mirror of
https://github.com/lcn2/calc.git
synced 2025-08-16 01:03:29 +03:00
63 lines
2.0 KiB
Plaintext
63 lines
2.0 KiB
Plaintext
NAME
|
|
rcmul - REDC multiplication
|
|
|
|
SYNOPSIS
|
|
rcmul(x, y, m)
|
|
|
|
TYPES
|
|
x integer
|
|
y integer
|
|
m odd positive integer
|
|
|
|
return integer v, 0 <= v < m.
|
|
|
|
DESCRIPTION
|
|
Let B be the base calc uses for representing integers internally
|
|
(B = 2^16 for 32-bit machines, 2^32 for 64-bit machines)
|
|
and N the number of words (base-B digits) in the representation
|
|
of m. Then rcmul(x,y,m) returns the value of B^-N * x * y % m,
|
|
where the inverse implicit in B^-N is modulo m
|
|
and the modulus operator % gives the least non-negative residue.
|
|
|
|
The normal use of rcmul() may be said to be that of multiplying modulo m
|
|
values encoded by rcin() and REDC functions, as in:
|
|
|
|
rcin(x * y, m) = rcmul(rcin(x,m), rcin(y,m), m),
|
|
|
|
or with only one factor encoded:
|
|
|
|
x * y % m = rcmul(rcin(x,m), y, m).
|
|
|
|
RUNTIME
|
|
If the value of m in rcmul(x,y,m) is being used for the first time
|
|
in a REDC function, the information required for the REDC
|
|
algorithms is calculated and stored for future use, possibly
|
|
replacing an already stored valued, in a table covering up to 5
|
|
(i.e. MAXREDC) values of m. The runtime required for this is about
|
|
two times that required for multiplying two N-word integers.
|
|
|
|
Two algorithms are available for evaluating rcmul(x,y,m), the one
|
|
which is usually faster for small N is used when N <
|
|
config("redc2"); the other is usually faster for larger N. If
|
|
config("redc2") is set at about 90 and x and y have both been
|
|
reduced modulo m, the runtime required for rcmul(x,y,m) is at most
|
|
about f times the runtime required for an N-word by N-word
|
|
multiplication, where f increases from about 1.3 for N = 1 to near
|
|
3 for N > 90. More runtime may be required if x and y have to be
|
|
reduced modulo m.
|
|
|
|
EXAMPLE
|
|
Using a 64-bit machine with B = 2^32:
|
|
|
|
> print rcin(4 * 5, 9), rcmul(rcin(4,9), rcin(5,9), 9), rcout(8, 9);
|
|
8 8 2
|
|
|
|
LIMITS
|
|
none
|
|
|
|
LIBRARY
|
|
void zredcmul(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)
|
|
|
|
SEE ALSO
|
|
rcin, rcout, rcsq, rcpow
|