Re: 🏳️‍🌈Exploration of Diffie-Hellman Multiplication versus Exponentiation🏳️‍🌈

On Mon, 15 Apr 2024 14:59:20 -0500, LO AND BEHOLD; SugarBug
<3883@sugar.bug> determined that the following was of great importance
to SugarBug <3883@sugar.bug> and subsequently decided to freely share it
with us in <2d79d2a2ad4bb66ed7e75c60f482a03d$1@sybershock.com>:
 
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Notation --------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=        [^] = power or exponentiation, NOT exclusive or.         [*] =
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= multiplication.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Exploration -----------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= With simple Diffie-Hellman Key Exchange we sometimes see something like
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= this:
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= (x ^ y) mod M where x, M are primes and x, y are less than M
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= or x, y, M are primes and x, y are less than M
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= with attacker knowing x, M and solving to recover y.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Example 1 ---------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= (3301 ^ 1033) mod 9901 = 5711
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= with attacker knowing x, M and solving to recover y.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= we can do similarly with large integers using singular multiplication
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= instead of exponentiation:
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Example 2 ---------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= x, y are secret and M is public,
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= and such that (x * y) mod M is used instead of (x ^ y) mod M:
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= 7614733470835803029 * 7614733470835803029 mod 17387392922984910589 =
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= 5287290486333094173
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= with attacker knowing M and solving to recover x, y.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Of course the integers must be considerably larger for difficulty.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Besides the number of multiplications required for exponentiation with
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= large exponents, what other property of exponentiation makes it the
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= go-to choice over simple multiplication with Diffie-Hellman Key
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Exchange problems?
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Or asked another way, given the modulus remainder:
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Example 3 ---------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= public modulus: 666019804555242738147912491645368766591 modulus
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= remainder: 307023976544267555158714704165625061584
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= such that (x * y) mod M is used instead of (x ^ y) mod M:
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= (x * y) mod 666019804555242738147912491645368766591 =
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= 307023976544267555158714704165625061584
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= with attacker knowing M and solving to recover x, y.
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Question 1 ----------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= How difficult is it to find the two hidden prime factors, as opposed to
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= finding the secret exponent of a known factor as in the first example?
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Question 2 ----------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= What modern methods of attack are used to discover the hidden factors of
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= example 3 and where can I read about them with examples?
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Question 3 ----------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Do any cryptosystems actually use any variation of example 3 with
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= multiplication rather than exponentiation, and if so, which?
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Question 4 ----------
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= Per examples 1 through 3, if x, y are GREATER than modulus M, what are
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= the implications for security or attack domain parameters and how? Or
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= in other words, why are x and y usually chosen to be less than M, from
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?= a security perspective?
=?UTF-8?B?8J+Ps++4j+KAjfCfjIg=?=

Did they take away your AI account?


Most of the questions up there can be answered by a web search and research instead of begging for incorrect suggestions on usenet from mostly idiots.

--

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"You should see my archive on you" -James "Checkmate" Gorman teases us with his "dosser" in <MPG.3eea81b23145043b98c338@usnews.blocknews.net>

"Sorry, nothing to see here. The joint wasn't as bad as they say, but I'm not looking to go back. I'm a model citizen, clean as a whistle. I've owned my own home for 12 years, owned my own business almost as long, don't bother anyone and they don't bother me. You have nothing in any "police report" pertaining to me. Don't you think they would have "come a-knockin" a long time ago if they had any reason to? You're delusional and paranoid, and I have to wonder why.

Oh... I should mention that there are a LOT of trannies in prison. I don't know why, but there are. The State even has to give them hormone shots for their tiddies at taxpayer's expense, and they wear bras and panties. I found everything about them revolting. That's why the whole "Bubba" thing is almost completely a myth, except in cell living. That shit wouldn't fly in a 100-man dorm, but trust me, those little trollops find ways to serve the willing when the lights go out. You see something, you keep your mouth shut about it because that way you don't get in a wreck. I never partook in such activities because the whole idea is just repulsive. I think that's a big part of what I don't like about you. I've seen how they act and I've talked to a few... total drama queens in every sense." -James "Checkmate" Gorman reminisces about prison in <MPG.3ceb37c2eddbaff98b07a@usnews.blocknews.net>

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