blocks|key|824555|text|让f(x,y)+=+-y%5E2+%2B+x%5E3%2B17230x%2B22699在\Bbb+F_p上使用p=23981。曲线上的点是奇点当且仅当偏导数在这一点上消失。偏导数为；|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|824556|\frac{\partial+f}{\partial+x}+=+3x%5E2+%2B+17230+=0+\pmod+p和消失在x=\{185,23796\}，由WolframAlpha发现，也可以由托内利-香克找到。|unordered-list-item|824557|\frac{\partial+f}{\partial+y}+=+-2y+=+0+\pmod+p和消失在y=0|824558|消失点(185,0)不在曲线上，而(23796,0)在曲线上。|824559|因此，(23796,0)是曲线的一个奇点。|824560|然后，我们将原点转化为奇异点(23796,0)，即在E方程中用(x%2B23796,y%2B0代替D21)，得到位移参考中的曲线方程：y%5E2+=+x%5E3+%2B+23426x%5E2，其余都在OP有问题的答案中。|824561|注:在目前的答案中，@kelalaka得到了更好的信用，特别是介绍了f(x,y)及其偏导数。|entityMap|0|LINK|mutability|MUTABLE|url|https://www.wolframalpha.com/input/?i=solve%25203x%255E2%2520%252B%252017230%2520%253D0%2520mod%252023981%2520|1|https://en.wikipedia.org/wiki/Tonelli%25E2%2580%2593Shanks_algorithm|2|https://crypto.stackexchange.com/a/61434/555^0|1|W|Y|8|19|7|0|0|1J|1N|F|24|C|0|2N|6|1|0|0|1B|1F|3|0|3|7|H|9|0|3|9|0|E|9|Q|1|V|C|19|3|1R|K|2M|2|2|0|Y|6^^$0|@$1|2|3|4|5|6|7|13|8|@$9|14|A|15|B|C]|$9|16|A|17|B|C]|$9|18|A|19|B|C]]|D|@]|E|$]]|$1|F|3|G|5|H|7|1A|8|@$9|1B|A|1C|B|C]|$9|1D|A|1E|B|C]]|D|@$9|1F|A|1G|1|1H]|$9|1I|A|1J|1|1K]]|E|$]]|$1|I|3|J|5|H|7|1L|8|@$9|1M|A|1N|B|C]|$9|1O|A|1P|B|C]]|D|@]|E|$]]|$1|K|3|L|5|6|7|1Q|8|@$9|1R|A|1S|B|C]|$9|1T|A|1U|B|C]]|D|@]|E|$]]|$1|M|3|N|5|6|7|1V|8|@$9|1W|A|1X|B|C]]|D|@]|E|$]]|$1|O|3|P|5|6|7|1Y|8|@$9|1Z|A|20|B|C]|$9|21|A|22|B|C]|$9|23|A|24|B|C]|$9|25|A|26|B|C]|$9|27|A|28|B|C]]|D|@$9|29|A|2A|1|2B]]|E|$]]|$1|Q|3|R|5|6|7|2C|8|@$9|2D|A|2E|B|C]]|D|@]|E|$]]]|S|$T|$5|U|V|W|E|$X|Y]]|Z|$5|U|V|W|E|$X|10]]|11|$5|U|V|W|E|$X|12]]]]

Let $f(x,y) = -y^2 + x^3+17230x+22699$ over $\Bbb F_p$ with $p=23981$. A point on the curve is a singular point if and only if the partial derivatives are vanishes at that point. The partial derivatives are;

<ul>
<li>$\frac{\partial f}{\partial x} = 3x^2 + 17230 =0 \pmod p$ and vanishes at $x=\{185,23796\}$ found by <a href="https://www.wolframalpha.com/input/?i=solve%203x%5E2%20%2B%2017230%20%3D0%20mod%2023981%20" rel="nofollow noreferrer">WolframAlpha</a> or it can be found by <a href="https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm" rel="nofollow noreferrer">Tonelli-Shanks</a>.</li>
<li>$\frac{\partial f}{\partial y} = -2y = 0 \pmod p$ and vanishes at $y=0$ </li>
</ul>

The vanishing point $(185,0)$ is not on the curve, however, $(23796,0)$ is.

Therefore $(23796,0)$ is a singular point for the Curve.

Then we then translate the origin to this singular point $(23796,0)$, that is replace $(x,y)$ by $(x+23796,y+0$) in the equation of $E$, yielding the equation of the curve in the shifted referential:
$$y^2 = x^3 + 23426x^2$$
The rest is in <a href="https://crypto.stackexchange.com/a/61434/555">the answer</a> which the OP had trouble with.

Note: credit goes to @kelalaka for the better of the present answer, in particular introducing $f(x,y)$ and its partial derivatives.

I have seen <a href="https://crypto.stackexchange.com/q/61302/555">your example</a> about ecdlp solver :The Problem is as follows:

$$E\backslash GF(p):y^2=x^3+17230x+22699$$

where $p=23981$, point $G$ with prime order $|G| = 109$

Alice creates a public key by selecting a private key $d&lt;q$, public key $Q=[d]G = (3141,12767)$

Therefore;

<ul>
<li>public information : $a,b,p,G,q,Q$ </li>
<li>private key : $d$</li>
</ul>

However, this curve has the following characteristic:

$\Delta= −16( 4a^3+27b^2) \bmod p =0$
That is, the discriminant is 0. and embedding degree is 2

I, however, don't understand where comes out the value (23796,0) i.e how can I calculate it? and then the following formula equation: after which we obtain the corresponding curve
$$y^2=x^3+23426x^2$$

<ul>
<li>is it possible for a further explanation of how you determine the point $(27396,0)$?</li>
</ul>

how to solve discrete logarithm problem information requested

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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我看到了关于ecdlp解决程序的你的例子：问题如下：E\backslash GF(p):y^2=x^3+17230x+22699其中p=23981，具有素数阶|G| = 109的点G通过选择私钥d，公钥Q=[d]G = (3141,12767)创建公钥。因此；公共信息：a,b,p,G,q,Q私钥：d然而，这条曲线具有以下特点：\Delta= −16( 4a^3+27b^2) \bmod p =0，

问如何解决所要求的离散对数问题信息
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