3 ms·

Or some other number-theoretic advance that is significantly below exponential time on the particular type of field or curve being used. The reason that we use

Or some other number-theoretic advance that is significantly below exponential time on the particular type of field or curve being used. The reason that we use elliptic curves these days, or if we must then something like 8k bit keys to get 128 bits of security over finite fields, is that for the old Z^*_q/Z_p setup, such a faster algorithm exists (index calculus). Someone could in theory find a better calculus that works only for groups with some specific characteristics of Curve25519, for example. No quantum computers needed. EDIT: we know that no _generic_ faster algorithm exists, that is one independent of the representation of the group involved, for the traditional computing model. But that doesn't exclude algorithms, as I said above, that work for very particular cases.

ernesto95 2y ago
Do you have a personal book recommendation on the group and/or number theory of this type of cryptography?
- red_admiral 2y ago
  Most of what I've learnt here was less from books and more from colleagues/seminars and reading research papers. You can get a brief introduction at https://soatok.blog/2020/04/26/a-furrys-guide-to-digital-signature-algorithms/ https://soatok.blog/2020/04/26/a-furrys-guide-to-digital-sig... (your own choice if you want that open in a tab at work or not, but there's nothing NSFW in the usual sense in there), and then read the details of each scheme in the RFCs. Some of the RFCs even talk about security implications. "djb" as he is known in the crypto world has a good paper at https://eprint.iacr.org/2024/1265 https://eprint.iacr.org/2024/1265 , it's 68 pages so "almost a book". He also has a lot of resources on his page https://cr.yp.to https://cr.yp.to . Be aware that he is sometimes ... controversial (not racist or anything, just has strong opinions on FIPS and the NSA and has actually taken the US government to court in the past over this). He's the author of Curve25519.

ernesto95 2y ago

Do you have a personal book recommendation on the group and/or number theory of this type of cryptography?

red_admiral 2y ago
Most of what I've learnt here was less from books and more from colleagues/seminars and reading research papers. You can get a brief introduction at https://soatok.blog/2020/04/26/a-furrys-guide-to-digital-signature-algorithms/ https://soatok.blog/2020/04/26/a-furrys-guide-to-digital-sig... (your own choice if you want that open in a tab at work or not, but there's nothing NSFW in the usual sense in there), and then read the details of each scheme in the RFCs. Some of the RFCs even talk about security implications. "djb" as he is known in the crypto world has a good paper at https://eprint.iacr.org/2024/1265 https://eprint.iacr.org/2024/1265 , it's 68 pages so "almost a book". He also has a lot of resources on his page https://cr.yp.to https://cr.yp.to . Be aware that he is sometimes ... controversial (not racist or anything, just has strong opinions on FIPS and the NSA and has actually taken the US government to court in the past over this). He's the author of Curve25519.