[HN Gopher] A new rare high-rank elliptic curve, and an orchard ...
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A new rare high-rank elliptic curve, and an orchard of Diophantine
equations
Author : mathgenius
Score : 194 points
Date : 2024-09-07 17:43 UTC (1 days ago)
(HTM) web link (thehighergeometer.wordpress.com)
(TXT) w3m dump (thehighergeometer.wordpress.com)
| miovoid wrote:
| [deleted]
| madars wrote:
| Your E/Fp has order 2^3 * 3 * 37991 * 21183269 * 373015308871 *
| 16071902378831708724506232718210977087913221837027589 and thus
| you can't hope for more than 86 bits of security due to Pohlig-
| Hellman, never mind cofactor attacks. encrypt() is also
| insecure (xor every byte of the message with the same shared
| secret byte), even if you chose a better curve.
| tptacek wrote:
| This is much better version of the sibling comment but I'm a
| message board nerd and can't keep myself from pointing out
| that this code is probably a little bit tongue-in-cheek.
| leijurv wrote:
| `for char in message: encrypted_char = ord(char) ^
| (shared_secret[0] % 256)`
|
| This is not real encryption, it picks only one byte of shared
| secret and XORs it into the plaintext. Therefore, there are
| only 256 possible decryption keys to check, which is trivial.
|
| Instead, you'd want to use the shared secret as a key to
| something strong and symmetric like AES.
| tptacek wrote:
| I don't think it's meant to be real encryption.
| leijurv wrote:
| I suspect it was, given that they've now deleted their
| comment.
| thechao wrote:
| Any idiot knows not to use power-of-two! You gotta use "+13",
| which is prime and, therefore, *secure*.
| BobbyTables2 wrote:
| And Twice is nice!
| Xcelerate wrote:
| This sounds very similar to the same process for Turing machines:
| https://www.quantamagazine.org/amateur-mathematicians-find-f...
|
| Determining the halting behavior of each successive Turing
| machine generally becomes harder and harder until eventually we
| reach a machine with Collatz-like behavior.
|
| The two problems are equivalent in some sense, but I wonder if
| there's an easy way to "port" over the work between the two
| projects.
| yorwba wrote:
| The equivalence of Diophantine equations and Turing machines is
| established by the MRDP theorem:
| https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's...
|
| For every Diophantine equation _P_ ( _x_ , _y_ ) = 0, where _x_
| is a tuple of integer parameters and _y_ is a tuple of
| unknowns, there is a Turing machine that takes the parameters
| _x_ as input and halts iff there is an integer solution _y_
| satisfying the equation (this direction is easy, just have the
| Turing machine test all tuples of integers in some order and
| halt iff it found a solution)
|
| and for every Turing machine, there is a Diophantine equation
| so that if _x_ is an encoding of the Turing machine input,
| there is an integer solution _y_ to the equation iff the the
| Turing machine halts for input _x_. (This direction is hard,
| you need to have _y_ encode the execution history of the Turing
| machine somehow, and enforce the transition rules with the
| structure of the equation.)
| fny wrote:
| I feel like we've reached an era where information provenance is
| of paramount importance. This has always been an issue with
| fabricated data sets, but the ease at which anything can be
| fabricated--even a video--demands some new determinant of what is
| real and human born.
| fermigier wrote:
| I broke the record back in 1992: Fermigier, Stefane - Un exemple
| de courbe elliptique definie sur Q de rang >=19. (French) [An
| example of an elliptic curve defined over Q with rank >=19] C. R.
| Acad. Sci. Paris Ser. I Math. 315 (1992), no. 6, 719-722.
|
| And again in 1997: Fermigier, Stefane - Une courbe elliptique
| definie sur Q de rang >=22. (French) [An elliptic curve defined
| over Q of rank >=22] Acta Arith. 82 (1997), no. 4, 359-363.
|
| Noam Elkies was already a major contributor to the field at the
| time. I dropped math to do computer stuff :)
| ykonstant wrote:
| What kind of computer stuff are you doing now?
| alecco wrote:
| His HN about page has a lot of info.
| fermigier wrote:
| Running an open source software business... Founded
| https://nuxeo.com/ and now https://abilian.com/
| anyon_fusion wrote:
| what made you drop math for computer stuff, if you don't mind
| sharing? (I'm in a similar boat)
| ykonstant wrote:
| Great write-up; I need to review how GRH is used to prove bounds
| on the rank of elliptic curves.
|
| Also, the talk about polynomial parametrizations reminded me of
| the first Diophantine equation I solved in high school:
| (a^2+b^2)/(a+b) = (c^2+d^2)/(c+d). I had initially thought it had
| finitely many solutions, but then Nikos Tzanakis corrected me and
| told me I am missing many. So I toiled for two entire days and
| found the complete 2-parameter polynomial family of solutions.
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