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数学代写|数论代写Number Theory代考|MATH413 Baby step/giant step method

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数学代写|数论代写Number Theory代考|Baby step/giant step method

As above, suppose that $\gamma \in \mathbb{Z}p^$ generates a subgroup $G$ of $\mathbb{Z}_p^$ of order $q>1$ (not necessarily prime), and we are given $p, q, \gamma$, and $\alpha \in G$, and wish to compute $\log \gamma \alpha$.

A faster algorithm than brute-force search is the baby step/giant step method. It works as follows.

Let us choose an approximation $m$ to $q^{1 / 2}$. It does not have to be a very good approximation – we just need $m=\Theta\left(q^{1 / 2}\right)$. Also, let $m^{\prime}=\lfloor q / m\rfloor$, so that $m^{\prime}=\Theta\left(q^{1 / 2}\right)$ as well.

The idea is to compute all the values $\gamma^i$ for $i=0, \ldots, m-1$ (the “baby steps”) and to build a “lookup table” $L$ that contains all the pairs $\left(\gamma^i, i\right)$, and that supports fast lookups on the first component of these pairs. That is, given $\beta \in \mathbb{Z}_p^*$, we should be able to quickly determine if $\beta=\gamma^i$ for some $i=0, \ldots, m-1$, and if so, determine the value of $i$. Let us define $L(\beta):=i$ if $\beta=\gamma^i$ for some $i=0, \ldots, m-1$; otherwise, define $L(\beta):=-1$.

Using an appropriate data structure, we can build the table $L$ in time $O\left(q^{1 / 2} \operatorname{len}(p)^2\right)$ (just compute successive powers of $\gamma$, and insert them in the table), and we can perform a lookup in time $O(\operatorname{len}(p))$. One such data structure is a radix tree (also called a search trie); other data structures may be used (for example, a hash table or a binary search tree), but these may yield slightly different running times for building the table and/or for table lookup.

数学代写|数论代写Number Theory代考|Groups of order qe

Suppose that $\gamma \in \mathbb{Z}p^$ generates a subgroup $G$ of $\mathbb{Z}_p^$ of order $q^e$, where $q>1$ and $e \geq 1$, and we are given $p, q, e, \gamma$, and $\alpha \in G$, and wish to compute $\log \gamma \alpha$

There is a simple algorithm that allows one to reduce this problem to the problem of computing discrete logarithms in the subgroup of $\mathbb{Z}_p^$ of order $q$. It is perhaps easiest to describe the algorithm recursively. The base case is when $e=1$, in which case, we use an algorithm for the subgroup of $\mathbb{Z}_p^$ of order $q$. For this, we might employ the algorithm in $\S 11.2 .2$, or if $q$ is very small, the algorithm in $\S 11.2 .1$.

Suppose now that $e>1$. We choose an integer $f$ with $0<f<e$. Different strategies for choosing $f$ yield different algorithms-we discuss this below. Suppose $\alpha=\gamma^x$, where $0 \leq x<q^e$. Then we can write $x=q^f v+u$, where $u$ and $v$ are integers with $0 \leq u<q^f$ and $0 \leq v<q^{e-f}$. Therefore,
$$\alpha^{q^{e-f}}=\gamma^{q^{e-f} u} .$$
Note that $\gamma^{q^{e-f}}$ has multiplicative order $q^f$, and so if we recursively compute the discrete logarithm of $\alpha^{q^{e-f}}$ to the base $\gamma^{q^{e-f}}$, we obtain $u$.
Having obtained $u$, observe that
$$\alpha / \gamma^u=\gamma^{q^f v} .$$

数学代写数论代写Number Theory代考|Groups of order qe

$$\alpha^{q^{e-f}}=\gamma^{q^{e-f_u}} .$$

$$\alpha / \gamma^u=\gamma^{q^{f_v}} .$$

MATLAB代写

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