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How are the cipher, the key and the initial message (that is not encrypted) are releted?

cd flag

Suppose that $m$ is a message that someone player $i$ wants to send to a network of other players $j\neq -i$. The player to prevent his message from cheating by others uses an encyrpstion scheme. Say $$g:M\times Y \to X$$ denotes a cipher where $Y$ is the key and $X$ a code that makes the message to look random. The standard assumptions to be made are that $|Y|\geq |M|$ and $g(\cdot,y)$ is a bijection namely every pair of $(m,y)$ is associated with only one $x$. My question is how are the key $y$, the code $x$, and the message $m$ are associated? for example if we could make some operations among $g$, $y$ and $m$, what would that be? could we claim that $x\oplus y \underbrace{=}_{?}m$? or somehting like this?

kelalaka avatar
in flag
What is the origin of this Question? You did not define $g(\cdot,y)$ other than saying it is a bijection. What is the aim of this?
Nav89 avatar
cd flag
@kelalaka what do you mean what is the aim of this?
kelalaka avatar
in flag
Just use RSA-KEM to encapsulate random key per user and encrypt with AES-GCM or see Libsodium...
Nav89 avatar
cd flag
@kelalaka I have no idea what is RSA-KEM and AES-GCM... cryptography is not my field, so explain to me what are these schemes. I only know group theory that I was taught in an introductory course as undergraduate
Nav89 avatar
cd flag
@kelalaka also I care about the maths...the mathematical formulation...nothing less
kelalaka avatar
in flag
[RSA-KEM, AES-GCM](https://crypto.stackexchange.com/a/76857/18298) I don't what you mean by anything less. This is Cryptography and Math, CS, EE, Stat, Prob, and some all other in!
Nav89 avatar
cd flag
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/132871/discussion-between-nav89-and-kelalaka).
Nav89 avatar
cd flag
@kelalaka w8 a minute and let me explain. For example you use a function in the point $2.$ Use a Key Derivation Function (KDF) on x etc. How is this function defined? What is the domain, the co-domain and what are the properties that it has. Also, for me maybe there is some other cosntruction is of major interest for specifir reasons...but let's say that I understand your encryption-decryption technique...there seem to be many things that I need to know about this mathematical structure
kelalaka avatar
in flag
Well, domain and codomain are really dependent on the KDF: Just a Hash, HKDF, Password based... My humble advice for you reading some into dictionary books? A heavily math based [An Introduction to Mathematical Cryptography](https://www.amazon.com/Introduction-Mathematical-Cryptography-Undergraduate-Mathematics/dp/1441926747) and/or [Introduction to Modern Cryptography: Third Edition](https://www.amazon.com/Introduction-Cryptography-Chapman-Network-Security/dp/0815354363/) and/or [A Graduate Course in Applied Cryptography ( free book)](https://toc.cryptobook.us/) and some free lectures?
Nav89 avatar
cd flag
@kelalaka i will take a look in the later one. However some tools are not useful when you are not doing cryptoghraphy. For example, even the economists use tools from cryptography to make communication equilibria in game thery, but they do not need to know or mention what a Hash, HKDF etc....they need to translate the semantics to their field or ignore them and search for anything close to them...this i why it is difficult to reconcile alo the kind of questionts that are made here...
Nav89 avatar
cd flag
@kelalaka the last of the three books that you mention is quite good as I see it...It will help me. Thanks.\
Score:0
ua flag

Taking into account the book. I write here an example. Suppose, that we have a mechanism of communication $\mathcal{M}=(g,h)$ such that $\mathcal{M}$ is defined over $(Y,M,X)$, where $Y$ is the key, $M$ the message and $X$ the cipher spaces respectively. To simplify the problem even more I assume that $Y=M=L=\{0,1\}^l=G$ instead of an arbitrarily finite field $\mathbb{F}^n$ and write below

$$g(y,m)=x,\quad\text{is the encrypted message, which by definition equals $x$}$$

$$h(y,x)=m,\quad\text{is the decrypted message, which by definition equals $m$}$$

So, indeed $(y,x)$ is defined to be associated with only one $m$ and hence $g(y,\cdot)$ is bijective by definition. To anser the question how are they associated, when someone knows both $x$ and $y$, then indeed $x\oplus_{G} y=m$

In order to decrypt the message we have that

$$h(y,x)=h(y,g(y,m))=y\oplus_G x=m$$

where $\oplus_{G}$ is the operation of $+$ as it is defined in the finite field $G$. And hence we have show that the calculation that you ask for, it holds by definition.

ph flag
If $g$ is just some bijection, how can you conclude that it has any connection to a field operation? The original question doesn't talk about fields at all.
Hunger Learn avatar
ua flag
Anyone who has to add a comment or thinks that I am understanding something wrong you can say this to me. But I think that this is the simplest explanation under the Shannon mechanism for perfect security.
Nav89 avatar
cd flag
Well, it seems ok to me...and after taking a look at the books mentions by @kelalaka I think that this is the case. So if the specialists here think that your answer is fine, I will accept it as the answer that solved my problem.
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ph flag

You can't really say much beyond what is true by definition, i.e. that $g(m, y) = x$. The problem as stated intentionally leaves $g$ minimally restricted, so there are any number of concrete functions that would satisfy the statement. Further, $M$, $X$, and $Y$ are also minimally restricted by being defined as sets, and as such you can't assume the existence of any such function $\oplus$.

Remember that the purpose of using such general definitions is to prove theorems with the broadest implications. Then you know that the results apply to any particular function that satisfies the conditions.

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