Release calc version 2.10.2t30

help/rcsq

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+NAME
+    rcsq - REDC squaring
+
+SYNOPSIS
+    rcsq(x, m)
+
+TYPES
+    x		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 rcsq(x,m) returns the value of B^-N * x^2 % 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 rcsq() may be said to be that of squaring modulo m a
+    value encoded by rcin() and REDC functions, as in:
+
+	    rcin(x^2, m) = rcsq(rcin(x,m), m)
+
+    from which we get:
+
+	    x^2 % m = rcout(rcsq(rcin(x,m), m), m)
+
+    Alternatively, x^2 % m may be evaluated usually more quickly by:
+
+	    x^2 % m = rcin(rcsq(x,m), m).
+
+RUNTIME
+    If the value of m in rcsq(x,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 rcsq(x, 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 0 <= x < m, the runtime
+    required for rcsq(x, m)i is at most about f times the runtime
+    required for an N-word by N-word multiplication, where f increases
+    from about 1.1 for N = 1 to near 2.8 for N > 90.  More runtime may
+    be required if x has to be reduced modulo m.
+
+EXAMPLE
+    Using a 64-bit machine with B = 2^32:
+
+    > for (i = 0; i < 9; i++) print rcsq(i,9),:; print;
+    0 7 1 0 4 4 0 1 7
+
+    > for (i = 0; i < 9; i++) print rcin((rcsq(i,9),:; print;
+    0 1 4 0 7 7 0 4 1
+
+LIMITS
+    none
+
+LIBRARY
+    void zredcsquare(REDC *rp, ZVALUE z1, ZVALUE *res)
+
+SEE ALSO
+    rcin, rcout, rcmul, rcpow