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9.9 KiB
9.9 KiB
section 2eW, lite number theory
++egcd
GCD
++ egcd :: schneier's egcd
|= [a=@ b=@]
=+ si
=+ [c=(sun a) d=(sun b)]
=+ [u=[c=(sun 1) d=--0] v=[c=--0 d=(sun 1)]]
|- ^- [d=@ u=@ v=@]
?: =(--0 c)
[(abs d) d.u d.v]
:: ?> ?& =(c (sum (pro (sun a) c.u) (pro (sun b) c.v)))
:: =(d (sum (pro (sun a) d.u) (pro (sun b) d.v)))
:: ==
=+ q=(fra d c)
%= $
c (dif d (pro q c))
d c
u [(dif d.u (pro q c.u)) c.u]
v [(dif d.v (pro q c.v)) c.v]
==
::
Greatest common denominator
~zod/try=> (egcd 20 15)
[d=5 u=2 v=1]
~zod/try=> (egcd 24 16)
[d=8 u=2 v=1]
~zod/try=> (egcd 7 5)
[d=1 u=3 v=6]
~zod/try=> (egcd (shaf ~ %ham) (shaf ~ %sam))
[ d=1
u=59.983.396.314.566.203.239.184.568.129.921.874.787
v=38.716.650.351.034.402.960.165.718.823.532.275.722
]
++pram
Probable prime
++ pram :: rabin-miller
|= a=@ ^- ?
?: ?| =(0 (end 0 1 a))
=(1 a)
=+ b=1
|- ^- ?
?: =(512 b)
|
?|(=+(c=+((mul 2 b)) &(!=(a c) =(a (mul c (div a c))))) $(b +(b)))
==
|
=+ ^= b
=+ [s=(dec a) t=0]
|- ^- [s=@ t=@]
?: =(0 (end 0 1 s))
$(s (rsh 0 1 s), t +(t))
[s t]
?> =((mul s.b (bex t.b)) (dec a))
=+ c=0
|- ^- ?
?: =(c 64)
&
=+ d=(~(raw og (add c a)) (met 0 a))
=+ e=(~(exp fo a) s.b d)
?& ?| =(1 e)
=+ f=0
|- ^- ?
?: =(e (dec a))
&
?: =(f (dec t.b))
|
$(e (~(pro fo a) e e), f +(f))
==
$(c +(c))
==
::
Probable prime test
~zod/try=> (pram 31)
%.y
~zod/try=> =+(a=2 |-(?:(=(a 31) ~ [i=(mod 31 a) t=$(a +(a))])))
~[1 1 3 1 1 3 7 4 1 9 7 5 3 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1]
~zod/try=> =+(a=2 |-(?:(=(a 31) ~ [i=(mod 30 a) t=$(a +(a))])))
~[0 0 2 0 0 2 6 3 0 8 6 4 2 0 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0]
~zod/try=> (pram 256)
%.n
~zod/try=> (pram (dec (bex 127)))
%.y
++ramp
r-m prime
++ ramp :: make r-m prime
|= [a=@ b=(list ,@) c=@] ^- @ux :: [bits snags seed]
=> .(c (shas %ramp c))
=+ d=_@
|-
?: =((mul 100 a) d)
~|(%ar-ramp !!)
=+ e=(~(raw og c) a)
?: &((levy b |=(f=@ !=(1 (mod e f)))) (pram e))
e
$(c +(c), d (shax d))
::
Random a bit prime, which isn't 1 modulo a list of other numbers,
using salt c.
~zod/try=> (ramp 20 ~ %hamelok)
0xf.1f0d
~zod/try=> (ramp 20 ~ %hameloe)
0x2.d341
~zod/try=> (ramp 5 ~ %kole)
0x1f
~zod/try=> (ramp 7 ~ %kole)
0x4f
~zod/try=> (ramp 7 ~[0x4e] %kole)
0x43
~zod/try=> `@uw`(ramp 128 ~ %late)
0w3y.irKIL.l-pp1.2CkG4.3lsTF
++fo
Prime engine
++ fo :: modulo prime
|_ a=@
XX DO NOT RERUN GET.LS, THERE EXIST ARM COLLISIONS
Core for performing arithmetic modulo a prime number
~zod/try=> ~(. fo 79)
<7.get [@ud <373.jdd 100.kzl 1.ypj %164>]>
++dif
Difference
++ dif
|= [b=@ c=@]
(sit (sub (add a b) (sit c)))
::
Subtract
~zod/try=> (~(dif fo 79) 10 5)
5
~zod/try=> (~(dif fo 79) 5 10)
74
++exp
Exponent
++ exp
|= [b=@ c=@]
?: =(0 b)
1
=+ d=$(b (rsh 0 1 b))
=+ e=(pro d d)
?:(=(0 (end 0 1 b)) e (pro c e))
::
Exponent
~zod/try=> (~(exp fo 79) 3 5)
46
++fra
Divide
++ fra
|= [b=@ c=@]
(pro b (inv c))
::
Divide
~zod/try=> (~(fra fo 79) 20 4)
5
~zod/try=> (~(fra fo 79) 7 11)
15
++inv
Inverse
++ inv
|= b=@
=+ c=(dul:si u:(egcd b a) a)
c
::
Multiplicative inverse
~zod/try=> (~(inv fo 79) 12)
33
~zod/try=> (~(pro fo 79) 12 33)
1
~zod/try=> (~(inv fo 79) 0)
0
++pro
Product
++ pro
|= [b=@ c=@]
(sit (mul b c))
::
Product
~zod/try=> (~(pro fo 79) 5 10)
50
~zod/try=> (~(pro fo 79) 5 20)
21
++sit
Bounds
++ sit
|= b=@
(mod b a)
::
Bounds check
~zod/try=> (~(sit fo 79) 9)
9
~zod/try=> (~(sit fo 79) 99)
20
++sum
Sum
++ sum
|= [b=@ c=@]
(sit (add b c))
--
Add
~zod/try=> (~(sum fo 79) 9 9)
18
~zod/try=> (~(sum fo 79) 70 9)
0
++ga
++ ga :: GF (bex p.a)
|= a=[p=@ q=@ r=@] :: dim poly gen
=+ si=(bex p.a)
=+ ma=(dec si)
=> |%
RSA internals
XX document
++dif
++ dif :: add and sub
|= [b=@ c=@]
~| [%dif-ga a]
?> &((lth b si) (lth c si))
(mix b c)
::
XX document
++dub
++ dub :: mul by x
|= b=@
~| [%dub-ga a]
?> (lth b si)
?: =(1 (cut 0 [(dec p.a) 1] b))
(dif (sit q.a) (sit (lsh 0 1 b)))
(lsh 0 1 b)
::
XX document
++pro
++ pro :: slow multiply
|= [b=@ c=@]
?: =(0 b)
0
?: =(1 (dis 1 b))
(dif c $(b (rsh 0 1 b), c (dub c)))
$(b (rsh 0 1 b), c (dub c))
::
XX document
++toe
++ toe :: exp/log tables
=+ ^= nu
|= [b=@ c=@]
^- (map ,@ ,@)
=+ d=*(map ,@ ,@)
|-
?: =(0 c)
d
%= $
c (dec c)
d (~(put by d) c b)
==
=+ [p=(nu 0 (bex p.a)) q=(nu ma ma)]
=+ [b=1 c=0]
|- ^- [p=(map ,@ ,@) q=(map ,@ ,@)]
?: =(ma c)
[(~(put by p) c b) q]
%= $
b (pro r.a b)
c +(c)
p (~(put by p) c b)
q (~(put by q) b c)
==
::
XX document
++sit
++ sit :: reduce
|= b=@
(mod b (bex p.a))
--
XX document
++fra
++ fra :: divide
|= [b=@ c=@]
(pro b (inv c))
::
XX document
++inv
++ inv :: invert
|= b=@
~| [%inv-ga a]
=+ c=(~(get by q) b)
?~ c !!
=+ d=(~(get by p) (sub ma u.c))
(need d)
::
XX document
++pow
++ pow :: exponent
|= [b=@ c=@]
=+ [d=1 e=c f=0]
|-
?: =(p.a f)
d
?: =(1 (cut 0 [f 1] b))
$(d (pro d e), e (pro e e), f +(f))
$(e (pro e e), f +(f))
::
XX document
++pro
++ pro :: multiply
|= [b=@ c=@]
~| [%pro-ga a]
=+ d=(~(get by q) b)
?~ d 0
=+ e=(~(get by q) c)
?~ e 0
=+ f=(~(get by p) (mod (add u.d u.e) ma))
(need f)
--
XX document