CS6260 Cryptography课程简介

Cryptographic schemes are used to protect sensitive information and to ensure the integrity and authenticity of communications. Here are some of the most commonly used cryptographic schemes and their applications:

1. AES (Advanced Encryption Standard) – AES is a symmetric encryption algorithm used to protect data in transit or at rest. It is widely used in applications such as file encryption, email encryption, and VPNs. AES works by transforming plaintext into ciphertext using a secret key, which is used to encrypt and decrypt the data.
2. CBC (Cipher Block Chaining) – CBC is a mode of operation for block ciphers, such as AES. CBC works by breaking the plaintext into blocks and encrypting each block using the previous block’s ciphertext as an input. This ensures that the same plaintext block does not always result in the same ciphertext block, making the encryption more secure.

PREREQUISITES

No previous knowledge of cryptography is necessary. This course is about applying theory to practical problems, but it is still a theory course. The main requirement is basic “mathematical maturity”. You have to be able to read and write mathematical definitions, statements and proofs.

It is expected that you were successful in your undergraduate discrete math class and took basic algorithms and computability/complexity theory classes. In particular, you have to know how to measure the running time of an algorithm and how to do proofs by contradiction and contraposition. You also have to know the basics of probability theory and modular arithmetic. You should also have familiarity with Python for the coding portions of the course.

If you cannot recall what terms like permutation, sample space, random variable, conditional probability, big-O notation mean, you should consider taking the course in a later semester and refresh your knowledge of the above topics in the meanwhile. I recommend you review an undergraduate textbook on discrete math.
All necessary elements of number theory will be presented during the course.

CS6260 Cryptography HELP（EXAM HELP， ONLINE TUTOR）

Exercise 5.1. Let $\omega=\exp (2 \pi i / N) \in \mathbb{C}$, which is a primitive $N$ th root of unity. Let $\mathbf{V}=\left(v_{k j}\right)$ be the Vandermonde matrix with $v_{k j}=\omega^{k j}, 0 \leq k, j<N$.
(a) Show that the $\overline{\omega^j}=\omega^{N-j}$, where $\bar{z}$ denotes complex conjugation.
(b) Show that $\sum_{j=0}^{N-1}\left(\omega^k\right)^j=0$ for any $k \not 00(\bmod N)$.
(c) Show that $\mathbf{V} \mathbf{V}^=N I$, where $\mathbf{V}^$ denotes the conjugate transpose of $\mathbf{V}$.
(d) Show that when $\beta$ is the discrete Fourier transform of $\alpha$, then $\beta=$ $\frac{1}{\sqrt{N}} \mathbf{V} \boldsymbol{\alpha}$.

(a) We have $\overline{\omega^j}=\overline{\cos(2\pi j/N)+i\sin(2\pi j/N)}=\cos(2\pi j/N)-i\sin(2\pi j/N)=\omega^{N-j}$, where we used the fact that $\cos$ is an even function and $\sin$ is an odd function.

(b) When $k$ is not divisible by $N$, we have $\sum_{j=0}^{N-1}\left(\omega^k\right)^j=\frac{1-\left(\omega^k\right)^N}{1-\omega^k}=0$, where we used the fact that $\omega^N=1$ and $\omega^k\neq 1$ because $k$ is not divisible by $N$.

(c) We have $(\mathbf{V}\mathbf{V}^*){k,l}=\sum{j=0}^{N-1}v_{k,j}\overline{v_{l,j}}=\sum_{j=0}^{N-1}\omega^{kj}\omega^{-lj}=\sum_{j=0}^{N-1}\omega^{j(k-l)}$. When $k=l$, this is equal to $N$, because every term is equal to $1$. When $k\neq l$, we have $\sum_{j=0}^{N-1}\omega^{j(k-l)}=\frac{1-\omega^{N(k-l)}}{1-\omega^{k-l}}=0$, because $k-l$ is not divisible by $N$.

(d) Let $\beta$ be the discrete Fourier transform of $\alpha$, i.e., $\beta_k=\sum_{j=0}^{N-1}\alpha_j\omega^{-kj}$. Then, we have \begin{align*} \frac{1}{\sqrt{N}}\mathbf{V}\boldsymbol{\alpha}k &= \frac{1}{\sqrt{N}}\sum{j=0}^{N-1}\omega^{kj}\alpha_j \ &= \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}\omega^{-jk}\alpha_j \ &= \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}\beta_j\omega^{jk} \ &= \beta_k, \end{align*} where we used the fact that $\omega^{-k}=\omega^{N-k}$ in the second step, and the fact that $\mathbf{V}$ is unitary in the third step. Therefore, we have $\beta=\frac{1}{\sqrt{N}}\mathbf{V}\boldsymbol{\alpha}$, as claimed.

Exercise 5.2. Let $\mathrm{U}$ be an $N \times N$ unitary matrix. Show that for any vector $\boldsymbol{\alpha} \in \mathbb{C}^N,|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$

We want to study the Fourier coefficients of a very special complex vector. Let $t_0$ be an integer such that $0 \leq t_0<r$, and let $m$ be minimal such that $m r+t_0 \geq N$. Let $\alpha \in \mathbb{C}^N$ be given by
$$\alpha_k= \begin{cases}\frac{1}{\sqrt{m}} & k=t_0+j r, \text { and } \ 0 & \text { otherwise. }\end{cases}$$
Note that $|\boldsymbol{\alpha}|=1$ and that $1-m r / N=1-\left(N-t_0\right) / N \leq r / N \leq 1 / r$.

Let $\mathbf{U}$ be an $N \times N$ unitary matrix and $\boldsymbol{\alpha} \in \mathbb{C}^N$. We want to show that $|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$.

Let $\boldsymbol{\beta}=\mathbf{U} \boldsymbol{\alpha}$, then we have: \begin{align*} |\boldsymbol{\beta}|^2 &=\boldsymbol{\beta}^* \boldsymbol{\beta} \ &=\left(\mathbf{U} \boldsymbol{\alpha}\right)^\left(\mathbf{U} \boldsymbol{\alpha}\right) \ &=\boldsymbol{\alpha}^ \mathbf{U}^* \mathbf{U} \boldsymbol{\alpha} \ &=\boldsymbol{\alpha}^* \boldsymbol{\alpha} && (\because \mathbf{U} \text{ is unitary, so } \mathbf{U}^* \mathbf{U} = \mathbf{I}) \ &=|\boldsymbol{\alpha}|^2. \end{align*}

Therefore, we have shown that $|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供gatech.edu CS6260 Cryptography密码学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

CS6260 Cryptography课程简介

Cryptographic schemes are used to protect sensitive information and to ensure the integrity and authenticity of communications. Here are some of the most commonly used cryptographic schemes and their applications:

1. AES (Advanced Encryption Standard) – AES is a symmetric encryption algorithm used to protect data in transit or at rest. It is widely used in applications such as file encryption, email encryption, and VPNs. AES works by transforming plaintext into ciphertext using a secret key, which is used to encrypt and decrypt the data.
2. CBC (Cipher Block Chaining) – CBC is a mode of operation for block ciphers, such as AES. CBC works by breaking the plaintext into blocks and encrypting each block using the previous block’s ciphertext as an input. This ensures that the same plaintext block does not always result in the same ciphertext block, making the encryption more secure.

PREREQUISITES

No previous knowledge of cryptography is necessary. This course is about applying theory to practical problems, but it is still a theory course. The main requirement is basic “mathematical maturity”. You have to be able to read and write mathematical definitions, statements and proofs.

It is expected that you were successful in your undergraduate discrete math class and took basic algorithms and computability/complexity theory classes. In particular, you have to know how to measure the running time of an algorithm and how to do proofs by contradiction and contraposition. You also have to know the basics of probability theory and modular arithmetic. You should also have familiarity with Python for the coding portions of the course.

If you cannot recall what terms like permutation, sample space, random variable, conditional probability, big-O notation mean, you should consider taking the course in a later semester and refresh your knowledge of the above topics in the meanwhile. I recommend you review an undergraduate textbook on discrete math.
All necessary elements of number theory will be presented during the course.

CS6260 Cryptography HELP（EXAM HELP， ONLINE TUTOR）

Problem 1. A supersingular curve and its embedding degree
Consider the elliptic curve $E$ defined by
$$E: y^2+y=x^3 \quad \text { over } \mathbb{F}2$$ (1) Prove that $$\left|E\left(\mathbb{F}{2^n}\right)\right|= \begin{cases}2^n+1 & \text { if } n \text { is odd, } \ 2^n+1-2(-2)^{n / 2} & \text { if } n \text { is even }\end{cases}$$
Conclude that $E$ is supersingular.
(2) For any elliptic curve $E$ over a field $\mathbb{F}q$ with group order $N=\left|E\left(\mathbb{F}_q\right)\right|$, the embedding degree $^1$ is the smallest integer $d$ such that $\mu_N \subseteq \mathbb{F}{q^d}^$, i.e. $\mathbb{F}{q^d}$ contains the $N$-th roots of unity. Equivalently, $d$ is the order of $q$ in the multiplicative group $(\mathbb{Z} / N \mathbb{Z})^$. Fix $n$ and consider the elliptic curve $(*)$ over $\mathbb{F}{2^n}$. Determine the embedding degree for $E$ over $\mathbb{F}_{2^n}$. (Note: it is dependent on the parity of $n$ ).

(1) To prove the cardinality of $E(\mathbb{F}_{2^n})$, we can use Hasse’s theorem which states that for an elliptic curve $E$ over a finite field $\mathbb{F}_q$, the cardinality of $E(\mathbb{F}_q)$ is given by $|N – q – 1| \leq 2\sqrt{q}$, where $N$ is the order of the group $E(\overline{\mathbb{F}_q})$.

In this case, $q=2^n$ and the equation of the curve $E$ is $y^2+y=x^3$ over $\mathbb{F}2$. To find the order $N$ of $E(\overline{\mathbb{F}{2^n}})$, we can use the Frobenius endomorphism, which maps a point $(x,y)$ on $E$ to its $2^n$-th power $(x^{2^n}, y^{2^n})$. It can be shown that the eigenvalues of the Frobenius endomorphism on $E$ are $\pm 2^{n/2}$, which means that $N=2^n+1-2(-2)^{n/2}$ if $n$ is even, and $N=2^n+1$ if $n$ is odd.

To show that $E$ is supersingular, we need to show that the characteristic polynomial of the Frobenius endomorphism has no roots in $\mathbb{F}_{2^n}$. This is equivalent to showing that the trace of the Frobenius endomorphism is equal to zero. The trace of the Frobenius endomorphism is given by $t = N – q$, which is equal to $1$ if $n$ is odd, and $-2(-2)^{n/2}$ if $n$ is even. Therefore, $E$ is supersingular if and only if $n$ is even.

(2) The embedding degree of $E$ over $\mathbb{F}_{2^n}$ is given by the order of $2^n$ in the multiplicative group $(\mathbb{Z}/N\mathbb{Z})^*$. Since $N=2^n+1$ if $n$ is odd, and $N=2^n+1-2(-2)^{n/2}$ if $n$ is even, we need to compute the order of $2^n$ modulo $N$.

If $n$ is odd, then $2^n$ has order $2^{n+1}$ modulo $N$. This can be shown using the fact that $2^{n+1} \equiv -1 \pmod{N}$ and the Chinese Remainder Theorem. Therefore, the embedding degree of $E$ over $\mathbb{F}_{2^n}$ is $2$.

If $n$ is even, then $2^n$ has order $2^{n-1}$ modulo $N$. This can be shown using the fact that $2^{n} \equiv -1 \pmod{N}$ and the Chinese Remainder Theorem. Therefore, the embedding degree of $E$ over $\mathbb{F}_{2^n}$ is $4$.

Problem 2. Anomalous Curves We say that an elliptic curve $E$ over $\mathbb{F}_q$ is anomalous over $\mathbb{F}_{q^n}$ if $\left|E\left(\mathbb{F}_{q^n}\right)\right|=q^n$. (1) Consider a prime power $q \geq 7$. Show that if an elliptic curve $E$ over $\mathbb{F}_q$ contains a point of order $q$, then it is anomalous over $\mathbb{F}_q$. (2) Show that if $E$ is anomalous over $\mathbb{F}_q$, then $E$ is not anomalous over $\mathbb{F}_{q^2}$. (3) Show that if $E$ is anomalous over $\mathbb{F}_2$, then $E$ is anomalous over $\mathbb{F}_{16}$.

(1) Suppose $E(\mathbb{F}_q)$ contains a point $P$ of order $q$. Then, by Lagrange’s theorem, the order of the group $E(\mathbb{F}_q)$ must be a multiple of $q$. Let $k$ be the smallest positive integer such that $kq \geq #E(\mathbb{F}_q)$. Then we have $$q^2 \leq #E(\mathbb{F}_q^2) = q(q+1) + kq$$ where the inequality follows from the Hasse-Weil bound. Rearranging, we get $#E(\mathbb{F}_q^2) – q^2 \geq kq \geq q$, so $#E(\mathbb{F}_q^2) \geq q^2 + q$.

On the other hand, the Frobenius endomorphism of $E$ over $\mathbb{F}_q$ satisfies $\phi_q^2 – q \phi_q + 1 = 0$ by the Weil conjectures. Therefore, the number of $\mathbb{F}_q$-rational points on $E$ is $#E(\mathbb{F}q) = q+1-\phi_q$, and the number of $\mathbb{F}q^2$-rational points is $$#E(\mathbb{F}q^2) = q^2+1-\phi{q^2} = q^2+1-q\phi_q+q$$ using the same formula for $\phi{q^2}$ and the fact that $\phi{q^2} = \phi_q^2 – q\phi_q + 1$. Since $#E(\mathbb{F}_q^2) \leq q^2 + q$ by the Hasse-Weil bound, we conclude that equality must hold, so $E$ is anomalous over $\mathbb{F}_q$.

(2) Suppose $E$ is anomalous over $\mathbb{F}q$, so $#E(\mathbb{F}q) = q$. Then, by the same argument as above, we have $#E(\mathbb{F}{q^2}) = q^2+1-q\phi_q+q$ and $\phi{q^2} = \phi_q^2 – q\phi_q + 1$. We want to show that $#E(\mathbb{F}_{q^2}) \neq q^2$.

If $#E(\mathbb{F}{q^2}) = q^2$, then we must have $\phi{q^2} = q$, which implies $\phi_q = 1$ by the quadratic formula. But this means that $E$ has a $\mathbb{F}q$-rational point of order $q$, which by part (1) implies that $E$ is anomalous over $\mathbb{F}q$, a contradiction. Therefore, $#E(\mathbb{F}{q^2}) \neq q^2$, so $E$ is not anomalous over $\mathbb{F}{q^2}$.

(3) The elliptic curve $E$ over $\mathbb{F}_2$ defined by $y^2 + y = x^3$ has $#E(\mathbb{F}_2) = 5$. Therefore, $E$ is anomalous over $\mathbb{F}_2$. By part (2),

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供gatech.edu CS6260 Cryptography密码学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。