Release calc version 2.10.2t30

help/rcpow

@@ -0,0 +1,73 @@
+NAME
+    rcpow - REDC powers
+
+SYNOPSIS
+    rcpow(x, k, m)
+
+TYPES
+    x		integer
+    k		nonnegative 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
+    rcpow(x,k,m) returns the value of B^-N * (B^N * x)^k % m, w here
+    the inverse implicit in B^-N is modulo m and the modulus operator %
+    gives the least nonnegative residue.  Note that rcpow(x,0,m) =
+    rcin(1,m), rcpow(x,1,m) = x % m; rcpow(x,2,m) = rcsq(x,m).
+
+    The normal use of rcpow() may be said to be that of finding the
+    encoded value of the k-th power of an integer modulo m:
+
+	    rcin(x^k, m) = rcpow(rcin(x,m), k, m),
+
+    from which one gets:
+
+	    x^k % m  = rcout(rcpow(rcin(x,m), k, m), m).
+
+    If x^k % m is to be evaluated for the same k and m and several
+    values of x, it may be worth while to first evaluate:
+
+	    a = minv(rcpow(1, k, m), m);
+
+    and use:
+
+	x^k % m = a * rcpow(x, k, m) % m.
+
+RUNTIME
+    If the value of m in rcpow(x,k,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 rcpow(x,k,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 rcpow(x,k,m) is at most about f times the runtime
+    required for ilog2(k) N-word by N-word multiplications, where f
+    increases from about 1.3 for N = 1 to near 4 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:
+
+    > m = 1234567;
+    > x = 15;
+    > print rcout(rcpow((rcin(x,m), m - 1, m), m), pmod(x, m-1, m)
+    783084 783084
+
+LIMITS
+    none
+
+LIBRARY
+    void zredcpower(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)
+
+SEE ALSO
+    rcin, rcout, rcmul, rcsq