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    1 /////////////////////////////////////////////////////////////////////////////
    2 version="version ringgb.lib 4.2.0.0 Dec_2020 "; // $Id: aafb860223a7a67603e5b8a8ecb7d84f41373cbc $
    3 category="Miscellaneous";
    4 info="
    5 LIBRARY:  ringgb.lib     Functions for coefficient rings
    6 AUTHOR:  Oliver Wienand, email: wienand@mathematik.uni-kl.de
    7 
    8 KEYWORDS: vanishing polynomial; zeroreduce; polynomial functions; library, ringgb.lib; ringgb.lib, functions for coefficient rings
    9 
   10 PROCEDURES:
   11  findZeroPoly(f);        finds a vanishing polynomial for reducing f
   12  zeroReduce(f);          normal form of f concerning the ideal of vanishing polynomials
   13  testZero(poly f);       tests f defines the constant zero function
   14  noElements(def r);      the number of elements of the coefficient ring, if of type (integer, ...)
   15 ";
   16 
   17 LIB "general.lib";
   18 ///////////////////////////////////////////////////////////////////////////////
   19 
   20 proc findZeroPoly (poly f)
   21 "USAGE:   findZeroPoly(f); f - a polynomial
   22 RETURN:  zero polynomial with the same leading term as f if exists, otherwise 0
   23 EXAMPLE: example findZeroPoly; shows an example
   24 "
   25 {
   26   list data = getZeroCoef(f);
   27   if (data[1] == 0)
   28   {
   29     return(0);
   30   }
   31   number q = leadcoef(f) / data[1];
   32   if (q == 0)
   33   {
   34     return(0);
   35   }
   36   poly g = getZeroPolyRaw(data[2]);
   37   g = leadmonom(f) / leadmonom(g) * g;
   38   return(q * data[1] * g);
   39   //return(system("findZeroPoly", f));
   40 }
   41 example
   42 { "EXAMPLE:"; echo = 2;
   43   ring r = (integer, 65536), (y,x), dp;
   44   poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x;
   45   findZeroPoly(f);
   46 }
   47 
   48 proc zeroReduce(poly f, list #)
   49 "USAGE:   zeroReduce(f, [i = 0]); f - a polynomial, i - noise level (if != 0 prints all steps)
   50 RETURN:  reduced normal form of f modulo zero polynomials
   51 EXAMPLE: example zeroReduce; shows an example
   52 "
   53 {
   54    int i = 0;
   55    if (size(#) > 0)
   56    {
   57      i = #[1];
   58    }
   59    poly h = f;
   60    poly n = 0;
   61    poly g = findZeroPoly(h);
   62    if (i <> 0) {
   63      printf("reducing polyfct  : %s", h);
   64    }
   65    while ( h <> 0 ) {
   66       while ( g <> 0 ) {
   67          h = h - g;
   68          if (i <> 0) {
   69             printf(" reduce with: %s", g);
   70             printf(" to: %s", h);
   71          }
   72          g = findZeroPoly(h);
   73       }
   74       n = lead(h) + n;
   75       if (i <> 0) {
   76          printf("head irreducible  : %s", lead(h));
   77          printf("irreducible start : %s", n);
   78          printf("remains to check  : %s", h - lead(h));
   79       }
   80       h = h - lead(h);
   81       g = findZeroPoly(h);
   82    }
   83    return(n);
   84 }
   85 example
   86 { "EXAMPLE:"; echo = 2;
   87   ring r = (integer, 65536), (y,x), dp;
   88   poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x;
   89   zeroReduce(f);
   90   kill r;
   91   ring r = (integer, 2, 32), (x,y,z), dp;
   92   // Polynomial 1:
   93   poly p1 = 3795162112*x^3+587202566*x^2*y+2936012853*x*y*z+2281701376*x+548767119*y^3+16777216*y^2+268435456*y*z+1107296256*y+4244635648*z^3+4244635648*z^2+16777216*z;
   94   // Polynomial 2:
   95   poly p2 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z+1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z;
   96   zeroReduce(p1-p2);
   97 }
   98 
   99 proc testZero(poly f)
  100 "USAGE:   testZero(f); f - a polynomial
  101 RETURN:  returns 1 if f is zero as a function and otherwise a counterexample as a list [f(x_1, ..., x_n), x_1, ..., x_n]
  102 EXAMPLE: example testZero; shows an example
  103 "
  104 {
  105   poly g;
  106   int j;
  107   bigint i = 0;
  108   bigint modul = noElements(basering);
  109   printf("Teste %s Belegungen ...", modul^nvars(basering));
  110   for (; i < modul^nvars(basering); i = i + 1)
  111   {
  112     if ((i + 1) % modul^(nvars(basering) div 2) == 0)
  113     {
  114       printf("bisher: %s", i);
  115     }
  116     g = f;
  117     for (j = 1; j <= nvars(basering); j++)
  118     {
  119       g = subst(g, var(j), number((i / modul^(j-1)) % modul));
  120     }
  121     if (g != 0)
  122     {
  123       list counter = g;
  124       for (j = 1; j <= nvars(basering); j++)
  125       {
  126         counter = insert(counter, (i / modul^(j-1)) % modul);
  127       }
  128       return(counter);
  129     }
  130   }
  131   return(1);
  132 }
  133 example
  134 { "EXAMPLE:"; echo = 2;
  135   ring r = (integer, 12), (y,x), dp;
  136   poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x;
  137   //zeroReduce(f);
  138   testZero(f);
  139   poly g = findZeroPoly(x2y3);
  140   g;
  141   testZero(g);
  142 }
  143 
  144 proc noElements(def r)
  145 "USAGE:   noElements(r); r - a ring with a finite coefficient ring of type integer
  146 RETURN:  returns the number of elements of the coefficient ring of r
  147 EXAMPLE: example noElements; shows an example
  148 "
  149 
  150 {
  151   list l = ringlist(basering);
  152   return(l[1][2][1]^l[1][2][2]);
  153 }
  154 example
  155 { "EXAMPLE:"; echo = 2;
  156   ring r = (integer, 233,6), (y,x), dp;
  157   noElements(r);
  158 }
  159 
  160 static proc getZeroCoef(poly f)
  161 {
  162   if (f == 0)
  163   {
  164     return(0, leadexp(f))
  165   }
  166   list data = sort(leadexp(f));
  167   intvec exp = data[1];
  168   intvec index = data[2];
  169   intvec nec = 0:size(exp);
  170   int i = 1;
  171   int j = 2;
  172   bigint g;
  173   bigint G = 1;
  174   bigint modul = noElements(basering);
  175   bigint B = modul;
  176 
  177   for (; exp[i] < 2; i++) {if (i == size(exp)) break;}
  178   for (; i <= size(exp); i++)
  179   {
  180     g = gcd(B, G);
  181     G = G * g;
  182     B = B / g;
  183     if (g != 1)
  184     {
  185       nec[index[i]] = j - 1;
  186     }
  187     if (B == 1)
  188     {
  189       return(B, nec);
  190     }
  191     for (; j <= exp[i]; j++)
  192     {
  193       g = gcd(B, bigint(j));
  194       G = G * g;
  195       B = B / g;
  196       if (g != 1)
  197       {
  198         nec[index[i]] = j;
  199       }
  200       if (B == 1)
  201       {
  202         return(B, nec);
  203       }
  204     }
  205   }
  206   if (B == modul)
  207   {
  208     nec = 0;
  209     return(0, nec);
  210   }
  211   return(B, nec);
  212 }
  213 
  214 static proc getZeroPolyRaw(intvec fexp)
  215 {
  216   list data = sort(fexp);
  217   intvec exp = data[1];
  218   intvec index = data[2];
  219   int j = 0;
  220   poly res = 1;
  221   poly tillnow = 1;
  222   int i = 1;
  223   for (; exp[i] < 2; i++) {if (i == size(exp)) break;}
  224   for (; i <= size(exp); i++)
  225   {
  226     for (; j < exp[i]; j++)
  227     {
  228       tillnow = tillnow * (var(1) - j);
  229     }
  230     res = res * subst(tillnow, var(1), var(index[i]));
  231   }
  232   return(res);
  233 }
  234 
  235 static proc getZeroPoly(poly f)
  236 {
  237   list data = getZeroCoef(f);
  238   poly g = getZeroPolyRaw(data[2]);
  239   g = leadmonom(f) / leadmonom(g) * g;
  240   return(data[1] * g);
  241 }
  242 
  243 static proc findZeroPolyWrap (poly f)
  244 "USAGE:   findZeroPolyWrap(f); f - a polynomial
  245 RETURN:  zero polynomial with the same leading term as f if exists, otherwise 0
  246 NOTE:    just a wrapper, work only in Z/2^n with n < int_machine_size - 1
  247 EXAMPLE: example findZeroPoly; shows an example
  248 "
  249 {
  250   return(system("findZeroPoly", f));
  251 }
  252 example
  253 { "EXAMPLE:"; echo = 2;
  254   ring r = (integer, 2, 16), (y,x), dp;
  255   poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x;
  256   findZeroPoly(f);
  257 }
  258 
  259 ///////////////////////////////////////////////////////////////////////////////
  260 
  261 /*
  262                            Examples:
  263 
  264 
  265 // POLYNOMIAL EXAMPLES (Singular ready)
  266 // ===================
  267 //
  268 // For each of the examples below, there are three equivalent polynomials. 'm' indicates the bit-widths of the
  269 // input/output variables. For some of the polynomials, I have attached the RTL as well.
  270 //
  271 //
  272 // 1) VOLTERRA MODELS:
  273 //
  274 //        A) CUBIC FILTER: (m = 32, 3 Vars)
  275 
  276 LIB "ringgb.lib";
  277 ring r = (integer, 2, 32), (x,y,z), dp;
  278 poly p1 = 3795162112*x^3+587202566*x^2*y+2936012853*x*y*z+2281701376*x+548767119*y^3+16777216*y^2+268435456*y*z \
  279          +1107296256*y+4244635648*z^3+4244635648*z^2+16777216*z;
  280 poly p2 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z \
  281          +1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z;
  282 poly p3 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z \
  283          +1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z;
  284 zeroReduce(p1-p2);
  285 zeroReduce(p1-p3);
  286 zeroReduce(p2-p3);
  287 
  288 //        B) DEGREE-4 FILTER: (m=16 , 3 Vars)
  289 
  290 LIB "ringgb.lib";
  291 ring r = (integer, 2, 16), (x,y,z), dp;
  292 poly p1 = 16384*x^4+y^4+57344*z^4+64767*x*y^3+16127*y^2*z^2+8965*x^3*z+19275*x^2*y*z+51903*x*y*z+32768*x^2*y  \
  293          +40960*z^2+32768*x*y^2+49152*x^2+4869*y;
  294 poly p2 = 8965*x^3*z+19275*x^2*y*z+31999*x*y^3+51903*x*y*z+32768*x*y+y^4+32768*y^3+16127*y^2*z^2+32768*y^2 \
  295          +4869*y+57344*z^4+40960*z^2;
  296 poly p3 = 8965*x^3*z+19275*x^2*y*z+31999*x*y^3+51903*x*y*z+32768*x*y+y^4+16127*y^2*z^2+4869*y+16384*z^3+16384*z;
  297 zeroReduce(p1-p2);
  298 zeroReduce(p1-p3);
  299 zeroReduce(p2-p3);
  300 
  301 
  302 // 2) Savitzsky Golay filter(m=16,5 Vars)
  303 
  304 LIB "ringgb.lib";
  305 ring r = (integer, 2, 16), (v,w,x,y,z), dp;
  306 poly p1 = 25000*v^2*y+37322*v^2+22142*v*w*z+50356*w^3+58627*w^2+17797*w+17797*x^3+62500*x^2*z+41667*x \
  307          +22142*y^3+23870*y^2+59464*y+41667*z+58627;
  308 poly p2 = 25000*v^2*y+4554*v^2+22142*v*w*z+32768*v+17588*w^3+25859*w^2+17797*w+17797*x^3+29732*x^2*z+32768*x^2 \
  309          +32768*x*z+8899*x+22142*y^3+23870*y^2+59464*y+41667*z+58627;
  310 poly p3 = 25000*v^2*y+4554*v^2+22142*v*w*z+32768*v+17588*w^3+25859*w^2+17797*w+17797*x^3+29732*x^2*z+32768*x*z \
  311          +41667*x+22142*y^3+23870*y^2+59464*y+41667*z+58627;
  312 zeroReduce(p1-p2);
  313 zeroReduce(p1-p3);
  314 zeroReduce(p2-p3);
  315 
  316 
  317 // 3) Anti-alias filter:(m=16, 1 Var)
  318 
  319 LIB "ringgb.lib";
  320 ring r = (integer, 2, 16), c, dp;
  321 poly p1 = 156*c^6+62724*c^5+17968*c^4+18661*c^3+43593*c^2+40224*c+13281;
  322 poly p2 = 156*c^6+5380*c^5+1584*c^4+43237*c^3+27209*c^2+40224*c+13281;
  323 poly p3 = 156*c^6+5380*c^5+1584*c^4+10469*c^3+27209*c^2+7456*c+13281;
  324 zeroReduce(p1-p2);
  325 zeroReduce(p1-p3);
  326 zeroReduce(p2-p3);
  327 
  328 
  329 // 4) PSK:(m=16, 2 Var)
  330 
  331 LIB "ringgb.lib";
  332 ring r = (integer, 2, 16), (x,y), dp;
  333 poly p1 = 4166*x^4+16666*x^3*y+25000*x^2*y^2+15536*x^2+16666*x*y^4+31072*x*y+4166*y^4+15536*y^2+34464;
  334 poly p2 = 4166*x^4+16666*x^3*y+8616*x^2*y^2+16384*x^2*y+15536*x^2+282*x*y^4+47456*x*y+53318*y^4+31920*y^2+34464;
  335 poly p3 = 4166*x^4+16666*x^3*y+8616*x^2*y^2+16384*x^2*y+15536*x^2+282*x*y^4+47456*x*y+4166*y^4+15536*y^2+34464;
  336 zeroReduce(p1-p2);
  337 zeroReduce(p1-p3);
  338 zeroReduce(p2-p3);
  339 
  340 // Ref: A. Peymandoust G. De Micheli, "Application of Symbolic Computer Algebra in High-Level Data-Flow
  341 // Synthesis," IEEE Transactions on CAD/ICAS, Vol. 22, No. 9, September 2003, pp.1154-1165.
  342 
  343 */