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From: Ethan Carter <ec1828@gmail.com>
Newsgroups: comp.misc
Subject: Re: Truly Random Numbers On A Quantum Computer??
Date: Sat, 29 Mar 2025 20:25:23 -0300
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Lawrence D'Oliveiro <ldo@nz.invalid> writes:

> On Fri, 28 Mar 2025 21:16:29 -0000 (UTC), I wrote:
>
>> The definition of “randomness” is “you don’t know what’s coming next”.
>> How do you prove you don’t know something? You can’t. There are various
>> statistical tests for randomness, but remember that a suitably encrypted
>> message can pass every one of them, and a person who knows the message
>> knows that the bitstream is not truly random.

Knuth gives a nice lecture about the definition of randomness in TAoCP,
volume 2, section 3.5---what is a random sequence?  He gives a nice
definition (definition R1, page 152), which doesn't quite work, though
it's quite simple; he then patches it various times, reaching definition
R6, which he claims it works against all criticisms.  It's quite a
precise definition, so it's worthy of mention.

There's also an interesting paper by Anna Johnston on entropy, in which
she makes the (correct, in my opinion) remark that entropy really is a
relative notion.

--8<-------------------------------------------------------->8---
Note that entropy is relative. It is not a solid, physical
entity. Entropy depends on perspective or what is known and unknown
about the data to a given entity. Once viewed, all information in the
data is known to the viewer (zero entropy in the viewers perspective),
but the data still contains entropy to non-viewers. The belief that
entropy is something that has a classical, fixed measure is false and
causes many interpretation issues. -- Anna Johnston, ``Comments on
Cryptographic Entropy Measurement'', 2019, section 2, page 3.

Source:
<https://eprint.iacr.org/2019/1263.pdf>
--8<-------------------------------------------------------->8---

> Here’s an even simpler proof, by reductio ad absurdum.
>
> Suppose you have a sequence of numbers which is provably random. Simply 
> pregenerate a large bunch of numbers according to that sequence, and store 
> them. Then supply them one by one to another party. The other party 
> doesn’t know what’s coming next, but you do. Therefore they are not random 
> to you.
>
> Which contradicts the original assumption of provable randomness. QED.

I get the feeling here that, by the same token, you could never have a
provably secure cryptosystem because someone knows the private key?