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Bill Cox
2014-09-29 22:30:14 -04:00
parent 73290eabb3
commit 7d2863fb1a
1 changed files with 58 additions and 22 deletions
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@@ -258,23 +258,24 @@ Fortunately, it is possible to reuse the same multplier for each stage, without
slowing down the circuit. In our long chain of 2X modular multipliers, we computed:
V1(1) = 2*Vzener(0) mod Vsupply
V2(2) = 2*V1 mod Vsupply
V3(3) = 2*V2 mod Vsupply
V2(2) = 2*V1(1) mod Vsupply
V3(3) = 2*V2(2) mod Vsupply
...
V30(30) = 2*V29 mod Vsupply
V30(30) = 2*V29(29) mod Vsupply
Here, Vzener(0) is Vzener when sampled at the first clock pulse. Vzener(n) is the voltage
sampled on the nth clock pulse. Vn(t) is the output of the nth 2X modular multiplier at
clock cycle t. Instead of using 30 stages, what would happen if we added V(1) to
Vzener(1), and used the same exact multiplier stage?
clock cycle t. Instead of using 30 stages, what would happen if we simply fed the output
of the 2X modular multiplier stage back on itself? We'd just have V1 instead of V1 ..
V30:
V1(1) = 2*Vzener(0) mod Vsupply
V2(2) = 2*(Vzener(1) + V1(1) mod Vsupply
V1(2) = 2*(Vzener(1) + V1(1) mod Vsupply
= 2*Vzener(1) + 4*Vzener(0) mod Vsupply
V3(3) = 2*(Vzener(2) + V2(2)) mod Vsupply
V1(3) = 2*(Vzener(2) + V1(2)) mod Vsupply
= 2*Vzener(2) + 4*Vzener(1) + 8*Vzener(0) mod Vsupply
...
V30(30) = 2*Vzener(29) + 4*Vzener(28) + 8*Vzener(27) + ... + 2^30*Vzener(0) mod
V1(30) = 2*Vzener(29) + 4*Vzener(28) + 8*Vzener(27) + ... + 2^30*Vzener(0) mod
Vsupply
If Vzener(t) samples are truely unpredictable and uncorrelated, then 2^30*Vzener(0) mod
@@ -282,27 +283,62 @@ Vsupply is an unpredictable value almost uniform between 0V and Vsupply. The ot
can in no way reduce this unpredicability. What if Mallory attacks? In that case, at
step 30, we have:
V30(30) = 2*(Vzener(29) + Vmallory(29)) + ... + 2^30*(Vzener(0) + Vmallory(0) mod
V1(30) = 2*(Vzener(29) + Vmallory(29)) + ... + 2^30*(Vzener(0) + Vmallory(0) mod
Vsupply
= [2*Vzener(29) + 4Vzener(28) + ... + 2^30*Vzener(0)] +
= [2*Vzener(29) + 4*Vzener(28) + ... + 2^30*Vzener(0)] +
[2*Vmallory(29) + 4Vmallory(28) + ... + 2^30*Vmallory(0)] mod Vsupply
This is just the signal we had before without Mallory's influence, plus a predictable
value injected by Mallory. Once again if Mallory does not know what the value of V30(30)
would have been, he cannot control the result. He can only make it more random.
This is just the signal we had before without Mallory's influence, plus a value injected
by Mallory. Once again if Mallory does not know what the value of V1(30) would have
been, he cannot control the result. He can only make it more random.
The value of V1 actually acts as an entropy pool, collecting entropy from all signals
that impact it's value.
### We Don't Need the Zener
In reality, there are many sources of noise in every circuit, and some will not be
correlated with others. Thermal noise at the fempto-volt level in the comparator is
unlikely to be correlated with fempto-volt level noise in the amplifier. About 40 clocks
later, these two noise sources will be on the same order of magnatude as Vsupply, and
their entropy will combine to make the output even less predictable.
In reality, there are many sources of unpredictable noise in every circuit. There's large
predictable and controlable noise, like power supply noise, and tiny 1/f noise in the
multi-gigahertz range. Shot noise, thermal noise, EMI, cross-talk... you name it, no
matter where we look, there's noise. Infinite noise multipliers amplify them all in
parallel, and adds them together effectively in an entropy pool. Zener noise would be
just one more source of noise in a symphony of noise sources.
It is exactly like what happened when Mallory tried to control the output. He made it
more random instead! *Every* source of entropy in the entire circuit contributes to the
unpredictability of the output. The individual noise sources contribute a power-of-two
sequence to the total, just like Mallory did.
An INM will amplify *every* source of niose until it is larger than Vsupply. It adds them
together and amplifies them in parallel. Every device in the signal path in the loop
contributes, most of them in several ways. The board level version has literally hundreds
of noise sources, any of which would be a fine noise source on its own.
With N sources of noise, the output looks like:
V1(1) = Vnoise1(1) + Vnoise2(1) + Vnoise3(1) ... VoinseN(1)
...
V1(30) = [2*Vnoise1(29) + 4*Vnoise1(28) + ... + 2^30*Vnoise1(0)] +
[2*Vnoise2(29) + 4*Vnoise2(28) + ... + 2^30*Vnoise2(0)] +
...
[2*VnoiseN(29) + 4VnoiseN(28) + ... + 2^30*VnoiseN(0)] mod Vsupply
Each individual noise sources contributes its own power-of-two sequence to the total.
A fempto-volt noise source is just as powerful as a Vsupply/10 amplitude noise source.
The mashing together of noise source data with unbounded modular multiplicationes leads to
awesome entropy.
### Non-Power-of-Two Multiplication
The circuit shown in infnoise_fast multiplies by 1.82 every clock rather than 2.0. As
stated above, this reduces the entropy per output bit to log(1.82)/log(2) = 0.86 bits of
entropy per output bit.
A reasonable model for thinking of the voltage V1 in a INM is as a binary floating point
real number between 0 and Vsupply. The number runs into quantum effects at it's lowest
bits, where things are really fuzzy. More solid digits are part of the entropy pool,
collecting noise from every source, and shifting the data, one bit at a time, towards
higher positions. If there are for example effectively 50 bits of precision that count as
measurable state in the system, then there is a 50 bit entropy pool collecting noise.
That's 2^50 different states the pool can be in. To determine which of those 2^50 states
we are in, we can do multiplication by 1.8X, modulo Vsupply, over and over again. Each
time we shift out a 0 or 1 let's us estimate the actual voltage we started with by
### Free As in Freedom