# 数学代写|密码学代写cryptography theory代考|CISS3341

## 数学代写|密码学代写cryptography theory代考|The Revolution: A Number Field Sieve

Pollard [474] invented the number field sieve (NFS) hy suggesting raising the degree of the polynomial in the QS, although only for numbers with special form. He factored the Fermat number $F_7=2^{2^7}+1$ (which had been factored earlier by Morrison and Brillhart with CFRAC) using the cubic polynomial $2 x^3+2$ on a small computer. Manasse and the Lenstra brothers [381] soon extended Pollard’s ideas to higher-degree polynomials, but still only for numbers of the form $r^e-s$, for small integers $r$ and $|s|$. Their goal was to factor $F_9$, the smallest Fermat number with no known prime factor. They hoped to use the special form of $F_9=2^{512}-(-1)$ to make the numbers that had to be smooth smaller than those needed for the QS. After they factored $F_9$ in 1990 , they and others extended the NFS to general numbers. See Lenstra and Lenstra [360] for more details of the early history of this factoring algorithm. See Pomerance [479] for a summary of the algorithm. Crandall and Pomerance [144] describe the modern algorithm.

Recall that the QS produces many relations $x_i^2 \equiv q_i(\bmod n)$ with $q_i$ factored. After we have enough relations, we match the prime factors of the $q_i$ and create a subset of the $q_i$ whose product is square. In this way, we find congruences $x^2 \equiv y^2(\bmod n)$ which may factor $n$ by Theorem $3.7$.

Now drop the requirement that the left side of a relation must be square. Instead seek relations $r_i \equiv q_i(\bmod n)$ in which both $r_i$ and $q_i$ have been factored completely. Use linear algebra to match the prime factors of $r_i$ and the prime factors of $q_i$ and select a subset of the relations for which both the product of the $r_i$ and the product of the $q_i$ are square. This is a fine idea, but too slow to be practical. The main difficulty is that at least one of $\left|r_i\right|,\left|q_i\right|$ must exceed $n / 2$, so it has little chance of being smooth.

The NFS solves this problem by letting the numbers on one side of each relation be algebraic integers from an algebraic number field. The idea is to match the irreducible factors so that each occurs an even number of times and hope the product of the algebraic integers in the selected subset of the relations might be a square in the algebraic number field.

## 数学代写|密码学代写cryptography theory代考|Number Fields

See the books $[172,272,284,288]$ for more about number fields. An algebraic number is the zero of a polynomial with integer coefficients. If the polynomial is monic, then the algebraic number is called an algebraic integer. An algebraic number field is a field that contains only algebraic numbers. The smallest algebraic number field containing the algebraic number $\alpha$ is written $\mathbb{Q}(\alpha)$.

The set of all algebraic integers in $\mathbb{Q}(\alpha)$ is written $\mathbb{Z}(\alpha)$. This set forms a commutative ring with unity. A unit in $\mathbb{Z}(\alpha)$ is an element having a multiplicative inverse in $\mathbb{Z}(\alpha)$. A non-zero, non-unit element $\gamma$ of $\mathbb{Z}(\alpha)$ is irreducible if it can be factored in $\mathbb{Z}(\alpha)$ only as $\gamma=u \beta$ where $u$ is a unit. When $\gamma=u \beta$, where $u$ is a unit, $\beta$ is called an associate of $\gamma$ (and $\gamma$ is an associate of $\beta$ ). An algebraic integer $\gamma$ has unique factorisation (in $\mathbb{Z}(\alpha)$ ) if any two factorisations of $\gamma$ into the product of irreducible elements and units are the same except for replacing irreducibles by their associates and using different units.

The polynomial of lowest degree having an algebraic number $\alpha$ as a zero must be irreducible, that is, it does not factor into the product of two polynomials of lower degree. If an algebraic number $\alpha$ is a zero of the irreducible polynomial $f(x) \in \mathbb{Z}[x]$, then the conjugates of $\alpha$ are all of the zeros of $f(x)$. The norm $\mathcal{N}(\alpha)$ of $\alpha$ is the product of all of the conjugates of $\alpha$ including $\alpha$. The norm of an algebraic integer is a rational integer. The norm function is multiplicative: $\mathcal{N}(\alpha \beta)=\mathcal{N}(\alpha) \mathcal{N}(\beta)$. Thus if $\beta=\gamma^2$ for some $\gamma \in \mathbb{Z}(\alpha)$, then $\mathcal{N}(\beta)$ is the square of the integer $\mathcal{N}(\gamma)$. If the algebraic integer $\alpha$ is a zero of the irreducible polynomial $f(x)=x^d+c_{d-1} x^{d-1}+\cdots+c_1 x+c_0$ and $a$ and $b$ are integers, then the norm of $a-b \alpha$ is $\mathcal{N}(a-b \alpha)=F(a, b)$, where $F$ is the homogeneous polynomial
$$F(x, y)=x^d+c_{d-1} x^{d-1} y+\cdots+c_1 x y^{d-1}+c_0 y^d=y^d f(x / y) .$$

# 密码学代考

## 数学代写|密码学代写cryptography theory代考|The Revolution: A Number Field Sieve

Pollard [474] 发明了数域笑法 (NFS)，建议提高 QS 中多项式的次数，尽管仅适用于具有特殊形式的数 字。他分解了费马数 $F_7=2^{2^7}+1$ (Morrison 和 Brillhart 早先用 CFRAC 分解了它) 使用三次多项式 $2 x^3+2$ 在小型计算机上。Manasse 和 Lenstra 兄弟 [381] 很快将 Pollard 的想法扩展到高阶多项式，但 仍然只适用于以下形式的数字 $r^e-s$, 对于小整数 $r$ 和 $|s|$. 他们的目标是考虑因溸 $F_9$ ，没有已知质因数的 最小艴马数。他们布望使用特殊的形式 $F_9=2^{512}-(-1)$ 使必须平滑的数字小于 QS 所需的数字。在他 们考虑之后 $F_9 1990$ 年，他们和其他人将 NFS 扩展到一般数字。有关此分解算法早期历史的更多详细信 息，请参阅 Lenstra 和 Lenstra [360]。有关该算法的摘要，请参见 Pomerance [479]。Crandall 和 Pomerance [144] 描述了现代算法。

NFS 通过让每个关系一们的数字是代数数域中的代数整数来解决这个问题。这个想法是匹配不可约因 䋤，使每个因溸出现偶数次，并苃望所选关系子集中的代数整数的乘积可能是代数数域中的平方。

## 数学代写|密码学代写cryptography theory代考|Number Fields

$$F(x, y)=x^d+c_{d-1} x^{d-1} y+\cdots+c_1 x y^{d-1}+c_0 y^d=y^d f(x / y)$$

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