# How Does the Equation D x E mod Φ(N) = 1 Relate to RSA Encryption?

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• brajesh
In summary, RSA encryption is a method of securely encrypting messages using a public key and a private key. The public key, which is known to everyone, is used to encrypt the message, while the private key, which is only known to the intended recipient, is used to decrypt the message. The encryption process involves multiplying the plain text message by an encoding key and taking the result modulo a public key. To decrypt the message, the recipient uses a decoding key and takes the result modulo the same public key. The relevance of the equation D x E mod Φ(N) = 1 is to find the appropriate values for the decoding and encoding keys, which are related to the prime factors of the public key. This ensures that only the
brajesh
TL;DR Summary
Understanding the RSA mathematics
I understand part of it, then I don't understand.

Definitions:

M=plain text message
C=encrypted message
N=public key
E=encoding key
D=decoding key
P,Q are prime factors on N

I get that
P x Q = N (where P and Q are primes).

I get that

Φ(N) = (P-1) x (Q-1)

I get that

MEmodN = C

I get that

CDmodN = M

But after this I don't understand the relevance of this equation to solve D and E

D x E mod Φ(N) = 1

Seems to be some step or something obvious I'm missing?

## 1. What is RSA mathematics?

RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem used for secure communication and digital signatures. It involves the use of mathematical algorithms to encrypt and decrypt messages, making it a popular method for securing sensitive information.

## 2. How does RSA work?

RSA uses a public and private key pair to encrypt and decrypt messages. The public key is used to encrypt the message, and the private key is used to decrypt it. The keys are generated using large prime numbers and complex mathematical calculations.

## 3. What is the significance of prime numbers in RSA?

Prime numbers play a crucial role in RSA as they are used to generate the public and private keys. The security of RSA relies on the difficulty of factoring large numbers into their prime factors. This makes it challenging for hackers to break the encryption and access the information.

## 4. Is RSA considered a secure method of encryption?

Yes, RSA is considered a secure method of encryption as long as the keys are kept secret. However, advancements in computing power have made it possible to break RSA encryption with enough time and resources. As a result, it is important to regularly update the key size to maintain its security.

## 5. Can RSA be used for all types of data?

RSA can be used to encrypt any type of data, as long as it can be represented as a number. This includes text, images, and other digital files. However, for larger files, it is more efficient to use a hybrid encryption system, where RSA is used to encrypt a symmetric key, which is then used to encrypt the actual data.

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