Last questions: Something you said was interesting ..

Actually you said something I thought was interesting.
Please allow me to query on last aspect of your obvious knowledge of the
reasoning tools
that you have exhibited to me - and actually explained pretty well at the
end.

You seemed to place emphasis on:
S1 has been generated by (say) a true Bernoulli process

Suppose this constraint is "not quite" true.
(a) Did you imply the possible ability to distinguish?
(b) If yes, what possible approaches would you use in addition to this
obvious one:
Use statistical measurements on the bit strings starting with something as
simple as the
number of 1's and 0's?
(c) Under these circumstances do you think that knowing that p is a fixed
prime
could be helpful knowledge?

[I will take a look at the references that you all mentioned to get a better
understanding of the reasoning tools that you exhibited to me.
I thank you all for that]

P.S. My use of statistical tools is not at your level.
Thus the value of your comments.

That's all for now.

"Pubkeybreaker" <Robert_silverman@xxxxxxxxxxxx> wrote in message
news:1151061711.462857.293080@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

kentucky wrote:
I have a simple thought experiment that I would like to ask you about.

A. My purpose is:
To learn analysis methods from the experts on this group.

There is no way that a few messages back and forth can replace the
courses you need to take. See below.

Please teach and guide me - so I can learn from your expertise.

(1) You need to learn some math. It is clear that your math background
is
not sufficient for you to understand what you are trying to do. [not
intended as
a flame; this is merely an observation]

(2) You need to learn how to discuss this subject without trying to
use common everyday English in place of rigorously defined mathematical
terminology.



B. I will now try to convey the experiment as clearly as I can:
Assume I have a prime number like the ones that satisfy RSA key
requirements
(p).

B.1 Say I had a good random source S (like the outputs of AES
encryption?)
I do not tell you what p is.
I run the source and generate a bit pattern S1 the same length as p.

"bit pattern" is gibberish. You have either generated an integer S1
or
a bit STRING of length log_2(p). [under certain conditions,
depending
on the method used to generate them, they are equivalent].

You also fail to discuss the pdf you are sampling when you generate
S1. Saying 'random source' is not sufficient. Do you even know what
a
probability density function is???

I embed p in S1

This is meaningless nonsense. How does one embed one number
inside another of the same length??? You CAN'T unless the numbers
are
EQUAL.

The general interpretation of the word 'embed' here, is that when p
and S1 are viewed as bitstrings, then p is a substring of S1. But
this
is impossible if they have the same length, unless they are equal.


I run the source again and generate a bit pattern S2 (S1 different from
S2).
I embed p in S2

More nonsense.


I repeat this process N times to get N bit patterns.


B.2 Now I run the source and generate N more S bit patterns.
This time I do not embed p in any members of this set.

Note: I am thinking that "embed" for this experiment
could be as simple as bitwise xor
or it could be some kind of digital signal convolution
(if that makes any sense?).

It does not make any sense at all.
If by "embed", you mean that an embedding is a non-linear
function (say E) of p and S1, then SAY SO.
Thus, for example, an embedding is E(p, S1) = p ^ S1 where ^ is
bitwise
xor. Or an embedding is simply E(p, S1) = p + S1 modulo 2^(length
of S1)
etc.

The idea is a simply to simulate hiding p in a noise.

Huh? Stop bandying words. One either hides p in a noisy signal
or one doesn't. The word 'simulate' in the above sentence has no
meaning.



C. Question:
C. 1 What is the feasibility of you deciding which bits patterns
have prime p embedded and which do not have p embedded.

It depends entirely on the selection of the embedding function E. If
S1
has been generated by (say) a true Bernoulli process, and E is the
simple xor function I defined above then there is NO WAY TO DISTINGUISH
between S1 and E(p, S1). The xor of a constant to a uniform
random
variable leaves the pdf invariant.

You are trying to disinguish the elements of two sets: Let L =
log_2(p)

A = {S_i | i=1,N, S_i is a uniformly generated random integer of
length L}

B = {E(S_i, p) | i=1,N, p is a fixed prime and S_i is the same
as in A}

Whether you can distinguish the elements of these two sets depends upon
ONLY one thing:

(1) The pdf for E(S_i, p)


Applying the function E is the same as doing a transformation on a
function
of a random variable. You question amounts to asking whether the
transformed
pdf is the same as the original. This question has nothing to do
with primes,
cryptography, or anything else except basic statistical theory of
distributions.

See, for example, Chapters 4 and 5 of:

Hogg & Craig: Introduction to Mathematical Statistics

I suggest that before you continue with this, that you take the
following:

(1) A course in number theory
(2) A course in mathematical statistics

Until then, you simply don't have the background to even discuss this
subject.


You also need to learn how to discuss mathematics.

I mean no flames in what I wrote, but you are trying to run before you
can walk. May I know the level of your math education?



.

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