I think you are wrong because the probability of finding a valid block at trial number N should follow a Geometric Distribution.
The Geometric Distribution has the property of being "memoryless".
Discussion
Correct, it makes no difference. This indifference is part of what makes a hash function a "cryptographic hash function". The distribution is nearly gaussian, close enough to call it random.
you probably wanted to say uniform distribution.
Hash function have the design goal of inducing a probability distribution on their image which is closer to a uniform distribution as possible.
So the probability that h(X) = k is the same for every k.
continuous uniform distribution
https://en.wikipedia.org/wiki/Probability_distribution#Absolutely_continuous_probability_distribution which is commonly associated with gaussian normal distributions.
of course it's discrete integers but continuous implies total randomness since all bits are at play and not any holes in it or symmetries.
I don't understand why you say Gaussian, when it actual fact is uniform.
Also, it's discrete and not continuous, because the Image of all hash functions is discrete (countable infinity of rational numbers).
If you found a distribution pattern in a hash function, it's no longer a cryptographic hash function.
There is no such thing in the real world as continuous, it is impossible to measure or quantify. Saying that this disqualifies the use of "gaussian" for a random distribution over a finite field is not a tenable position due to the nature of computation.
The expression "cryptographic hash function" itself implies apparent continuity of distribution. As soon as the discontinuity is found in the distribution its security is busted.
The other thing that you can't say either is that it's not deterministic, because without determinism the hash function is not useful.
Yeah yeah, in this we agree. I was just lazy.
One should say:
X is a random variable representing the choice of a message at random (under probability distribution PX).
Then h(X) is a random variable, that induce a different probably distribution bla bla.
it's super magical stuff anyway. non-crypto people don't get it.
Yes, no such thing as continuity in the real world, real number are not physically real.
Buuut, they are very useful for making computations easiers.
I never disqualified the utility of the gaussian distribution, I simply pointed out the obvious that Gaussian != Uniform
> the expression "cryptographic hash function" itself implies apparent continuity of distribution.
No, it implies apparent uniformity of disitribution.
Continuity is another (topological) property.
f^(-1) (A) is an open set for every open set A.
gaussian is continuous uniform distribution. aliasing is an inherent property of finite fields. if this error of precision says it's not gaussian than what use is the expression "gaussian distribution" anyway?
perhaps ergodicity is a more accurate expression, since this doesn't carry baggage of theoretical and impossible things with it?
Cool thanks. I don't Really understand what that means. But I think I get what you are saying.
My brain was thinking if it's Ramdom, wouldn't Be easier to hit a still target rather than a moving one.
AApparently doesn't Make a difference
%20-%20Profi.png/:/cr=t:7.63%25,l:7.63%25,w:84.75%25,h:84.75%25/rs=w:1536,cg:true,m)
You have to keep iterating changes, then hashing with SHA256, to keep trying to find a hash under the current target.
A few of these levers are to include/exclude a transaction, reorder transactions, or iterate the nonce.
All hash attempts are equiprobable in finding a valid block, irrespective of which levers you're pulling.
Thanks. Yeah, that’s where my brain was going, why introduce more variables like change in tx or tx order, if you can keep these fixed and just iterate on nonce.
But I have no idea how this works, so with every comment I’m learning something new.
Might help you grok this if you mess around with a SHA256 calculator, like https://xorbin.com/tools/sha256-hash-calculator
Paste in the first paragraph (the description under the calculator, starting with "The SHA") and it'll give you a hash that starts with ae8b.
If I told you to make changes to that paragraph until you generate a hash that starts with 0, you should hit that target once every 16 modifications on average, regardless of if you start adding letters to the beginning, subtracting from the middle, or changing punctuation at the end.
Instead of that paragraph, hashers are using block templates and nonces as inputs. Both can change every time, or just one can change, and the odds of finding a hash below the target isn't affected by which one is changing (or if it's one or both).