数学代写|编码理论代写Coding theory代考|ELEC5507

2022年7月2日

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数学代写|编码理论代写Coding theory代考|A GENERAL ALGORITHM

We present here an algorithm for factoring a given polynomial, $f(x)=$ $\sum_{k=0}^{m} f_{k} x^{k}, f_{k} \in \mathrm{GF}(q)$, into powers of irreducible polynomials. If $q$ is small, this algorithm proves effective even if $m$ is quite large.

1. We construct the $m \times m$ matrix $Q$ over $\mathrm{GF}(q)$, whose $i$ th row represents $x^{q(i-1)}$ reduced modulo $f(x)$. Specifically,
$$x^{q i} \equiv \sum_{k=0}^{m-1} Q_{i+1, k+1} x^{k} \bmod f(x)$$
The $Q$ matrix may be computed with a shift register wired to multiply by $x \bmod f(x)$. The register is started at 1 , which is the first row of $Q$. After $q$ shifts, it contains the second row of $Q, \ldots$, etc. After $q(m-1)$ shifts, it contains the last row of $Q$.

Given any polynomial $g(x)$ of degree $<m$ over GF $(q), g(x)=$ $\sum_{i=0}^{m-1} g_{i} x^{i}$, we may compute the residue of $[g(x)]^{\alpha} \bmod f(x)$ by multi-plying the row vector $\left[g_{0}, g_{1}, \ldots, g_{m-1}\right]$ by the $Q$ matrix. This follows from the observation that
\begin{aligned} {[g(x)]^{q}=g\left(x^{q}\right)=\sum_{i=0}^{m-1} g_{i} x^{q i} \equiv \sum_{i=0}^{m-1}\left(\sum_{k=0}^{m-1} g_{i} \Theta_{i+1, k+1} x^{k}\right) } \ &=\sum_{k=0}^{m-1}\left(\sum_{i=0}^{m-1} g_{i} \Theta_{i+1, k+1}\right) x^{k} \end{aligned}
Similarly, we could compute $[g(x)]^{\alpha}-g(x) \bmod f(x)$ by multiplying the row vector $\left[g_{0}, g_{1}, \ldots, g_{m-1}\right]$ by the matrix $Q-g$, where $g$ is the $m \times m$ identity matrix over GF $(q)$.

1. We find a set of row vectors which span the null space of $Q-g$. This may be done by appropriate column operations on the matrix $Q-g$, as described in Secs. $2.5$ and 2.6. Each such row vector in the null space of $Q-g$ represents a polynomial $g(x)$ which satisfies the equation $[g(x)]^{a}-g(x) \equiv 0 \bmod f(x)$, and, conversely, each $g(x)$ which satisfies this equation is represented by a row vector in the null space of $Q-g$.
2. We select any of the polynomials $g(x)$ found in step 2 and apply Euclid’s algorithm to determine the greatest common divisor of $f(x)$ and $g(x)-s$ for each $s \in \mathrm{GF}(q) . \dagger \quad$ We then have the factorization of Theorem 6.11.

数学代写|编码理论代写Coding theory代考|DETERMINING THE PERIOD OF A POLYNOMIAL

Given a polynomial $f(x)$, we are often interested in determining its period, i.e., the smallest $j$ for which $f(x)$ divides $x^{j}-1$. If $f(x)$ is an irreducible polynomial of degree $m$ over GF $(q)$, then the period of $f(x)$ is equal to the multiplicative order of its roots, which is a factor of $q^{m}-1$.

How is the period of a factorable polynomial related to the periods of its irreducible factors? Let us first suppose that $f(x)$ is an irreducible polynomial of period $n$ and try to find the period of a power of $f(x)$, $[f(x)]^{m}$. Since $n$ is a factor of $q^{\text {deg } f}-1$, it is a nonmultiple of $p$, the characteristic of $\operatorname{GF}(q)$. Since the period of $f(x)$ is $n, f(x)$ divides $x^{k}-1$ iff $k$ is a multiple of $n$. If $k=p^{i} j$, where $j$ is a nonmultiple of $p$, then $x^{k}-1=\left(x^{j}-1\right)^{p^{i}}$. The polynomial $x^{j}-1$ has no repeated roots, because it divides $x^{q^{m-1}}-1$ for that $m$ which is the multiplicative order of the integer $q$ mod $j$. Thus, every irreducible factor of $x^{k}-1$ has multiplicity $p^{i}$. We conclude that if the irreducible polynomial $f(x)$ has period $n$, then the period of $[f(x)]^{m}$ is $n$ times the least power of $p \geq m$.

Next suppose that $f(x)$ is the product of several irreducible factors $f^{(1)}(x), f^{(2)}(x), \ldots$ which have multiplicities $m_{1}, m_{2}, \ldots$ Since $x^{k}-1$ divides $x^{N}-1$ iff $k$ divides $N$, we conclude that the period of $f(x)$ is the least common multiple of the periods of $\left[f^{(1)}(x)\right]^{m_{1}},\left[f^{(2)}(x)\right]^{m_{2}}, \ldots . .$ This, in turn, is the laast common multiple of the periods of the $f^{(i)}(x)$ times the least power of $p$ which is greater than or equal to all the $m_{i}$.

数学代写|编码理论代写Coding theory代考|TRINOMIALS OVER

The methods of Secs. $6.1$ and $6.2$ may be easily programmed. Using a modern computer, it is quite feasible to factor any binary polynomial of degree up to 1,000 .
† Using special techniques introduced by Lucas (1878), it is relatively easy for moderate values of $m$ to determine whether or not the number $2^{m}-1$ is prime. If $m$ is prime, this number is called a Mersenne number; if $2^{m}-1$ is prime, it is called a Mersenne prime. Unfortunately, the Lucas procedure often shows that a Mersenne number is composite without giving any information about the factors. In other čases, however, the proceduré has ennabléd Gillies (1064) to prove that cêrtain enormous numbers, including $2^{11,213}-1$, are prime.

Among the most interesting binary polynomials are the trinomials, $x^{n}+x^{k}+1$. If $\alpha$ is a root in $\mathrm{GF}\left(2^{n}\right)$ of an irreducible binary trinomial of degree $n$, then the first $n$ powers of $\alpha$ provide a particularly attractive basis for representing $\mathrm{GF}\left(2^{n}\right)$, since it is then possible to multiply by $\alpha$ using an $n$-bit feedback shift register whose connections require only one two-input adder (see Sec. 2.41). In coding theory, most practical interest centers on fields of order $2^{12}$ or smaller. From this viewpoint, the interest in $\mathrm{GF}\left(2^{n}\right)$ for $n>12$ is largely academic. However, there are certain other applications of feedback shift registers, such as the generation of sequences of pseudorandom bits, where there is practical interest in finding primitive binary trinomials of as large degree as possible.

数学代写|编码理论代写Coding theory代考|A GENERAL ALGORITHM

1. 我们构建 $m \times m$ 矩阵 $Q$ 超过 $\mathrm{GF}(q)$ ，谁的 $i$ 第行代表 $x^{q(i-1)}$ 减少模 $f(x)$. 具体来 说，
$$x^{q i} \equiv \sum_{k=0}^{m-1} Q_{i+1, k+1} x^{k} \bmod f(x)$$
这 $Q$ 矩阵可以用连接到乘以的移位寄存器计算 $x \bmod f(x)$. 寄存器从 1 开始，即 第一行 $Q$. 后 $q$ 班次，它包含第二行 $Q, \ldots$ 等之后 $q(m-1)$ 班次，它包含最后一行 $Q .$
给定任何多项式 $g(x)$ 学位 $<m$ 过GF $(q), g(x)=\sum_{i=0}^{m-1} g_{i} x^{i}$ ，我们可以计算 $[g(x)]^{\alpha} \bmod f(x)$ 通过乘以行向量 $\left[g_{0}, g_{1}, \ldots, g_{m-1}\right]$ 由 $Q$ 矩阵。这是从以下观崇得出 的:
$$[g(x)]^{q}=g\left(x^{q}\right)=\sum_{i=0}^{m-1} g_{i} x^{q i} \equiv \sum_{i=0}^{m-1}\left(\sum_{k=0}^{m-1} g_{i} \Theta_{i+1, k+1} x^{k}\right)=\sum_{k=0}^{m-1}\left(\sum_{i=0}^{m-1} g_{i} \Theta_{i+1, k+1}\right) x^{k}$$
同样，我们可以计算 $[g(x)]^{\alpha}-g(x) \bmod f(x)$ 通过乘以行向量 $\left[g_{0}, g_{1}, \ldots, g_{m-1}\right]$ 由矩 阵 $Q-g$ ，在哪里 $g$ 是个 $m \times m \mathrm{GF}$ 上的单位矩阵 $(q)$.
2. 我们找到一组跨越雾空间的行向量 $Q-g$. 这可以通过对矩阵进行适当的列操作来完 成 $Q-g$ ，如 Secs 中所述。 $2.5$ 和 2.6。霎空间中的每个这样的行向量 $Q-g$ 表示 多项式 $g(x)$ 满足方程 $[g(x)]^{a}-g(x) \equiv 0 \bmod f(x)$ ，并且相反，每个 $g(x)$ 满足 这个方程的由零空间中的行向量表示 $Q-g$.
3. 我们选择任何多项式 $g(x)$ 在步骤 2 中找到并应用欧几里得算法来确定最大公约数 $f(x)$ 和 $g(x)-s$ 对于每个 $s \in \mathrm{GF}(q)$. $\dagger$ 然后我们得到定理 $6.11$ 的因式分解。

数学代写|编码理论代写Coding theory代考|TRINOMIALS OVER

†使用 Lucas (1878) 弓入的特殊技术，对于中等值相对容易 $m$ 判断昊码是否 $2^{m}-1$ 是溸 数。如果 $m$ 是牡数，这个数称为梅森数；如果 $2^{m}-1 \mathrm{~ 是 㲅 数 ， 称 为 梅 森 拝}$ 卢卡斯过程经常表明梅棌数是合数，而没有提供任何有关因子的信息。然而，在其他情况 下，程序使 Gillies (1064) 能够证明某些巨大的数字，包括 $2^{11,213}-1$ ，是嗉数。

有限元方法代写

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MATLAB代写

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