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68 lines
2.1 KiB
Plaintext
68 lines
2.1 KiB
Plaintext
NAME
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rcsq - REDC squaring
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SYNOPSIS
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rcsq(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)
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and N the number of words (base-B digits) in the representation
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of m. Then rcsq(x,m) returns the value of B^-N * x^2 % m,
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where the inverse implicit in B^-N is modulo m
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and the modulus operator % gives the least non-negative residue.
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The normal use of rcsq() may be said to be that of squaring modulo m a
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value encoded by rcin() and REDC functions, as in:
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rcin(x^2, m) = rcsq(rcin(x,m), m)
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from which we get:
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x^2 % m = rcout(rcsq(rcin(x,m), m), m)
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Alternatively, x^2 % m may be evaluated usually more quickly by:
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x^2 % m = rcin(rcsq(x,m), m).
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RUNTIME
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If the value of m in rcsq(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 rcsq(x, 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 rcsq(x, m)i is at most about f times the runtime
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required for an N-word by N-word multiplication, where f increases
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from about 1.1 for N = 1 to near 2.8 for N > 90. More runtime may
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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 rcsq(i,9),:; print;
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0 7 1 0 4 4 0 1 7
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> for (i = 0; i < 9; i++) print rcin((rcsq(i,9),:; print;
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0 1 4 0 7 7 0 4 1
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LIMITS
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none
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LIBRARY
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void zredcsquare(REDC *rp, ZVALUE z1, ZVALUE *res)
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SEE ALSO
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rcin, rcout, rcmul, rcpow
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