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<TITLE>CLN, a Class Library for Numbers - 7. Modular integers</TITLE></HEAD><BODY>Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_6.html">previous</A>, <A HREF="cln_8.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.<P><HR><P>
<H1><A NAME="SEC49" HREF="cln_toc.html#TOC49">7. Modular integers</A></H1><P><A NAME="IDX240"></A>
<H2><A NAME="SEC50" HREF="cln_toc.html#TOC50">7.1 Modular integer rings</A></H2><P><A NAME="IDX241"></A>
<P>CLN implements modular integers, i.e. integers modulo a fixed integer N.The modulus is explicitly part of every modular integer. CLN doesn'tallow you to (accidentally) mix elements of different modular rings,e.g. <CODE>(3 mod 4) + (2 mod 5)</CODE> will result in a runtime error.(Ideally one would imagine a generic data type <CODE>cl_MI(N)</CODE>, but C++doesn't have generic types. So one has to live with runtime checks.)
<P>The class of modular integer rings is
<PRE> Ring cl_ring <cl_ring.h> | | Modular integer ring cl_modint_ring <cl_modinteger.h></PRE>
<P><A NAME="IDX242"></A>
<P>and the class of all modular integers (elements of modular integer rings) is
<PRE> Modular integer cl_MI <cl_modinteger.h></PRE>
<P>Modular integer rings are constructed using the function
<DL COMPACT>
<DT><CODE>cl_modint_ring cl_find_modint_ring (const cl_I& N)</CODE><DD><A NAME="IDX243"></A>This function returns the modular ring <SAMP>`Z/NZ'</SAMP>. It takes careof finding out about special cases of <CODE>N</CODE>, like powers of twoand odd numbers for which Montgomery multiplication will be a win,<A NAME="IDX244"></A>and precomputes any necessary auxiliary data for computing modulo <CODE>N</CODE>.There is a cache table of rings, indexed by <CODE>N</CODE> (or, more precisely,by <CODE>abs(N)</CODE>). This ensures that the precomputation costs are reducedto a minimum.</DL>
<P>Modular integer rings can be compared for equality:
<DL COMPACT>
<DT><CODE>bool operator== (const cl_modint_ring&, const cl_modint_ring&)</CODE><DD><A NAME="IDX245"></A><DT><CODE>bool operator!= (const cl_modint_ring&, const cl_modint_ring&)</CODE><DD><A NAME="IDX246"></A>These compare two modular integer rings for equality. Two different callsto <CODE>cl_find_modint_ring</CODE> with the same argument necessarily return thesame ring because it is memoized in the cache table.</DL>
<H2><A NAME="SEC51" HREF="cln_toc.html#TOC51">7.2 Functions on modular integers</A></H2>
<P>Given a modular integer ring <CODE>R</CODE>, the following members can be used.
<DL COMPACT>
<DT><CODE>cl_I R->modulus</CODE><DD><A NAME="IDX247"></A>This is the ring's modulus, normalized to be nonnegative: <CODE>abs(N)</CODE>.
<DT><CODE>cl_MI R->zero()</CODE><DD><A NAME="IDX248"></A>This returns <CODE>0 mod N</CODE>.
<DT><CODE>cl_MI R->one()</CODE><DD><A NAME="IDX249"></A>This returns <CODE>1 mod N</CODE>.
<DT><CODE>cl_MI R->canonhom (const cl_I& x)</CODE><DD><A NAME="IDX250"></A>This returns <CODE>x mod N</CODE>.
<DT><CODE>cl_I R->retract (const cl_MI& x)</CODE><DD><A NAME="IDX251"></A>This is a partial inverse function to <CODE>R->canonhom</CODE>. It returns thestandard representative (<CODE>>=0</CODE>, <CODE><N</CODE>) of <CODE>x</CODE>.
<DT><CODE>cl_MI R->random(cl_random_state& randomstate)</CODE><DD><DT><CODE>cl_MI R->random()</CODE><DD><A NAME="IDX252"></A>This returns a random integer modulo <CODE>N</CODE>.</DL>
<P>The following operations are defined on modular integers.
<DL COMPACT>
<DT><CODE>cl_modint_ring x.ring ()</CODE><DD><A NAME="IDX253"></A>Returns the ring to which the modular integer <CODE>x</CODE> belongs.
<DT><CODE>cl_MI operator+ (const cl_MI&, const cl_MI&)</CODE><DD><A NAME="IDX254"></A>Returns the sum of two modular integers. One of the arguments may also bea plain integer.
<DT><CODE>cl_MI operator- (const cl_MI&, const cl_MI&)</CODE><DD><A NAME="IDX255"></A>Returns the difference of two modular integers. One of the arguments may also bea plain integer.
<DT><CODE>cl_MI operator- (const cl_MI&)</CODE><DD>Returns the negative of a modular integer.
<DT><CODE>cl_MI operator* (const cl_MI&, const cl_MI&)</CODE><DD><A NAME="IDX256"></A>Returns the product of two modular integers. One of the arguments may also bea plain integer.
<DT><CODE>cl_MI square (const cl_MI&)</CODE><DD><A NAME="IDX257"></A>Returns the square of a modular integer.
<DT><CODE>cl_MI recip (const cl_MI& x)</CODE><DD><A NAME="IDX258"></A>Returns the reciprocal <CODE>x^-1</CODE> of a modular integer <CODE>x</CODE>. <CODE>x</CODE>must be coprime to the modulus, otherwise an error message is issued.
<DT><CODE>cl_MI div (const cl_MI& x, const cl_MI& y)</CODE><DD><A NAME="IDX259"></A>Returns the quotient <CODE>x*y^-1</CODE> of two modular integers <CODE>x</CODE>, <CODE>y</CODE>.<CODE>y</CODE> must be coprime to the modulus, otherwise an error message is issued.
<DT><CODE>cl_MI expt_pos (const cl_MI& x, const cl_I& y)</CODE><DD><A NAME="IDX260"></A><CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
<DT><CODE>cl_MI expt (const cl_MI& x, const cl_I& y)</CODE><DD><A NAME="IDX261"></A>Returns <CODE>x^y</CODE>. If <CODE>y</CODE> is negative, <CODE>x</CODE> must be coprime to themodulus, else an error message is issued.
<DT><CODE>cl_MI operator<< (const cl_MI& x, const cl_I& y)</CODE><DD><A NAME="IDX262"></A>Returns <CODE>x*2^y</CODE>.
<DT><CODE>cl_MI operator>> (const cl_MI& x, const cl_I& y)</CODE><DD><A NAME="IDX263"></A>Returns <CODE>x*2^-y</CODE>. When <CODE>y</CODE> is positive, the modulus must be odd,or an error message is issued.
<DT><CODE>bool operator== (const cl_MI&, const cl_MI&)</CODE><DD><A NAME="IDX264"></A><DT><CODE>bool operator!= (const cl_MI&, const cl_MI&)</CODE><DD><A NAME="IDX265"></A>Compares two modular integers, belonging to the same modular integer ring,for equality.
<DT><CODE>cl_boolean zerop (const cl_MI& x)</CODE><DD><A NAME="IDX266"></A>Returns true if <CODE>x</CODE> is <CODE>0 mod N</CODE>.</DL>
<P>The following output functions are defined (see also the chapter oninput/output).
<DL COMPACT>
<DT><CODE>void fprint (cl_ostream stream, const cl_MI& x)</CODE><DD><A NAME="IDX267"></A><DT><CODE>cl_ostream operator<< (cl_ostream stream, const cl_MI& x)</CODE><DD><A NAME="IDX268"></A>Prints the modular integer <CODE>x</CODE> on the <CODE>stream</CODE>. The output may dependon the global printer settings in the variable <CODE>cl_default_print_flags</CODE>.</DL>
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