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# 数学代写|密码学代写Cryptography Theory代考|Security of RSA

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## 数学代写|密码学Cryptography Theory代考|Security of RSA

There are two obvious ways of trying to break RSA. Indeed, these apply to any public-key cryptosystem. An attacker can either attempt to:

Decrypt a ciphertext without knowledge of the private key; or

Determine the private key directly from the public key.
Clearly the second attack is more powerful than the first, since an attacker who can perform the second attack can then decrypt subsequent ciphertexts. We now consider these two attack strategies.
DECRYPTING ACIPHERTEXT WITHOUT KNOWLEDGE OF THE PRIVATE KEY
Consider trying to decrypt an RSA ciphertext without the private key. Recall that we specified in Section 5.1.4 that a public-key encryption function should be a trapdoor one-way function. By assuming we do not know the private key, we are thus assuming we do not know the trapdoor. Thus, to assess the difficulty of determining the plaintext directly from the ciphertext, we need to assess the effectiveness of the one-way function which lies at the heart of RSA.

We need to take a closer look at the function being used for RSA encryption. The encryption process in RSA involves computing the function:
$$C=P^e \bmod n .$$
An attacker who observes $C$, and has knowledge of $e$ and $n$ (but not $d$ ), needs to work out what the value $P$ is. Computing $P$ from $C, e$, and $n$ is regarded as a hard problem (fortunately!), and thus the encryption function of RSA is believed to be a one-way function.

## 数学代写|密码学Cryptography Theory代考|RSA in practice

As with most of the cryptographic primitives we discuss, our explanation of RSA has been simplified in order to emphasise the main design aspects. It is essential that RSA is not deployed in any real implementation in exactly the way we have described. Rather, the latest best-practice guidelines outlined in the relevant standards should be consulted and followed. Perhaps the most critical alteration to the ‘textbook’ version of RSA which is often made in practice is to introduce randomisation into the encryption process. We now look at why this is important.
PROBABILISTIC ENCRYPTION
The version of RSA we presented in Section 5.2.2 is an example of deterministic encryption, which means that each time the same plaintext is encrypted using the same public key, the resulting ciphertext will be the same.

A significant disadvantage of deterministic public-key encryption is that the following attack is possible. Suppose a ciphertext sent to a known recipient has been observed by an attacker, who then proceeds as follows:

1. The attacker makes an informed guess as to the value of the plaintext;
2. The attacker encrypts the guessed plaintext using the known recipient’s public key; and
3. If the result matches the observed ciphertext, then the guess was correct; if not, the attacker tries another guess of the plaintext.

This attack is particulary effective in situations where there are limited choices for the plaintext (for example, if the plaintext is a database entry from a limited range). We will refer to this attack as an informed exhaustive plaintext search.

Note this attack does not apply to symmetric encryption. This is because the encryption key is secret. Even if the attacker knows that the plaintext comes from a small set of potential values (perhaps even just two), the attacker cannot conduct this attack because any encryption key could have been used. This is why the attacker has to exhaustively search through all the potential symmetric keys instead.

## 数学代写|密码学Cryptography Theory代考|Security of RSA

$$C=P^e \bmod n .$$

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