permanently smart,
for the same reason that nothing you do in PE class will make you
permanently strong. The best we can hope for is that the classes
will motivate you to develop a lifelong /habit/ of physical exercise
and mental exercise, so that over time, you get stronger and stronger
... and smarter and smarter.

I'm ashamed to admit it, but when I first learned about symplectic
integrators, I did not immediately see the connection to geometric
optics. I really should have. Phase space is phase space, after
all. Much later I was thinking about lenses, and HEY WAIT A MINUTE,
where have I seen this before??? So I started thinking about the
connection to symplectic integrators.

Another of my rules is: When you find a bug, don't just fix that
one bug; check for other bugs of the same kind. So I started
thinking about what's special about the update rule in these cases,
and looking for other examples. This is tricky, because almost
all thought and memory processes are subconscious, and it's hard
to tell your subconscious mind what to do. It took me about an
hour of racking my brain to come up with the cryptography example.

Multitasking helps with this. If I spend an hour swimming or
doing the dishes and/or various other mindless tasks, I can use
the /same/ hour to rack my brain looking for entropy-preserving
phase-space-preserving update rules.

Having done my homework, I feel better now. I feel guardedly
optimistic that the next time I see a situation where an entropy-
preserving update rule is used (or could be used!) I will recognize
it as such.