# Why do we get like that the message?

• MHB
• evinda
In summary, when encrypting with RSA, we choose two large arbitrary prime numbers and use them to compute the public and private keys. We can encrypt a message by raising it to the power of the public key modulo n, and decrypt it by raising the encrypted message to the power of the private key modulo n. This works because of Euler's theorem, which states that if two numbers are coprime, then raising one to the power of the Euler's totient of the other will result in 1 modulo the other number.
evinda
Gold Member
MHB
Hello! (Wave)

When we encrypt with RSA, we choose two large arbitrary prime numbers $p$ and $q$.
We choose $n=pq$. We compute $\phi(n)=(p-1)(q-1)$.
We choose a number $e>1$ such that $e^{\phi(n)}\equiv 1 \pmod{n}$.

We compute the inverse of $e$, $d \equiv e^{-1}\pmod{\phi(n)}$.
The public key is $(n,e)$ and the private key $(n,d)$.

We encrypt a message $M$ as follows:

$$C(M)=M^e \mod{n}$$

We decrypt the message as follows:

$$M(C)=C^{d} \mod{n}$$

My question is the following:

Why do we have that $M^{ed}=M \mod{n}$, given that $ed=1 \mod{\phi{n}}$ and not modulo $n$ ? (Thinking)

Hey evinda!

It's a consequence of Euler's theorem that says that if $a$ is coprime with $n$ that then $a^{\phi(n)}=1\bmod n$.

Let $k$ be such that $ed = k\phi(n) + 1$.
If $M$ is coprime with $n$ then it follows that:
$$M^{ed} = M^{k\phi(n) + 1} = (M^{\phi(n)})^k\cdot M = M\bmod n$$

Klaas van Aarsen said:
Hey evinda!

It's a consequence of Euler's theorem that says that if $a$ is coprime with $n$ that then $a^{\phi(n)}=1\bmod n$.

Let $k$ be such that $ed = k\phi(n) + 1$.
If $M$ is coprime with $n$ then it follows that:
$$M^{ed} = M^{k\phi(n) + 1} = (M^{\phi(n)})^k\cdot M = M\bmod n$$

Ah, I see... Thanks a lot! (Sun)

## 1. Why do we sometimes misunderstand or misinterpret messages?

There are a few reasons why we may misunderstand or misinterpret messages. One common reason is that we bring our own biases and assumptions into the conversation, which can lead us to interpret the message in a way that aligns with our preconceived notions. Additionally, communication is a complex process and there are many factors that can affect how a message is received, such as tone, body language, and cultural differences.

## 2. How does the sender's communication style impact the message?

The sender's communication style can greatly impact how the message is received. For example, if the sender is using vague or ambiguous language, the receiver may have a difficult time understanding the intended message. Similarly, if the sender is using a confrontational or aggressive tone, the receiver may feel defensive and be less receptive to the message.

## 3. Can our emotions affect how we interpret a message?

Yes, our emotions can definitely affect how we interpret a message. When we are feeling angry, sad, or anxious, we may be more likely to perceive a message in a negative way. On the other hand, if we are feeling happy or content, we may be more open to receiving and understanding the message.

## 4. How does our past experiences influence our understanding of a message?

Our past experiences can greatly influence how we understand and interpret a message. For example, if we have had negative experiences in the past with a certain topic or individual, we may be more likely to perceive the message in a negative light. On the other hand, if we have positive experiences, we may be more open and receptive to the message.

## 5. What are some strategies for improving our communication and understanding of messages?

Some strategies for improving communication and understanding of messages include active listening, clarifying any misunderstandings, and being aware of our own biases and assumptions. It can also be helpful to ask questions and seek clarification if we are unsure of the message's meaning. Additionally, being open-minded and considering the perspective of the sender can lead to better understanding and communication.

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