# 数学代写|密码学代写cryptography theory代考|CS388H

## 数学代写|密码学代写cryptography theory代考|What Is Asymmetric Cryptography?

Asymmetric cryptography, as the name suggests, is a form of cryptography wherein one key is used to encrypt a message, and a different (but related) key is used to decrypt. This concept often baffles those new to cryptography and students in network security courses. How can it be that a key used to encrypt will not also decrypt? This will be clearer to you once we examine a few algorithms and you see the actual mathematics involved. For now, set that issue to one side and simply accept that one key encrypts but cannot decrypt the message. Another key is used to decrypt.

The reason asymmetric cryptography is such a powerful concept is because symmetric cryptography (that you studied in Chaps. 6 and 7) has a serious problem. That problem is key exchange. Most cryptography books and papers prefer to use the fictitious characters Alice, Bob, and Eve to explain how asymmetric cryptography works, and I will continue that tradition. This stems from the original paper describing the RSA algorithm.

Let’s assume Alice would like to send Bob a message. But Alice is concerned that Eve might eavesdrop (thus her name!) on the communication. Now let us further assume that we don’t have asymmetric cryptography, that all you have available to you are the symmetric ciphers that you learned in Chaps. 6 and 7. And assume Bob and Alice do not live in the same location. How can they exchange a key so that they might encrypt messages? Any method you might think of has the very real chance of being compromised, short of a secure/trusted courier manually taking keys between the two parties. If a courier was needed to exchange keys every time secure communication was required, then we would not have online banking, e-commerce, or a host of other useful technologies.

With public key/asymmetric cryptography, Alice will get Bob’s public key and use that to encrypt the message she sends to Bob. Now should Eve intercept the message and have access to Bob’s public key, it is ok. That key won’t decrypt the message. Only Bob’s private key will, and this he safeguards. You can see this in Fig. 10.1.

If Bob wishes to respond to Alice, he reverses the process. He gets Alice’s public key and encrypts a message to her, which only her private key will decrypt. Thus, asymmetric cryptography solves the problem of key exchange. It does not impede security if literally every person on the planet has both Bob and Alice’s public keys. Those keys can only be used to encrypt messages to Bob and Alice (respectively) and cannot decrypt the messages. So as long as Bob and Alice keep their private keys secret, secure communication is achieved with no problems in key exchange.

This basic concept of one key being public and another being private is why asymmetric cryptography is often referred to as public key cryptography. Unfortunately, it is as far as many security courses go with explaining asymmetric cryptography. Of course, we will be delving into the actual algorithms.

## 数学代写|密码学代写cryptography theory代考|The Rabin Cryptosystem

This algorithm was created in 1979 by Michael Rabin. Michael Rabin is an Israeli cryptographer and a recipient of the Turing Award. The Rabin cryptosystem can be thought of as an RSA cryptosystem in which the value of e and d are fixed (Hinek 2009).
$$e=2 \text { and } d=1 / 2$$
The encryption is $\mathrm{C} \equiv \mathrm{P}^2(\bmod \mathrm{n})$ and the decryption is $\mathrm{P} \equiv \mathrm{C}^{1 / 2}(\bmod \mathrm{n})$.
Here is a very trivial example to show the idea.
Bob selects $p=23$ and $q=7$.
Bob calculates $n=p \times q=161$.
Bob announces $n$ publicly; he keeps $p$ and $q$ private.
Alice wants to send the plaintext message $M=24$. Note that 161 and 24 are relatively prime; 24 is in the group selected, $\mathrm{Z}_{161}$.
Encryption: $\mathrm{C}=24^2 \bmod 161=93$ and sends the ciphertext 93 to Bob.
This algorithm is not as widely used as RSA or Diffie Hellman but is presented to give you a general overview of alternative asymmetric algorithms.

# 密码学代考

## 数学代写|密码学代写cryptography theory代考|The Rabin Cryptosystem

Bob选择 $p=23$ 和 $q=7$

Alice想发送明文消息 $M=24$。注意，161和24是相对质数;24在被选中的组中， $\mathrm{Z}_{161}$.

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