## 数学代写|密码学作业代写Cryptography代考|CS171

2023年2月1日

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## 数学代写|密码学作业代写Cryptography代考|DEFINING SECURITY

The many issues involved in defining security notions for encryption have already been extensively discussed in Section 7.1.

The major difference between symmetric cryptography and public key cryptography is that there is no need for chosen plaintext queries in the security games, since public key encryption implies that the adversary must have the public encryption key, and can therefore encrypt ciphertexts without any

secret information. Obviously, the adversary must get the encryption key, so we simply start off the games by giving the encryption key to the adversary. Before we discuss security, we shall define public key cryptosystems including associated data. Just like for symmetric cryptosystems, including associated data extends the functionality of public key cryptosystems and makes it easier to design larger systems.

Definition 8.1. A public key encryption scheme PKE consists of three algo$\operatorname{rithms}(\mathcal{K}, \mathcal{E}, \mathcal{D})$

• The key generation algorithm $\mathcal{K}$ takes no input and outputs an encryption key ek and a decryption key $d k$. To each encryption key ek there is an associated message set $\mathfrak{M}{e k}$ and set of associated data $\mathfrak{F}{e k}$.
• The encryption algorithm $\mathcal{E}$ takes as input an encryption key, associated data and a message. It outputs a ciphertext.
• The decryption algorithm $\mathcal{D}$ takes as input a decryption key, associated data and a ciphertext and outputs either a message or the special symbol $\perp$ indicating decryption failure.

We require that for any key pair $(e k, d k)$ output by $\mathcal{K}$, any associated data $a d \in \mathfrak{F}{e k}$ and any message $m \in \mathfrak{M}{e k}$
$$\mathcal{D}(d k, a d, \mathcal{E}(e k, a d, m))=m .$$
While the concept does not matter much, it is convenient for bookkeeping reasons to define a value for a public key cryptosystem, namely the probability of getting a collision among a set of encryption keys, and the probability of getting a collision among a set of ciphertexts. This value must be small if our cryptosystem is to be secure. In most cases, it will be very small and easy to determine, so we shall not bother computing it for most cryptosystems.

## 数学代写|密码学作业代写Cryptography代考|A Single Challenge Suffices – Maybe

We said that sometimes security is defined for a single challenge query. We shall now prove that in some sense, it is sufficient to prove security for a single challenge query. However, this generic theorem is not tight in the sense that the advantage bound contains a factor $l_c$. Proving security for multiple challenges directly, without this non-tightness, would be better better. We begin with the generic result and illustrate later with two examples.

Proposition 8.5. Let $\mathcal{A}$ be a $\left(\tau, l_c, l_d\right)$-adversary against indistinguishability for PKE. Then there exists a $\left(\tau^{\prime}, 1, l_d\right)$-adversary $\mathcal{B}$ against indistinguishability for PKE, where $\tau^{\prime}$ is essentially $\tau$, such that
$$\operatorname{Adv}{\mathrm{PKE}}^{\mathrm{ind}}(\mathcal{A}) \leq l_c \mathbf{A d v}{\mathrm{PKE}}^{\mathrm{ind}}(\mathcal{B}) .$$
Exercise 8.4. Prove Proposition 8.5. Hint: Look at Proposition 7.5.
Example 8.2. Propositions $8.4$ and $8.5$ say that any $\left(\tau, l_c, 0\right)$-adversary $\mathcal{A}$ against real-or-random security for ElGamal can be turned into a $\tau^{\prime}$-adversary $\mathcal{B}$ against DDH, where $\tau^{\prime}$ is essentially equal to $\tau$, and
$$\operatorname{Adv}_{\text {ELGGAMAL }}^{\text {ror-cpa }}(\mathcal{A}) \leq l_c \operatorname{Adv}_G^{\mathrm{DDH}}(\mathcal{B}) .$$

Example 8.3. Consider ElGamal encryption as in Example 8.1. Observe that if $\left(x_1, w_1\right)$ and $\left(x_2, w_2\right)$ decrypt to $m_1$ and $m_2$, respectively, then $\left(x_1 x_2, w_1 w_2\right)$ decrypts to $m_1 m_2$, and $\left(x_2^r, w_2^r\right)$ decrypts to $m_1^r$.

Next, consider a tuple $(x, y, z) \in G^3$. If this is a DDH tuple, then with $y$ as the ElGamal encryption key, both $(g, y)$ and $(x, z)$ are encryptions of 1 . Then for $r, t$ sampled from the uniform distribution on ${0,1, \ldots, p-1}$ we have that
$$\left(g^r x^t, y^r z^t\right)$$
is an encryption of 1 , distributed identically to the output of $\mathcal{E}$.

# 密码学代写

## 数学代写|密码学作业代写Cryptography代考|DEFINING SECURITY

• 密钥生成算法 $\mathcal{K}$ 不接受输入并输出加密密钥 ek 和解密密钥 $d k$. 对于每个加密密钥 ek 都有一个关联的消息集 M $2 e k$ 和一组相关 数据 Fek.
• 加密算法 $\mathcal{E}$ 将加密密钥、相关数据和消息作为输入。它输出一 个密文。
• 解密算法 $\mathcal{D}$ 将解密密钥、相关数据和密文作为输入，并输出消
我们要求任何密钥对 $(e k, d k)$ 输出方式 $\mathcal{K}$ ，任何相关数据 $a d \in \mathfrak{F} e k$ 和 任何消息 $m \in \mathfrak{M} e k$
$$\mathcal{D}(d k, a d, \mathcal{E}(e k, a d, m))=m .$$
虽然这个概念并不重要，但出于簿记的原因，为公钥密码系统定义个值很方便，即一组加密密钥之间发生冲突的概率，以及一组密文之 间发生冲突的概率. 如果我们的密码系统是安全的，这个值必须很小。 在大多数情况下，它会非常小并且很容易确定，因此我们不会为大多 数密码系统费心计算它。

## 数学代写|密码学作业代写Cryptography代考|A Single Challenge Suffices – Maybe

$$\operatorname{Adv} \operatorname{PKE}^{\text {ind }}(\mathcal{A}) \leq l_c \mathbf{A d v P K E}{ }^{\text {ind }}(\mathcal{B})$$

$$\operatorname{Adv}_{\text {ELGGAMAL }}^{\text {ror-cpa }}(\mathcal{A}) \leq l_c \operatorname{Adv}_G^{\mathrm{DDH}}(\mathcal{B}) .$$

$$\left(g^r x^t, y^r z^t\right)$$

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## MATLAB代写

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