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# 数学代写|密码学代写Cryptography Theory代考|CMSC456 A Dozen Non-Factoring Attacks

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## 数学代写|密码学Cryptography Theory代考|A Dozen Non-Factoring Attacks

This attack was first demonstrated by Gus Simmons (Figure 15.1) in $1983 .{ }^1$
Imagine that a common message is sent to two individuals who share the same value for $n$, but use distinct values for $e$. Suppose Eve intercepts both enciphered messages:
$$C_1=M^{e_1}(\bmod n) \text { and } C_2=M^{c_2}(\bmod n) .$$
If $e_1$ and $e_2$ are relatively prime, she may then use the Euclidean algorithm to find integers $x$ and $y$ such that $x e_1+y e_2=1$. Exactly one of $x$ and $y$ must be negative. Assume it is $x$. Eve then calculates
$$\left(C_1^{-1}\right)^{-x} C_2^y=C_1^x C_2^y=\left(M^{e_1}\right)^x\left(M^{e_2}\right)^y=M^{x e_1+j e_2}=M^1=M(\bmod n) .$$
Thus, Eve, who hasn’t recovered $d$, can obtain $M$.

Imagine the malicious hacker Mallory controls Alice and Bob’s communication channel. When Alice requests Bob’s public key, Mallory changes the $e$ that Bob tries to send her by a single bit. Instead of $(e, n)$, Alice receives $\left(e^{\prime}, n\right)$. When Alice enciphers her message, Mallory lets it pass unchanged to Bob, who is unable to read it. After some confusion, Bob sends his public key to Alice again, since she clearly didn’t use the right values. Alice then sends the message again using $(e, n)$. Mallory may then use the attack described above to read $M .^2$

PATCH: Never resend the same message enciphered two different ways. If you must resend, alter the message first.

## 数学代写|密码学Cryptography Theory代考|Attack 2. Man-in-the-Middle

In the attack described above, where a hacker controls the communications, you may well ask why he doesn’t simply keep Bob’s public key and send Alice his own. When Alice encrypts a message, thinking Bob will get it, Mallory can read it using her own key and then re-encipher it with Bob’s key before passing it on. She can even make changes first, if she desires. Similarly, if Bob requests Alice’s key, Mallory can keep it and send Bob another key she has made for herself. In this manner, Mallory has complete control over the exchanges. For obvious reasons, this is known as a man-inthe-middle attack. Studying ways to prevent attacks like these falls under the “protocols” heading of cryptography. We do not pursue this line here, but the reader will find the subject treated nicely by Schneier.

Attack 3. Low Decryption Exponent
In 1990, Michael J. Wiener presented an attack for when the decryption exponent, $d$, is small. ${ }^4$ To be more precise, the attack applies when
$$q<p<2 q \text { and } d<\frac{\sqrt[4]{n}}{3}$$
In this case, $d$ may be computed efficiently. To see how this is done, ${ }^5$ we begin with $e d=1$ (mod $\varphi(n))$ and rewrite it as $e d-k \varphi(n)=1$ for some $k$ in the set of integers. We then divide both sides by $d \varphi(n)$ to get
$$\frac{e}{\varphi(n)}-\frac{k}{d}=\frac{1}{d \varphi(n)}$$
$\varphi(n) \approx n$, so we have
$$\frac{e}{\varphi(n)}-\frac{k}{d} \approx\left|\frac{e}{n}-\frac{k}{d}\right|=\left|\frac{e d-k n}{n d}\right|=\left|\frac{e d-k \varphi(n)-k n+k \varphi(n)}{n d}\right|$$
$\varphi(n)$ is actually a bit smaller than $n$, so we need to introduce absolute value signs following the $\approx$ to be sure the quantity remains positive.

## 数学代写|密码学Cryptography Theory代考|A Dozen Non-Factoring Attacks

$$C_1=M^{e_1}(\bmod n) \text { and } C_2=M^{c_2}(\bmod n)$$

$$\left(C_1^{-1}\right)^{-x} C_2^y=C_1^x C_2^y=\left(M^{e_1}\right)^x\left(M^{e_2}\right)^y=M^{x e_1+j e_2}=M^1=M(\bmod n) .$$

PATCH：永远不要重新发送以两种不同方式加密的同一条消息。如果您必须重新发送，请先更改消息。

## 数学代写|密码学Cryptography Theory代考|Attack 2. Man-in-theMiddle

1990 年，Michael J. Wiener 提出了解密指数为 $d$ ，是小。 ${ }^4$ 更准确地说，攻击适用于
$$q<p<2 q \text { and } d<\frac{\sqrt[4]{n}}{3}$$

$$\frac{e}{\varphi(n)}-\frac{k}{d}=\frac{1}{d \varphi(n)}$$
$\varphi(n) \approx n$, 所以我们有
$$\frac{e}{\varphi(n)}-\frac{k}{d} \approx\left|\frac{e}{n}-\frac{k}{d}\right|=\left|\frac{e d-k n}{n d}\right|=\left|\frac{e d-k \varphi(n)-k n+k \varphi(n)}{n d}\right|$$
$\varphi(n)$ 实际上比 $n$, 所以我们需要在后面引入绝对值符号 $\approx$ 以确保数量保持正数。

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