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74 lines
2.3 KiB
Plaintext
74 lines
2.3 KiB
Plaintext
NAME
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rcpow - REDC powers
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SYNOPSIS
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rcpow(x, k, m)
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TYPES
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x integer
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k nonnegative 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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rcpow(x,k,m) returns the value of B^-N * (B^N * x)^k % m, w here
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the inverse implicit in B^-N is modulo m and the modulus operator %
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gives the least nonnegative residue. Note that rcpow(x,0,m) =
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rcin(1,m), rcpow(x,1,m) = x % m; rcpow(x,2,m) = rcsq(x,m).
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The normal use of rcpow() may be said to be that of finding the
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encoded value of the k-th power of an integer modulo m:
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rcin(x^k, m) = rcpow(rcin(x,m), k, m),
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from which one gets:
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x^k % m = rcout(rcpow(rcin(x,m), k, m), m).
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If x^k % m is to be evaluated for the same k and m and several
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values of x, it may be worth while to first evaluate:
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a = minv(rcpow(1, k, m), m);
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and use:
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x^k % m = a * rcpow(x, k, m) % m.
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RUNTIME
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If the value of m in rcpow(x,k,m) is being used for the first time
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in a REDC function, the information required for the REDC
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algorithms is calculated and stored for future use, possibly
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replacing an already stored valued, in a table covering up to 5
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(i.e. MAXREDC) values of m. The runtime required for this is about
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two times that required for multiplying two N-word integers.
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Two algorithms are available for evaluating rcpow(x,k,m), the one
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which is usually faster for small N is used when N <
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config("redc2"); the other is usually faster for larger N. If
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config("redc2") is set at about 90 and 0 <= x < m, the runtime
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required for rcpow(x,k,m) is at most about f times the runtime
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required for ilog2(k) N-word by N-word multiplications, where f
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increases from about 1.3 for N = 1 to near 4 for N > 90. More
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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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> m = 1234567;
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> x = 15;
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> print rcout(rcpow((rcin(x,m), m - 1, m), m), pmod(x, m-1, m)
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783084 783084
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LIMITS
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none
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LIBRARY
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void zredcpower(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)
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SEE ALSO
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rcin, rcout, rcmul, rcsq
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