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ag.algebraic geometry – What are possible applications of ‘fast arithmetic’ in the Jacobian (degree zero Picard group) of projective curves over fields?

It is well known that there are plenty possible applications of ‘fast arithmetic’ (that is, 1. having an algorithm at hand that actually computes in…, and 2. the running time of that algorithm is known and not too bad) in the degree zero Picard group (the ‘Jacobian’) of an integral, smooth and projective curve $C$ with small genus (for instance elliptic curves and hyperelliptic curves of genus 2) over a field $k$. For instance, there are cryptosystems whose safety relies on the discrete logarithm problem in the Jacobian of elliptic curves and thus it is crucial to know how fast the arithmetic in that group can be carried out.

I wonder whether there are applications of ‘fast arithmetic’ in the degree zero Picard group of more general projective curves $C$ over $k$. Especially, I am curious about the case when $C$ is at least one of the following: reducible, singular or has large genus.

For instance, in the paper of Michael Stoll and Peter Bruin the authors use the Mordell-Weil sieve to prove the non-existence of rational points on smooth projective curves $C$ over $mathbb{Q}$ of genus $g geq 2$. They use the embedding of the rational points on $C$ into the Mordell-Weil group (degree zero Picard group) and local data of reductions at primes $p$ of $mathbb{Q}$. And there is a speed up in their computation if they do not restrict themselves to primes $p$ of good reduction, but also consider $p$ with bad reduction. In that case the authors provide a variant of the Cantor algorithm to compute in the Jacobian in the case that the reduction has genus $2$. Here a generalization to higher genus may be useful to also gather information from bad primes whose corresponding reduction has large(er) genus.

I am interested if there are further applications of ‘fast arithmetic’ in the Jacobian/degree zero Picard group in the same manner as above or even in a complete different style (for instance, it might be valuable to know a lower bound for the running time of such an algorithm to provide statements about how hard a problem related to the Jacobian is).

Many thanks in advance for your help!

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  • GTX 1080 Ti 11GB ~ 59 – 61 Sol / s
  • RTX 2070 8GB ~ 52 – 58 Sol / s
  • RTX 2080 Ti 8GB ~ 90 – 96 Sol / s

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  • GTX 1080 8GB ~ 42 – 46 Sol / s
  • GTX 1080 Ti 11GB ~ 59 – 63 Sol / s
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  • GTX 1660 Ti 6GB ~ 187 – 201 Sol / s
  • GTX 1070 Ti 8GB ~ 227 – 264 Sol / s
  • GTX 1080 8GB ~ 242 – 273 Sol / s
  • GTX 1080 Ti 11GB ~ 352 – 375 Sol / s
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  • RTX 2080 Ti 8GB ~ 478 – 517 Sol / s

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  • GTX 1060 3GB ~ 14 – 15 Sol / s
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  • GTX 1080 8GB ~ 28 – 31 Sol / s
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  • GTX 1080 8GB ~ 69 – 75 Sol / s
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computational complexity – Fast division by 3 or 5

There was a question about this topic posted on StackExchange 12 years ago, see here. Basically, it says that there is nothing better than the 3000 years-old technique, and the suggestion in modern times is to “let the compiler do its job”.

Well, here is what mine is doing. Divide the following integer by 3:

222730506728397170591387079211836683557072632289734369842905278066498119541270318525897952142956554114412930297037751282

The result is this:

74243502242799056863795693070612227852360000000000000000000000000000000000000000000000000000000000000000000000000000000

Obviously it is totally wrong, despite using the BigNum library in Perl. I know you can compute the multiplication $3x$ very fast: $3x = 2x + x = (x$ << $1) + x$ using the bit shifting operator. But what about the division?

I came up with the following rudimentary algorithm, but I am wondering if there is a faster solution.

Algorithm

Let $x$ be the number to be divided by $3$. Pre-compute $3^k$ and $2cdot 3^k$ for $k=0,cdots,n$ with $n=lfloorlog_3 xrfloor$. In other words, $n$ is the largest integer such that $3^nleq x$. Very useful step, since I have a bunch of large integers (all equal to zero modulo $3$) that I need to divide by $3$ and by any power of $3$ that is also a divisor. Here I assume $x$ is the largest of the integers that I am dealing with. Then the initialization step consists of:

  • $z leftarrow x$
  • $d_{n+1} leftarrow 0$
  • $nleftarrow n+1$

The loop consists of:

  • $y=z-d_n 3^n$
  • If $2cdot 3^{n-1} < y$ then $d_{n-1}=2$ else if $3^{n-1} <
    y$
    then $d_{n-1}=1$ else $d_{n-1}=0$.
  • $z leftarrow y$
  • $nleftarrow n-1$

Repeat the loop until $n=0$.

The sequence $d_1,cdots,d_n$ is the digits of $x/3$ in base $3$. If $3$ divides $x$ then $d_0=0$. Thus
$$frac{x}{3}=sum_{k=1}^nd_{k}3^{k-1}.$$

Note that $2cdot 3^{k} = 3^{k}$ << $1$ (faster than the multiplication by $2$). In the final sum, use pre-computed values for $d_k 3^{k-1}$.

mining pools – Bminer: a fast Equihash/Ethash/Cuckaroo29m miner for AMD/NVIDIA GPUs 16.2.9

Bminer is a highly optimized cryptocurrency miner that runs on modern AMD / NVIDIA GPUs (Maxwell and Pascal, i.e. GPUs with a computing power of 5.0 or higher). Bminer is one of the fastest miners available today – we use a variety of techniques, including tiling and pipelining, to realize the full potential of the hardware.

Bminer also comes with REST APIs to simplify product deployments (like mining farms).
Bminer supports Equihash based coin mining (like Zcash) with 2% devfee.
Bminer supports Zhash / Equihash 144.5 based coin mining (e.g. BitcoinGold, BitcoinZ) with 2% from devfee.
Bminer supports Ethash based coin mining (like Ethereum) with 0.65% devfee.
Bminer also supports dual mining mode – simultaneous mining of Ethash based coins (like Ethereum) and Blake14r based coins (like Decred) / Blake2s (like Verge). Devfee for dual mining mode is 1.3%, and the second coin (e.g. Decred / Verge) is mined without devfee.
Bminer supports Tensority based coin mining (like Bytom (BTM)) with 2% devfee.
Bminer supports Grin (GRIN) mining with 1% devfee.

Download

Specifications

Fast

Grin31 mining on stock settings
2.60 G / s on RTX 2080Ti
1.65 G / s on RTX 2080
1.55 G / s on GTX 1080Ti
0.95 G / s on GTX 1070

Bminer 16.0.6 Grin29m mining on stock settings
8.32 G / s on GTX 2080Ti
5.18 G / s on GTX 2070
3.96 G / s on 2060
2.18 G / s on 1060
3.34 G / s on 1070
5.03 G / s on 1080ti
3.00 G / s on P104-4G
3.56 G / s on P104-8G
4.95 G / s on P102-10G

AE mining on stock settings:
11.8 Sol / s on GTX 2080Ti
8.90 Sol / s on GTX 2080
7.40 Sol / s on GTX 1080Ti
4.7 Sol / s on GTX 1070
3.4 Sol / s on GTX 1060 6G

Beam mining on stock settings
30 Sol / s on GTX 1080Ti
21 Sol / s on GTX 1070
12 Sol / s on GTX 1060 6G

Equihash mining on stock settings
735-745 Sol / s on GTX 1080Ti
450-460 Sol / s on GTX 1070
315-325 Sol / s on GTX 1060

Equihash 144.5 (Zhash) mining on stock settings
61 Sol / s on GTX 1080Ti
25 Sol / s on GTX 1060
Ethash mining on GTX 1080Ti stock settings (power: 250 W)

With OhGodAnETHlargementPill: 46.7 MH / s
Without OhGodAnETHlargementPill: 32.2 MH / s

Dual mining using automatic tuning (default) on GTX 1080Ti stock settings (power: 250 W)

With OhGodAnETHlargementPill:
ETH 46 MH / s and DCR 1000 MH / s
ETH 46 MH / s and XVG 1770 MH / s

Without OhGodAnETHlargementPill:
ETH 32 MH / s and DCR 2200 MH / s
ETH 32 MH / s and XVG 3750 MH / s

Bytom mining on stock settings:
4650 H / s on GTX 1080Ti
2850 H / s on GTX 1070
1800 H / s on GTX 1060 6G
Safe and reliable

SSL support
Automatic reconnection to recover from network failures.
Automatic restart when GPU freezes

Operation friendly

Comes with REST API to simplify deployment

Fast start

Depending on the coins you want to mine, find the appropriate script in the folder. For example, the corresponding script is mine_grin29.bat (on Windows) or mine_grin29.sh (on Linux) when mining Grin using the Cuckaroo29 algorithm.
Change the address and account information in the script.
Run the script and enjoy mining clear.png
Release Notes

16.2.9 (current)
Improve the performance of the Cuckaroo29z miner.