# wrting algorithm for torsion group elements

Yesterday,I took an exam. There are two questions I received very low points. I will write the first question in this post.

The question says let $$E:y^2:x^3+kx+1$$ in GF(p) be an elliptic curve where p is a prime number which is 512 bits long. Now, think another prime number b with same size that of p (it means b is a prime integer of size 512 bits), and there is a point on curve such that $$P(0,1)$$. Assume that $$P(0,1) \in E[b]$$. It is wanted to write a pseudo code of a method which gives other elements of $$E[b]$$.

Choose a random integer k such that $$1 \leq k \leq b-1$$

Compute the point $$R = kP$$ using point multiplication

If R is the identity element of the group, go back to step 1

Return R as an element of ($$E[b]$$)

I have received very low point from this answer. Because of there is no any answer paper, I dont know the correct way. Can you please help for writing the correct answer ? WHat is the algorithm ? I think my pseudo code was fine, but it is not.

It is absolutely your lecturer's/professor's responsibility to give feedback on what's wrong with your answer. We don't know what is the assumed knowledge for the course, what level of pseudocode is required, etc.

Therefore what I say below may not apply given the level expected from you:

What is the definition of $$E[b]$$?

Some more brief thoughts:

What you wrote does not look like pseudocode. Where is step 1? You refer to it but it's unspecified.

Do you define a loop? You are trying to hit the identity again and again for different random $$k$$, but the output is when you don't hit the identity? It's all extremely unclear.

• $E[b]$ is b-torsion group Apr 10 at 17:15
• @kelalaka yes, you are right
– user115992
Apr 10 at 17:54
• I stated that I dont have answer paper,so I am here for learning. you can be advanced however you want.but please don write unnecessary statements like "Where is step 1? " as you see in my post, ı wrote the process from above to below
– user115992
Apr 10 at 17:56
• You should check that $P$ is not the identity. If so, it can only generate the trivial groups, otherwise the whole $E[b]$. since the $b$ is prime. Next is, $P, [2]P, [3]P. \ldots, [b]P$ are those elements/ Apr 10 at 18:02
• @juniorstudent Things like not putting numbers in front of the steps and such can be seen as extremely sloppy, and this can bite you when grades are delivered. I mean, it's not just that your question isn't capitalized, the very first word is spelled incorrectly as well. Precision is important for mathematicians, developers and by extension also cryptographers. Practice makes perfect. Apr 10 at 22:05