[curves] The Pareto frontiers of sleeveless primes

Ben Harris mail at bharr.is
Wed Oct 29 21:44:12 PDT 2014


Finally - integer radix (minimum 4-bits of headroom for 32-bit and 64-bit)
primes, grouped into byte sizes, non-dominated set. 28/27 candidates (one
of the 114 bit candidates is dominated, not sure which). Are there
recommended limits on the small 'c' in Crandall primes? This list is only
up to 32, but many on the SafeCurves list are in the 100s.

Prime             mod  bytes  32-bit  64-bit
 2^850 -  3          5   128   34x25   17x50
 2^848 - 17          3   128   53x16   16x53
 2^810 -  5          3   128   30x27   15x54
 2^729 -  9          3    96   27x27   27x27
 2^689 -  3          5    96   53x13   13x53
 2^550 -  5          3    96   22x25   10x55
 2^546 - 11          5    96   21x26   13x42
 2^480 - 2^240 - 1   3    64   20x24    8x60
 2^468 - 17          3    64   18x26    9x52
*2^448 - 2^224 - 1   3    64   16x28    8x56
 2^336 -  3          5    48   12x28    6x56
 2^285 -  9          3    48   15x19    5x57
*2^255 - 19          5    32   15x17    5x51
 2^243 -  9          3    32   9x27     9x27
 2^230 - 27          5    32   10x23    5x46
 2^216 - 2^108 - 1   3    32   8x27     4x54
 2^190 - 11          5    24   10x19    5x38
 2^189 - 25          3    24   7x27     7x27
 2^171 - 19          5    24   9x19     3x57
 2^152 - 17          3    24   8x19     4x38
 2^150 -  3          5    24   6x25     3x50
 2^140 - 27          5    24   5x28     4x35
 2^125 -  9          3    16   5x25     5x25
 2^114 - 2^57  - 1   3    16   6x19     2x57
 2^114 - 11          5    16   6x19     2x57
 2^110 - 21          3    16   5x22     2x55
 2^104 - 17          3    16   4x26     2x52
 2^96  - 17          3    12   4x24     2x48


On 29 October 2014 10:37, Ben Harris <mail at bharr.is> wrote:

> Including Ridinghoods, and preferring 3 mod 4 over 1 mod 4 when all else
> is equal nets 57.
>
> prime            mod 4  28   32   58
> 2^96  - 17          3    4    3    2
> 2^110 - 21          3    4    4    2
> 2^114 - 2^57  - 1   3    5    4    2
> 2^116 -  3          1    5    4    2
> 2^127 -  1          3    5    4    3
> 2^137 - 13          3    5    5    3
> 2^140 - 27          1    5    5    3
> 2^152 - 17          3    6    5    3
> 2^158 - 15          1    6    5    3
> 2^166 -  5          3    6    6    3
> 2^174 -  3          1    7    6    3
> 2^189 - 25          3    7    6    4
> 2^191 - 19          1    7    6    4
> 2^196 - 15          1    7    7    4
> 2^216 - 2^108 - 1   3    8    7    4
> 2^221 -  3          1    8    7    4
> 2^226 -  5          3    9    8    4
> 2^230 - 27          1    9    8    4
> 2^251 -  9          3    9    8    5
> 2^255 - 19          1   10    8    5
> 2^266 -  3          1   10    9    5
> 2^285 -  9          3   11    9    5
> 2^291 - 19          1   11   10    6
> 2^322 - 2^161 - 1   3   12   11    6
> 2^336 -  3          1   12   11    6
> 2^338 - 15          1   13   11    6
> 2^369 - 25          3   14   12    7
> 2^383 - 31          1   14   12    7
> 2^389 - 21          3   14   13    7
> 2^401 - 31          1   15   13    7
> 2^416 - 2^208 - 1   3   15   13    8
> 2^448 - 2^224 - 1   3   16   14    8
> 2^450 - 2^225 - 1   3   17   15    8
> 2^452 -  3          1   17   15    8
> 2^468 - 17          3   17   15    9
> 2^480 - 2^240 - 1   3   18   15    9
> 2^489 - 21          3   18   16    9
> 2^495 - 31          1   18   16    9
> 2^521 -  1          3   19   17    9
> 2^529 - 31          1   19   17   10
> 2^537 -  9          3   20   17   10
> 2^550 -  5          3   20   18   10
> 2^563 -  9          3   21   18   10
> 2^583 - 27          1   21   19   11
> 2^607 -  1          3   22   19   11
> 2^610 - 27          1   22   20   11
> 2^620 - 15          1   23   20   11
> 2^664 - 17          3   24   21   12
> 2^694 -  3          1   25   22   12
> 2^699 -  9          3   25   22   13
> 2^717 - 25          3   26   23   13
> 2^729 -  9          3   27   23   13
> 2^810 -  5          3   29   26   14
> 2^848 - 17          3   31   27   15
> 2^850 -  3          1   31   27   15
> 2^869 - 21          3   32   28   15
> 2^923 - 31          1   33   29   16
>
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