Expect primitive support for modular inverse in an upcoming beta.
Henry Rich
On Tue, Apr 4, 2023, 1:49 PM 'Michael Day' via Programming <
programm...@jsoftware.com> wrote:
> Thanks, Chris, and Cliff too.
>
> Yes, a mod inverse helps a lot. Once you've got an inverse, it's easy
> to derive a modular divide, or vice
> versa.
>
> inversep in primutil.ijs is well defined for a prime modulus - the name
> "primutil" does of course imply
> a prime modulus.
> inversep also appears to work ok for those numbers in the ring of
> integers modulo non-prime modulus.
>
> eg members of the ring modulo 10 are {1 3 7 9}
> {{ y,:y (10 mtimes) (10&inversep)"0 y}} 1 3 7 9
> 1 3 7 9
> 1 1 1 1
> {{ y,:y (10 mtimes) (10&mrecip)"0 y}} 1 3 7 9
> 1 3 7 9
> 1 1 1 1
>
> I find rather better performance with my mrecip:
> ts =: 6!:2 , 7!:2@]
>
> 1 p: 1000000009
> 1
>
> ts'100000009 mrecip 999999000 + >:i.1007'
> 0.0061255 22592
> ts'100000009 inversep"0] 999999000 + >:i.1007'
> 0.0798054 97160
> ts'100000009 mi"0] 999999000 + >:i.1007'
> 0.0908823 174192
>
> A bit surprising as the Extended Euler Algorithm is supposed to be best
> for getting a modular inverse.
>
> 1000000009 (mrecip-:inversep"0) 999999000 + >:i.1007
> 1
> 1000000009 ((1000000009 mi)"0 @] -:inversep"0) 999999000 + >:i.1007
> 1
>
> Here's mrecip:
>
> mrecip =: {{
> y (x&|@^) <: 5 p: x
> }}"0
>
> As for inversep, and Cliff's mi, mrecip is well-defined for prime x,
> and also for composite
> x for y in x's ring, ie where 1 = x +. y
>
> Results are NOT reliable for arguments not coprime with the modulus.
>
> Thanks,
>
> Mike
>
> On 03/04/2023 16:22, chris burke wrote:
> > Cliff
> >
> > There are some mod functions in the math/misc addon, e.g. this gives
> > Mike Day's table
> >
> > load 'math/misc/primutil'
> > f=: (17 timesmod) (17&inversep)
> > 2 3 4 f"0 table >:i.8
> > +---+---------------------+
> > |f"0|1 2 3 4 5 6 7 8|
> > +---+---------------------+
> > |2 |2 1 12 9 14 6 10 13|
> > |3 |3 10 1 5 4 9 15 11|
> > |4 |4 2 7 1 11 12 3 9|
> > +---+---------------------+
> >
> > Any improvements welcome, thanks.
> >
> > Chris
> >
> > On Mon, Apr 3, 2023 at 5:49 AM Clifford Reiter<reit...@lafayette.edu>
> wrote:
> >> I dug up an old extended gcd to build an adverb for modular divide
> >>
> >> NB. Find the gcd of two numbers
> >>
> >> NB. and coef giving gcd as a linear combination of y
> >>
> >> gcd2x=: 3 : 0
> >>
> >> 'r0 r1'=.y
> >>
> >> 's0 s1'=.1 0x
> >>
> >> 't0 t1'=.0 1x
> >>
> >> while. r1 ~: 0 do.
> >>
> >> q=. r0 <.@% r1
> >>
> >> 'r0 r1'=. r1,r0-q*r1
> >>
> >> 's0 s1'=. s1,s0-q*s1
> >>
> >> 't0 t1'=. t1,t0-q*t1
> >>
> >> end.
> >>
> >> r0,s0,t0
> >>
> >> )
> >>
> >> gcd2x 51 119
> >>
> >> 17 _2 1
> >>
> >> _2 1 +/ . * 51 119
> >>
> >> 17
> >>
> >> NB. adverb giving divide (inverse) mod m
> >>
> >> mi=:1 : 0"0
> >>
> >> 'r0 s0 t0'=:gcd2x m,y
> >>
> >> if. r0=1 do. m|t0 else. 1r0 end.
> >>
> >> :
> >>
> >> m|x*m mi y
> >>
> >> )
> >>
> >> 17 mi 6
> >>
> >> 3
> >>
> >> NB. Mike Day's Table
> >>
> >> 2 3 4 (17 mi)table >:i.8
> >>
> >> +-+---------------------+
> >>
> >> | |1 2 3 4 5 6 7 8|
> >>
> >> +-+---------------------+
> >>
> >> |2|2 1 12 9 14 6 10 13|
> >>
> >> |3|3 10 1 5 4 9 15 11|
> >>
> >> |4|4 2 7 1 11 12 3 9|
> >>
> >> +-+---------------------+
> >>
> >>
> >> I have some questions regarding system solving modulo m that I will
> offer
> >> in a new thread in a few days.
> >>
> >> Best, Cliff
> >>
> >> On Thu, Mar 30, 2023 at 12:11 PM Clifford Reiter<reit...@lafayette.edu>
> >> wrote:
> >>
> >>> I think I recall a conversation, some decades ago, with Roger about
> >>> whether specifying a modulus for system solving makes sense for J. I
> >>> thought maybe that was a use for the fit conjunction but now think that
> >>> would be a poor choice for such a numeric function. I have vague
> memories
> >>> of J essays on guass-jordan row reduction and extended gcds but didn't
> find
> >>> them poking around J help.
> >>> They could be useful for what I had in mind and modular inverses would
> be
> >>> part of that. Perhaps someone has those handy and could offer an
> addon? New
> >>> adverbs giving b m %.: a and m %.: a anyone?
> >>> Best, Cliff
> >>>
> >>> On Wed, Mar 29, 2023 at 5:02 PM 'Michael Day' via Programming <
> >>> programm...@jsoftware.com> wrote:
> >>>
> >>>> While this primitve works nicely in an example:
> >>>>
> >>>> (2 3 4) (17&|@*)/ table >:i.8
> >>>> +-------+---------------------+
> >>>> |17&|@*/|1 2 3 4 5 6 7 8|
> >>>> +-------+---------------------+
> >>>> |2 |2 4 6 8 10 12 14 16|
> >>>> |3 |3 6 9 12 15 1 4 7|
> >>>> |4 |4 8 12 16 3 7 11 15|
> >>>> +-------+---------------------+
> >>>>
> >>>> I find this less satisfying:
> >>>> (2 3 4) (17&|@%)/ table >:i.8
> >>>> +-------+-----------------------------------------------+
> >>>> |17&|@%/|1 2 3 4 5 6 7 8|
> >>>> +-------+-----------------------------------------------+
> >>>> |2 |2 1 0.666667 0.5 0.4 0.333333 0.285714 0.25|
> >>>> |3 |3 1.5 1 0.75 0.6 0.5 0.428571 0.375|
> >>>> |4 |4 2 1.33333 1 0.8 0.666667 0.571429 0.5|
> >>>> +-------+-----------------------------------------------+
> >>>>
> >>>> I have a function which does what one would expect. I'll rename it as
> >>>> m17div here, details unimportant for this discussion:
> >>>> (2 3 4) m17div/ table >:i.8
> >>>> +-------+---------------------+
> >>>> |m17div/|1 2 3 4 5 6 7 8|
> >>>> +-------+---------------------+
> >>>> |2 |2 1 12 9 14 6 10 13|
> >>>> |3 |3 10 1 5 4 9 15 11|
> >>>> |4 |4 2 7 1 11 12 3 9|
> >>>> +-------+---------------------+
> >>>> ( eg 3 % 2 == 10 mod 17 because 3 = 17 | 2 * 10 )
> >>>>
> >>>> Would anyone else find this return of integer results useful or is it
> >>>> better
> >>>> to force a floating output?
> >>>>
> >>>> (Henry tells me that m&|@^ returns integer results, working ok when
> m^2
> >>>> can be represented as a non-extended integer.)
> >>>>
> >>>> Thanks,
> >>>>
> >>>> Mike
> >>>>
> >>>> ----------------------------------------------------------------------
> >>>> For information about J forums seehttp://www.jsoftware.com/forums.htm
> >>>>
> >> ----------------------------------------------------------------------
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>
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