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65 lines
2.1 KiB
Plaintext
65 lines
2.1 KiB
Plaintext
NAME
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rcout - decode for REDC algorithms
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SYNOPSIS
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rcout(x, m)
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TYPES
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x integer
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m odd positive integer
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return integer v, 0 <= v < m.
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DESCRIPTION
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Let B be the base calc uses for representing integers internally
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(B = 2^16 for 32-bit machines, 2^32 for 64-bit machines) and N the
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number of words (base-B digits) in the representation of m. Then
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rcout(x,m) returns the value of B^-N * x % m, where the inverse
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implicit in B^-N is modulo m and the modulus operator % gives the
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least non-negative residue. The functions rcin() and rcout() are
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inverses of each other for all x:
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rcout(rcin(x,m), m) = rcin(rcout(x,m),m) = x % m.
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The normal use of rcout() may be said to be that of decoding
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values encoded by rcin() and REDC functions, as in:
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x * y % m = rcout(rcmul(rcin(x,m),rcin(y,m),m),m),
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x^2 % m = rcout(rcsq(rcin(x,m),m),m),
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x ^ k % m = rcout(rcpow(rcin(x,m),k,m),m).
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RUNTIME
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If the value of m in rcout(x,m) is being used for the first time in
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a REDC function, the information required for the REDC algorithms
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is calculated and stored for future use, possibly replacing an
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already stored valued, in a table covering up to 5 (i.e. MAXREDC)
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values of m. The runtime required for this is about two times that
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required for multiplying two N-word integers.
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Two algorithms are available for evaluating rcout(x, m), the one
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which is usually faster for small N is used when N <
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config("pow2"); the other is usually faster for larger N. If
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config("pow2") is set at about 200, and x has been reduced modulo
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m, the runtime required for rcout(x, m) is at most about f times
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the runtime required for an N-word by N-word multiplication, where
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f increases from about 1 for N = 1 to near 2 for N >
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config("pow2"). More runtime may be required if x has to be
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reduced modulo m.
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EXAMPLE
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Using a 64-bit machine with B = 2^32:
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> for (i = 0; i < 9; i++) print rcout(i,9),:; print;
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0 7 5 3 1 8 6 4 2
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LIMITS
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none
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LIBRARY
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void zredcdecode(REDC *rp, ZVALUE z1, ZVALUE *res)
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SEE ALSO
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rcout, rcmul, rcsq, rcpow
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