Timeline for elementary question on ECDLP
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2010 at 13:52 | comment | added | athena | E is a random elliptic curve defined over $E(\mathbb{F}_p)$ such that ECDLP is hard in $E(\mathbb{F}_p)$. So $p$ is a large prime, not less than 159 bits. $m$ is the order of a subgroup generated by $T$ of $E(\mathbb{F}_p)$. $E(\mathbb{F}_q)$ is a finite field much smaller than $E(\mathbb{F}_p)$, where $q$ is a prime. There are no other limits. | |
| Dec 30, 2010 at 16:39 | comment | added | Victor Miller | My guess is that the OP meant that $q$ is a prime which divides the order of $E(\mathbb{F}_p)$. If that's the case there certainly can be torsion points whose order isn't divisible by $q$. | |
| Dec 30, 2010 at 15:55 | comment | added | Felipe Voloch | The order of the group cannot be $q$ as the OP said $q \ll p$. It's not up to us to figure out what the OP meant to ask. | |
| Dec 30, 2010 at 14:25 | comment | added | Chris Wuthrich | You are right, maybe I misread the question and $q$ just stands for the order of the group, in which case $T = (q+1)T$ is obvious. But then $m=q$. | |
| Dec 30, 2010 at 13:57 | comment | added | Felipe Voloch | The role of $q$ is unclear. What does it have to do with $E,p,m$? | |
| Dec 30, 2010 at 13:34 | comment | added | Chris Wuthrich | for any $n\in\mathbb{Z}$, we define $n\cdot P=P+P+\cdots +P$ using the addition law $+$ on the curve. There is absolutely no reason that $p\cdot P = O$, just because the coordinates of the points are in $\FF_p$. As I said, take $y^2 + y = x^3 + 1$ over $\FF_2$ it has $3$ rational points, so $2\cdot P = -P$. You should read the basics of elliptic curve in a book like Silverman;s "The arithmetic of elliptic curves". | |
| Dec 30, 2010 at 12:39 | comment | added | athena | $E(\mathbb{F}_p)$ consists of all the rational points on E over $\mathbb{F}_p$. Excuse me, but could you tell me why $E(\mathbb{F}_p)$ is a $\mathbb{Z}$ module or isn't a $\mathbb{F}_p$ vector space? | |
| Dec 30, 2010 at 10:29 | comment | added | Chris Wuthrich | the error is that $E(\mathbb{F}_p)$ is a $\mathbb{Z}$-module not a $\mathbb{F}_p$-vector space. Curves for which $p$ divides the order of $E(\mathbb{F}_p)$ are sometimes called anomalous. Pick your any elliptic curves and do some computations and you will see your error. (This should be closed.) | |
| Dec 30, 2010 at 8:36 | history | edited | athena | CC BY-SA 2.5 |
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| Dec 30, 2010 at 8:29 | history | edited | athena | CC BY-SA 2.5 |
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| Dec 30, 2010 at 8:19 | history | asked | athena | CC BY-SA 2.5 |