# 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|Finding Suitable Curves

We have shown that the points on an elliptic curve form a commutative group, and the set of rational points is a finite commutative group. As we saw in Section 2.2, we need a cyclic group whose order is divisible by a large prime.
Exercise 2.65. Show that an elliptic curve has 0,1 or 3 rational points of order 2.
Hint: Use Exercise 2.62.
Exercise 2.66. Let $E$ be the curve defined by $Y^2=X^3+12$ over the field $\mathbb{F}_{13}$. Show that $E$ is not cyclic.

The group of rational points is not cyclic in general, but there must be a large cyclic subgroup.

Fact 2.35. Let $E$ be an elliptic curve defined over $\mathbb{F}p$. Then there exists $n_1, n_2$, where $n_1$ divides both $n_2$ and $p-1$, such that $$E\left(\mathbb{F}_p\right) \simeq \mathbb{Z}{n_1}^{+} \times \mathbb{Z}_{n_2}^{+} .$$
But a large cyclic subgroup is not sufficient for our purposes, we also need to know that its order is divisible by a large prime.

Proposition 2.36. Let $p$ be a prime congruent to 2 modulo 3 , and let $E$ be an elliptic curve defined by $Y^2-X^3+B$ over $\mathbb{F}_p$. Then $\left|E\left(\mathbb{F}_p\right)\right|-p+1$.
Proof. Since $p \equiv 2(\bmod 3), 3$ is invertible modulo $p-1$. Then the $\operatorname{map} \zeta \mapsto \zeta^3$ is invertible. If $k \equiv 3^{-1}(\bmod p-1)$, then $\zeta \mapsto \zeta^k$ is the inverse map.

## 数学代写|密码学代写cryptography theory代考|Discrete Logarithms

In the group $\mathbb{F}_p^*$ the group operation requires two arithmetic operations (one integer multiplication and one integer division), while finding inverses is much more costly (using the extended Euclidian algorithm). Note that division in a finite field is usually done by multiplying with inverses.

In the group $E\left(\mathbb{F}_p\right)$, Proposition $2.34$ says that adding distinct non-inverse points requires one inversion, three multiplications and six additions. Adding a point to itself requires one inversion, two multiplications hy small eonstants, four multiplications, one addition of a constant and four additions.

At first glance, it would seem odd to consider the elliptic curve group, since the group operation there is much more complicated than the group operation in $\mathbb{F}_p^*$. But there is one more variable to consider: the size of the underlying field. Recall that we choose the size of the group such that the discrete logarithm problem in the group is sufficiently difficult.

The algorithms in Section $2.2$ work for any group. For $\mathbb{F}_p^*$ we also have much faster index calculus methods. For most elliptic curves, there are no equivalents of the small primes, so there are no useful index calculus methods.
We briefly describe some results on elliptic curve discrete logarithms.

• For so-called anomalous elliptic curves, curves defined over $\mathbb{F}_p$ with $p$ elements, there are very efficient algorithms for computing discrete logarithms. These curves are completely unsuitable for use in cryptography.

# 密码学代考

## 数学代写|密码学代写cryptography theory代考|Discrete Logarithms

• 对于所谓的反常椭圆曲线，定义在Fp和p元素，有非常有效的算法来计算离散对数。这些曲线完全不适合用于密码学。

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