Score:1

Crypto

Formal definition of mono-alphabetic Substitution Cipher

Saikat

2/19/23, 6:56 PM

I am trying to write the formal definition for mono-alphabetic Substitution Cipher. I have tried the following

$\mathcal{M}:=$ Set of all possible arbitrary length string of English language text, removing all punctuations, numerals, spaces in a string.

$\mathcal{C}=\mathcal{M}$

$\mathcal{K}:=S_{26}$

Let $m=m_1m_2\cdots m_l \in \mathcal{M}$ then

$Enc_{k}(m):=k(m_1)k(m_2)\cdots k(m_l)=c=c_1c_2\cdots c_l \ \ \text{where} \ \ c_i=k(m_i)$

$Dec_k(c):=k^{-1}(c_1)k^{-1}(c_2)\cdots k^{-1}(c_l)$

My question is how to define the Key generating algorithm $Gen$?

0 + 0

encryption

kelalaka

2/19/23, 6:59 PM

Select a random key from $\mathcal{K}$? And, $c_i=k(m_i)$ is not clear for substitution cipher. Should be $c_i = (m_i + k) \bmod 26$. Also message encoding and decoding is an impartant part $A=0, B=1,...$

Saikat

2/19/23, 7:01 PM

Yes this is ok. But is there any algorithm for this?

kelalaka

2/19/23, 7:03 PM

language-specific yes, algorithmic saying `uniformly select an element from` should be enough.

bmm6o

2/19/23, 9:46 PM

@kelalaka it's not addition mod 26, you're thinking of the Caesar cipher.

bmm6o

2/19/23, 9:46 PM

But I agree that key generation does not need to be part of the algorithm.

kelalaka

2/19/23, 9:49 PM

@bmm6o this is a substitution cipher where Caesar cipher is part of it. It contains only 26 of the permutation cipher's 26! keys.

Score:1

Crypto

Titanlord

2/21/23, 12:37 PM

First of all a monoalphabetic substitution cipher can be the concept of monoalphabetic substitution or an encryption only using this technique. That means, Ceaser is a monoalphabetic substitution, because it uses this concept. The difference to the "real" monoalphabetic substitution is, that the key generation is not completely random.

I personally would not define $\mathcal{M}, \mathcal{C}, \mathcal{K}$ the way you did, because I think it's not clear enough. A more mathematically definition could be better:

$\mathcal{M} = \{ a,b,c,...,z\}^*$ and $\mathcal{C} = \{ a,b,c,...,z\}^*$

I also want to avoid $\mathcal{M} = \mathcal{C}$ because it may be mathematically correct, but could imply a wrong understanding to the reader. I chose $\{ a,b,c,...,z\}^*$, because normally the message space is $\{0,1\}^*$ and I adapted that to the given context.

Now I think you can't define $\mathcal{K} = S_{26}$. What is $S$? I would define $\mathcal{K} = \{f_k\mid k \in \{a,...,z\}\}$, where $f_k$ is a bijective substitution function.

Encryption and Decryption looks good. I would do it like that for a message $m = m_1, ... m_n \in \mathcal{M}$:

$Enc_k(m)$: $c_i = f_{m_i}(m_i)\forall i \in \{1,...,n\}$
$Dec_k(c)$: $m_i = f^{-1}_{c_i}(c_i) \forall i \in \{1,...,n\}$

The Generation can then be defined as:

$Gen$: Choose a random substitution $f_k$ for every $k \in \{a,...,z\}$, so that $f^{-1}_k$ is the bijective inverse function

0 + 0

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Formal definition of mono-alphabetic Substitution Cipher

TH: คำจำกัดความอย่างเป็นทางการของรหัสแทนตัวอักษรแบบโมโน

RO: Definiția formală a cifrului de substituție monoalfabetic

RU: Формальное определение моноалфавитного шифра замены

VI: Định nghĩa chính thức của mật mã thay thế đơn chữ cái

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.