Explain the mathematical definition of a cryptosystem using words

So when googling "cryptosystem" I found a Wikipedia page which contained a mathematical definition of a cryptosystem. Find this definition below.

Could someone tell me what that what the last line means, in words instead of using mathematical signs? Maybe it will help my understanding of it.

Mathematically, a cryptosystem or encryption scheme can be defined as a tuple $(\mathcal{P}, \mathcal{C},\mathcal{K},\mathcal{E},\mathcal{D})$ with the following properties.

1. $\mathcal{P}$ is a set called the "plaintext space". Its elements are called plaintexts.
2. $\mathcal{C}$ is a set called the "ciphertext space". Its elements are called ciphertexts.
3. $\mathcal{K}$ is a set called the "key space". Its elements are called keys.
4. $\mathcal{E}= \left\lbrace E_k:k\in \mathcal{K}\right\rbrace$ is a set of functions $E_k:\mathcal{P}\rightarrow \mathcal{C}$. Its elements are called "encryption functions".
5. $\mathcal{D} = \left\lbrace D_k:k\in \mathcal{K}\right\rbrace$ is a set of functions $D_k:\mathcal{C}\rightarrow\mathcal{P}$. Its elements are called "decryption functions".

For each $e\in \mathcal{K}$, there is $d\in \mathcal{K}$ such that $D_d(E_e(p))=p$ for all $p \in\mathcal{P}$.

• "For every encryption key, there's a decryption key" – poncho Nov 21 '16 at 19:07
• ..."and if you apply them both with their respective algorithms you get back what you'd expect: the message for all allowed messages." – SEJPM Nov 21 '16 at 19:10
• "No matter what key $e$ you choose for encrypting a plaintext $p$, there is some key $d$ (which may or may not be equal to $e$) such that you can decrypt the resulting ciphertext successfully to recover the original plaintext." – Luis Casillas Nov 21 '16 at 19:19
• "Every plaintext is associated with a ciphertext by an encryption function that uses key $e$"."Every ciphertext is associated with a plaintext by a decryption function that uses key $d$". – kub0x Nov 21 '16 at 19:44

For each key $e$ in the set of keys $\mathcal{K}$, there is a key $d$ also in the set of keys $\mathcal{K}$ such that the decryption function using $d$ as a key, called $D_d$, when applied to the encryption function using $e$ as a key called $E_e$ is the identity function, so that when both are applied in a succession, starting with $E_e$, to a plaintext $p$, one will recover the same plaintext $p$.
So, for each encryption key $e$, there is a decryption key $d$ (possibly the same key) which allows to decrypt the content encrypted using the key $e$.
• I would comment on Lerys wonderfuly explanation but sadly I don't have enough reputation yet. One thing I like to add to Lery is that the definition also includes: [...] for all $p \in\mathcal{P}$. This means that the definition must hold for all plaintext $p$ from the set $\mathcal{P}$. If you find one $p$ that you can encrypt with a given key $e$ and the encryption function $E_e$ but there is no key $d$ to decrypt it back to $p$ with $D_d$ it's no longer a crypto system per definition. Even if it works for all other elements in $\mathcal{P}$. – c-pid Dec 6 '17 at 13:14