Primo Verifier... - Page 2

jack · 2013-09-09, 17:05

Quote:

Originally Posted by WraithX

Good job on finding a problem certificate.

Assuming the verifier correctly computes the values, i.e., assuming the Hasse interval is defined using integer square root and not real one... but is it the case?

Quote:

we now use the ceil(sqrt(n)) in the calculations instead of floor(sqrt(n)).

What about floor(2*sqrt(n))?

jack · 2013-09-09, 19:40

Quote:

Originally Posted by jack

Assuming the verifier correctly computes the values, i.e., assuming the Hasse interval is defined using integer square root and not real one... but is it the case?

What about floor(2*sqrt(n))?

Of course, I meant "What about floor(2*sqrt(n)) instead of 2*floor(sqrt(n))?" If the decimal part of sqrt(n) is greater than or equal to 1/2, this is not the same.

danaj · 2013-09-09, 22:36

Two independently written ECPP implementations generated similar M (R*S) values. I noticed that Primo uses the smaller value, which is probably an optimization (search the smallest m values first). I made my program sort the m values by size and I get the same value Primo uses.

Two separate verifiers end up doing the same calculations. I had looked at the conditions to generate m from u and v, but I missed this gotcha. Yes, both verifiers are doing all integer math. Using FP math would add a lot of complication.

Technically I don't believe these conditions are even required, as the theorem's conditions consist of "let m be an integer, let q be a prime dividing m and q > (N^1/4+1)^2, ..." (Atkin/Morain 5.2). But we know from the proof of the theorem and ECPP implementations that m is going to be within the Hasse interval (Goldwasser & Killian 3.3(1) or Atkin & Morain 4.2). m = N + 1 - t where |t| <= 2sqrt(N). The test is an easy sanity check that things look reasonable and the EC calculations have a chance of succeeding. So WraithX's solution of using 2*ceil(sqrt(N)) looks good to me. Perhaps ceil(sqrt(4N)) for a slightly tighter bound.

Edit: actually given the limits, I think floor(sqrt(4N)) should do it, which is basically what Jack just wrote. :)

jack · 2013-09-10, 05:13

Quote:

Originally Posted by danaj

Using FP math would add a lot of complication.

Yes but there is no need of FP math. Assuming you have an integer N, in order to know whether the decimal part (dp) of its integer square root (isqrt) is less than 0.5 or greater than or equal to 0.5, all you have to do is to compute isqrt(4N). Now, if the root is even then dp(sqrt(N)) < 0.5 whereas if it is odd then dp(sqrt(N)) >= 0.5. And, of course, if what you wanted was isqrt(N), there is no need to re-compute a root, just shift by one to the right the integer value isqrt(4N).

For instance, isqrt(4*7) = 5. Since 5 is odd, we know that the first digit of the decimal part of sqrt(7) is greater than or equal to 5 (and also that isqrt(7) = 5 >> 1 = 2). As a matter of fact, all this stuff is easy to understand with the binary representations of the numbers:
isqrt(4*7) --> 101
sqrt(7) --> 10.1xx...

danaj · 2013-09-10, 07:24

Jack, I wasn't sure if you were advocating FP or not. At the end of my post I came to the same idea -- isqrt(4n), and then did the edit from ceil to floor after looking at the code and trying it out. I also noted it was a long winded way of arriving at your floor(2*sqrt(n)) suggestion. mpz_root(4*N) should give us the right number.

frmky · 2022-07-07, 18:10

Is there a command line, parallel PRIMO certificate verifier available? WraithX's verifier doesn't support v4 certificates, and neither his or danaj's is parallel.

PRIMO will verify certificates quickly in parallel, but the GUI is a PITA for me to use. I have to use a couple tricks to forward the GUI from a cluster compute node, I can't seem to resize the window, and on my laptop the Load button on the bottom is below the bottom of the screen unless I muck with my screen resolution.

kruoli · 2022-07-07, 18:39

The verification is quite well documented in the Primo executable's archive, it is named verifier-f4.txt. This is not directly a program, but maybe 80 % of it...?

frmky · 2022-12-22, 06:32

Quote:

Originally Posted by frmky

Is there a command line, parallel PRIMO certificate verifier available?

I made danaj's verifier parallel using OpenMP. I also disabled MPU certificate support in the process. (Ironic, perhaps, since it's a part of MPU, but there's little demand for a parallel MPU certificate verifier.) All of the verification steps remain intact but required numerous changes to remove global variables. It has been lightly tested on Primo v3 and v4 certificates so consider it a work in progress for now. Please let me know if you find a bug. Compile it with something like...

Code:

gcc -O3 -fopenmp -o vcertomp vcertomp.c -lgmp -lm

https://github.com/gchilders/Math-Pr...les/vcertomp.c

chalsall · 2022-12-22, 06:51

Quote:

Originally Posted by frmky

I made danaj's verifier parallel using OpenMP. I also disabled MPU certificate support in the process. (Ironic, perhaps, since it's a part of MPU, but there's little demand for a parallel MPU certificate verifier.)

Excellent... 8^)

2013-09-09, 22:36	#14
danaj "Dana Jacobsen" Feb 2011 Bangkok, TH 3²×101 Posts	Two independently written ECPP implementations generated similar M (RS) values. I noticed that Primo uses the smaller value, which is probably an optimization (search the smallest m values first). I made my program sort the m values by size and I get the same value Primo uses. Two separate verifiers end up doing the same calculations. I had looked at the conditions to generate m from u and v, but I missed this gotcha. Yes, both verifiers are doing all integer math. Using FP math would add a lot of complication. Technically I don't believe these conditions are even required, as the theorem's conditions consist of "let m be an integer, let q be a prime dividing m and q > (N^1/4+1)^2, ..." (Atkin/Morain 5.2). But we know from the proof of the theorem and ECPP implementations that m is going to be within the Hasse interval (Goldwasser & Killian 3.3(1) or Atkin & Morain 4.2). m = N + 1 - t where \|t\| <= 2sqrt(N). The test is an easy sanity check that things look reasonable and the EC calculations have a chance of succeeding. So WraithX's solution of using 2ceil(sqrt(N)) looks good to me. Perhaps ceil(sqrt(4N)) for a slightly tighter bound. Edit: actually given the limits, I think floor(sqrt(4N)) should do it, which is basically what Jack just wrote. :) Last fiddled with by danaj on 2013-09-09 at 23:04

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2013-09-10, 07:24	#16
danaj "Dana Jacobsen" Feb 2011 Bangkok, TH 3²·101 Posts	Jack, I wasn't sure if you were advocating FP or not. At the end of my post I came to the same idea -- isqrt(4n), and then did the edit from ceil to floor after looking at the code and trying it out. I also noted it was a long winded way of arriving at your floor(2sqrt(n)) suggestion. mpz_root(4N) should give us the right number.

2022-07-07, 18:10	#17
frmky Jul 2003 So Cal 2×3×433 Posts	Is there a command line, parallel PRIMO certificate verifier available? WraithX's verifier doesn't support v4 certificates, and neither his or danaj's is parallel. PRIMO will verify certificates quickly in parallel, but the GUI is a PITA for me to use. I have to use a couple tricks to forward the GUI from a cluster compute node, I can't seem to resize the window, and on my laptop the Load button on the bottom is below the bottom of the screen unless I muck with my screen resolution.

2022-07-07, 18:39	#18
kruoli "Oliver" Sep 2017 Porta Westfalica, DE 10100101100₂ Posts	The verification is quite well documented in the Primo executable's archive, it is named verifier-f4.txt. This is not directly a program, but maybe 80 % of it...?