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74 lines
2.5 KiB
Plaintext
74 lines
2.5 KiB
Plaintext
NAME
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rcin - encode for REDC algorithms
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SYNOPSIS
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rcin(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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rcin(x,m) returns the value of B^N * x % m, where the modulus
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operator % here gives the least nonnegative residue.
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If y = rcin(x,m), x % m may be evaluated by x % m = rcout(y, m).
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The "encoding" method of using rcmul(), rcsq(), and rcpow() for
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evaluating products, squares and powers modulo m correspond to the
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formulae:
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rcin(x * y, m) = rcmul(rcin(x,m), rcin(y,m), m);
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rcin(x^2, m) = rcsq(rcin(x,m), m);
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rcin(x^k, m) = rcpow(rcin(x,m), k, m).
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Here k is any nonnegative integer. Using these formulae may be
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faster than direct evaluation of x * y % m, x^2 % m, x^k % m.
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Some encoding and decoding may be bypassed by formulae like:
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x * y % m = rcin(rcmul(x, y, m), m).
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If m is a divisor of B^N - h for some integer h, rcin(x,m) may be
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computed by using rcin(x,m) = h * x % m. In particular, if
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m is a divisor of B^N - 1 and 0 <= x < m, then rcin(x,m) = x.
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For example if B = 2^16 or 2^32, this is so for m = (B^N - 1)/d
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for the divisors d = 3, 5, 15, 17, ...
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RUNTIME
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The first time a particular value for m is used in rcin(x, m),
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the information required for the REDC algorithms is
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calculated and stored for future use in a table covering up to
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5 (i.e. MAXREDC) values of m. The runtime required for this is about
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two that required for multiplying two N-word integers.
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Two algorithms are available for evaluating rcin(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 both been reduced
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modulo m, the runtime required for rcin(x, m) is at most about f
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times the runtime required for an N-word by N-word multiplication,
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where f increases from about 1.3 for N = 1 to near 2 for N > 200.
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More runtime may be required if x has to be 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 rcin(x, 9),:; print;
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0 4 8 3 7 2 6 1 5
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LIMITS
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none
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LIBRARY
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void zredcencode(REDC *rp, ZVALUE z1, ZVALUE *res)
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SEE ALSO
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rcout, rcmul, rcsq, rcpow
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