/**************************************************************************\ MODULE: zz_pX SUMMARY: The class zz_pX implements polynomial arithmetic modulo p. Polynomial arithmetic is implemented using a combination of classical routines, Karatsuba, and FFT. \**************************************************************************/ #include "zz_p.h" #include "vec_zz_p.h" class zz_pX { public: zz_pX(); // initial value 0 zz_pX(const zz_pX& a); // copy zz_pX& operator=(const zz_pX& a); // assignment zz_pX& operator=(zz_p a); zz_pX& operator=(long a); ~zz_pX(); // destructor zz_pX(long i, zz_p c); // initialize to X^i*c zz_pX(long i, long c); }; /**************************************************************************\ Comparison \**************************************************************************/ long operator==(const zz_pX& a, const zz_pX& b); long operator!=(const zz_pX& a, const zz_pX& b); long IsZero(const zz_pX& a); // test for 0 long IsOne(const zz_pX& a); // test for 1 // PROMOTIONS: operators ==, != promote {long, zz_p} to zz_pX on (a, b) /**************************************************************************\ Addition \**************************************************************************/ // operator notation: zz_pX operator+(const zz_pX& a, const zz_pX& b); zz_pX operator-(const zz_pX& a, const zz_pX& b); zz_pX operator-(const zz_pX& a); // unary - zz_pX& operator+=(zz_pX& x, const zz_pX& a); zz_pX& operator+=(zz_pX& x, zz_p a); zz_pX& operator+=(zz_pX& x, long a); zz_pX& operator-=(zz_pX& x, const zz_pX& a); zz_pX& operator-=(zz_pX& x, zz_p a); zz_pX& operator-=(zz_pX& x, long a); zz_pX& operator++(zz_pX& x); // prefix void operator++(zz_pX& x, int); // postfix zz_pX& operator--(zz_pX& x); // prefix void operator--(zz_pX& x, int); // postfix // procedural versions: void add(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a + b void sub(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a - b void negate(zz_pX& x, const zz_pX& a); // x = -a // PROMOTIONS: binary +, - and procedures add, sub promote {long, zz_p} // to zz_pX on (a, b). /**************************************************************************\ Multiplication \**************************************************************************/ // operator notation: zz_pX operator*(const zz_pX& a, const zz_pX& b); zz_pX& operator*=(zz_pX& x, const zz_pX& a); zz_pX& operator*=(zz_pX& x, zz_p a); zz_pX& operator*=(zz_pX& x, long a); // procedural versions: void mul(zz_pX& x, const zz_pX& a, const zz_pX& b); // x = a * b void sqr(zz_pX& x, const zz_pX& a); // x = a^2 zz_pX sqr(const zz_pX& a); // PROMOTIONS: operator * and procedure mul promote {long, zz_p} to zz_pX // on (a, b). void power(zz_pX& x, const zz_pX& a, long e); // x = a^e (e >= 0) zz_pX power(const zz_pX& a, long e); /**************************************************************************\ Shift Operations LeftShift by n means multiplication by X^n RightShift by n means division by X^n A negative shift amount reverses the direction of the shift. \**************************************************************************/ // operator notation: zz_pX operator<<(const zz_pX& a, long n); zz_pX operator>>(const zz_pX& a, long n); zz_pX& operator<<=(zz_pX& x, long n); zz_pX& operator>>=(zz_pX& x, long n); // procedural versions: void LeftShift(zz_pX& x, const zz_pX& a, long n); zz_pX LeftShift(const zz_pX& a, long n); void RightShift(zz_pX& x, const zz_pX& a, long n); zz_pX RightShift(const zz_pX& a, long n); /**************************************************************************\ Division \**************************************************************************/ // operator notation: zz_pX operator/(const zz_pX& a, const zz_pX& b); zz_pX operator%(const zz_pX& a, const zz_pX& b); zz_pX& operator/=(zz_pX& x, const zz_pX& a); zz_pX& operator/=(zz_pX& x, zz_p a); zz_pX& operator/=(zz_pX& x, long a); zz_pX& operator%=(zz_pX& x, const zz_pX& b); // procedural versions: void DivRem(zz_pX& q, zz_pX& r, const zz_pX& a, const zz_pX& b); // q = a/b, r = a%b void div(zz_pX& q, const zz_pX& a, const zz_pX& b); // q = a/b void rem(zz_pX& r, const zz_pX& a, const zz_pX& b); // r = a%b long divide(zz_pX& q, const zz_pX& a, const zz_pX& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 long divide(const zz_pX& a, const zz_pX& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 // PROMOTIONS: operator / and procedure div promote {long, zz_p} to zz_pX // on (a, b). /**************************************************************************\ GCD's These routines are intended for use when p is prime. \**************************************************************************/ void GCD(zz_pX& x, const zz_pX& a, const zz_pX& b); zz_pX GCD(const zz_pX& a, const zz_pX& b); // x = GCD(a, b), x is always monic (or zero if a==b==0). void XGCD(zz_pX& d, zz_pX& s, zz_pX& t, const zz_pX& a, const zz_pX& b); // d = gcd(a,b), a s + b t = d // NOTE: A classical algorithm is used, switching over to a // "half-GCD" algorithm for large degree /**************************************************************************\ Input/Output I/O format: [a_0 a_1 ... a_n], represents the polynomial a_0 + a_1*X + ... + a_n*X^n. On output, all coefficients will be integers between 0 and p-1, amd a_n not zero (the zero polynomial is [ ]). On input, the coefficients are arbitrary integers which are reduced modulo p, and leading zeros stripped. \**************************************************************************/ istream& operator>>(istream& s, zz_pX& x); ostream& operator<<(ostream& s, const zz_pX& a); /**************************************************************************\ Some utility routines \**************************************************************************/ long deg(const zz_pX& a); // return deg(a); deg(0) == -1. zz_p coeff(const zz_pX& a, long i); // returns the coefficient of X^i, or zero if i not in range zz_p LeadCoeff(const zz_pX& a); // returns leading term of a, or zero if a == 0 zz_p ConstTerm(const zz_pX& a); // returns constant term of a, or zero if a == 0 void SetCoeff(zz_pX& x, long i, zz_p a); void SetCoeff(zz_pX& x, long i, long a); // makes coefficient of X^i equal to a; error is raised if i < 0 void SetCoeff(zz_pX& x, long i); // makes coefficient of X^i equal to 1; error is raised if i < 0 void SetX(zz_pX& x); // x is set to the monomial X long IsX(const zz_pX& a); // test if x = X void diff(zz_pX& x, const zz_pX& a); zz_pX diff(const zz_pX& a); // x = derivative of a void MakeMonic(zz_pX& x); // if x != 0 makes x into its monic associate; LeadCoeff(x) must be // invertible in this case. void reverse(zz_pX& x, const zz_pX& a, long hi); zz_pX reverse(const zz_pX& a, long hi); void reverse(zz_pX& x, const zz_pX& a); zz_pX reverse(const zz_pX& a); // x = reverse of a[0]..a[hi] (hi >= -1); // hi defaults to deg(a) in second version void VectorCopy(vec_zz_p& x, const zz_pX& a, long n); vec_zz_p VectorCopy(const zz_pX& a, long n); // x = copy of coefficient vector of a of length exactly n. // input is truncated or padded with zeroes as appropriate. /**************************************************************************\ Random Polynomials \**************************************************************************/ void random(zz_pX& x, long n); zz_pX random_zz_pX(long n); // x = random polynomial of degree < n /**************************************************************************\ Polynomial Evaluation and related problems \**************************************************************************/ void BuildFromRoots(zz_pX& x, const vec_zz_p& a); zz_pX BuildFromRoots(const vec_zz_p& a); // computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = // a.length() void eval(zz_p& b, const zz_pX& f, zz_p a); zz_p eval(const zz_pX& f, zz_p a); // b = f(a) void eval(vec_zz_p& b, const zz_pX& f, const vec_zz_p& a); vec_zz_p eval(const zz_pX& f, const vec_zz_p& a); // b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length() void interpolate(zz_pX& f, const vec_zz_p& a, const vec_zz_p& b); zz_pX interpolate(const vec_zz_p& a, const vec_zz_p& b); // interpolates the polynomial f satisfying f(a[i]) = b[i]. p should // be prime. /**************************************************************************\ Arithmetic mod X^n It is required that n >= 0, otherwise an error is raised. \**************************************************************************/ void trunc(zz_pX& x, const zz_pX& a, long n); // x = a % X^n zz_pX trunc(const zz_pX& a, long n); void MulTrunc(zz_pX& x, const zz_pX& a, const zz_pX& b, long n); zz_pX MulTrunc(const zz_pX& a, const zz_pX& b, long n); // x = a * b % X^n void SqrTrunc(zz_pX& x, const zz_pX& a, long n); zz_pX SqrTrunc(const zz_pX& a, long n); // x = a^2 % X^n void InvTrunc(zz_pX& x, const zz_pX& a, long n); zz_pX InvTrunc(const zz_pX& a, long n); // computes x = a^{-1} % X^n. Must have ConstTerm(a) invertible. /**************************************************************************\ Modular Arithmetic (without pre-conditioning) Arithmetic mod f. All inputs and outputs are polynomials of degree less than deg(f), and deg(f) > 0. NOTE: if you want to do many computations with a fixed f, use the zz_pXModulus data structure and associated routines below for better performance. \**************************************************************************/ void MulMod(zz_pX& x, const zz_pX& a, const zz_pX& b, const zz_pX& f); zz_pX MulMod(const zz_pX& a, const zz_pX& b, const zz_pX& f); // x = (a * b) % f void SqrMod(zz_pX& x, const zz_pX& a, const zz_pX& f); zz_pX SqrMod(const zz_pX& a, const zz_pX& f); // x = a^2 % f void MulByXMod(zz_pX& x, const zz_pX& a, const zz_pX& f); zz_pX MulByXMod(const zz_pX& a, const zz_pX& f); // x = (a * X) mod f void InvMod(zz_pX& x, const zz_pX& a, const zz_pX& f); zz_pX InvMod(const zz_pX& a, const zz_pX& f); // x = a^{-1} % f, error is a is not invertible long InvModStatus(zz_pX& x, const zz_pX& a, const zz_pX& f); // if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise, // returns 1 and sets x = (a, f) // for modular exponentiation, see below /**************************************************************************\ Modular Arithmetic with Pre-Conditioning If you need to do a lot of arithmetic modulo a fixed f, build zz_pXModulus F for f. This pre-computes information about f that speeds up subsequent computations. Required: deg(f) > 0 and LeadCoeff(f) invertible. As an example, the following routine computes the product modulo f of a vector of polynomials. #include "zz_pX.h" void product(zz_pX& x, const vec_zz_pX& v, const zz_pX& f) { zz_pXModulus F(f); zz_pX res; res = 1; long i; for (i = 0; i < v.length(); i++) MulMod(res, res, v[i], F); x = res; } Note that automatic conversions are provided so that a zz_pX can be used wherever a zz_pXModulus is required, and a zz_pXModulus can be used wherever a zz_pX is required. \**************************************************************************/ class zz_pXModulus { public: zz_pXModulus(); // initially in an unusable state ~zz_pXModulus(); zz_pXModulus(const zz_pXModulus&); // copy zz_pXModulus& operator=(const zz_pXModulus&); // assignment zz_pXModulus(const zz_pX& f); // initialize with f, deg(f) > 0 operator const zz_pX& () const; // read-only access to f, implicit conversion operator const zz_pX& val() const; // read-only access to f, explicit notation }; void build(zz_pXModulus& F, const zz_pX& f); // pre-computes information about f and stores it in F. // Note that the declaration zz_pXModulus F(f) is equivalent to // zz_pXModulus F; build(F, f). // In the following, f refers to the polynomial f supplied to the // build routine, and n = deg(f). long deg(const zz_pXModulus& F); // return deg(f) void MulMod(zz_pX& x, const zz_pX& a, const zz_pX& b, const zz_pXModulus& F); zz_pX MulMod(const zz_pX& a, const zz_pX& b, const zz_pXModulus& F); // x = (a * b) % f; deg(a), deg(b) < n void SqrMod(zz_pX& x, const zz_pX& a, const zz_pXModulus& F); zz_pX SqrMod(const zz_pX& a, const zz_pXModulus& F); // x = a^2 % f; deg(a) < n void PowerMod(zz_pX& x, const zz_pX& a, const ZZ& e, const zz_pXModulus& F); zz_pX PowerMod(const zz_pX& a, const ZZ& e, const zz_pXModulus& F); void PowerMod(zz_pX& x, const zz_pX& a, long e, const zz_pXModulus& F); zz_pX PowerMod(const zz_pX& a, long e, const zz_pXModulus& F); // x = a^e % f; deg(a) < n (e may be negative) void PowerXMod(zz_pX& x, const ZZ& e, const zz_pXModulus& F); zz_pX PowerXMod(const ZZ& e, const zz_pXModulus& F); void PowerXMod(zz_pX& x, long e, const zz_pXModulus& F); zz_pX PowerXMod(long e, const zz_pXModulus& F); // x = X^e % f (e may be negative) void PowerXPlusAMod(zz_pX& x, const zz_p& a, const ZZ& e, const zz_pXModulus& F); zz_pX PowerXPlusAMod(const zz_p& a, const ZZ& e, const zz_pXModulus& F); void PowerXPlusAMod(zz_pX& x, const zz_p& a, long e, const zz_pXModulus& F); zz_pX PowerXPlusAMod(const zz_p& a, long e, const zz_pXModulus& F); // x = (X + a)^e % f (e may be negative) void rem(zz_pX& x, const zz_pX& a, const zz_pXModulus& F); // x = a % f void DivRem(zz_pX& q, zz_pX& r, const zz_pX& a, const zz_pXModulus& F); // q = a/f, r = a%f void div(zz_pX& q, const zz_pX& a, const zz_pXModulus& F); // q = a/f // operator notation: zz_pX operator/(const zz_pX& a, const zz_pXModulus& F); zz_pX operator%(const zz_pX& a, const zz_pXModulus& F); zz_pX& operator/=(zz_pX& x, const zz_pXModulus& F); zz_pX& operator%=(zz_pX& x, const zz_pXModulus& F); /**************************************************************************\ More Pre-Conditioning If you need to compute a * b % f for a fixed b, but for many a's, it is much more efficient to first build a zz_pXMultiplier B for b, and then use the MulMod routine below. Here is an example that multiplies each element of a vector by a fixed polynomial modulo f. #include "zz_pX.h" void mul(vec_zz_pX& v, const zz_pX& b, const zz_pX& f) { zz_pXModulus F(f); zz_pXMultiplier B(b, F); long i; for (i = 0; i < v.length(); i++) MulMod(v[i], v[i], B, F); } Note that a (trivial) conversion operator from zz_pXMultiplier to zz_pX is provided, so that a zz_pXMultiplier can be used in a context where a zz_pX is required. \**************************************************************************/ class zz_pXMultiplier { public: zz_pXMultiplier(); // initially zero zz_pXMultiplier(const zz_pX& b, const zz_pXModulus& F); // initializes with b mod F, where deg(b) < deg(F) zz_pXMultiplier(const zz_pXMultiplier&); zz_pXMultiplier& operator=(const zz_pXMultiplier&); ~zz_pXMultiplier(); const zz_pX& val() const; // read-only access to b }; void build(zz_pXMultiplier& B, const zz_pX& b, const zz_pXModulus& F); // pre-computes information about b and stores it in B; deg(b) < // deg(F) void MulMod(zz_pX& x, const zz_pX& a, const zz_pXMultiplier& B, const zz_pXModulus& F); zz_pX MulMod(const zz_pX& a, const zz_pXMultiplier& B, const zz_pXModulus& F); // x = (a * b) % F; deg(a) < deg(F) /**************************************************************************\ vectors of zz_pX's \**************************************************************************/ NTL_vector_decl(zz_pX,vec_zz_pX) // vec_zz_pX NTL_eq_vector_decl(zz_pX,vec_zz_pX) // == and != NTL_io_vector_decl(zz_pX,vec_zz_pX) // I/O operators /**************************************************************************\ Modular Composition Modular composition is the problem of computing g(h) mod f for polynomials f, g, and h. The algorithm employed is that of Brent & Kung (Fast algorithms for manipulating formal power series, JACM 25:581-595, 1978), which uses O(n^{1/2}) modular polynomial multiplications, and O(n^2) scalar operations. \**************************************************************************/ void CompMod(zz_pX& x, const zz_pX& g, const zz_pX& h, const zz_pXModulus& F); zz_pX CompMod(const zz_pX& g, const zz_pX& h, const zz_pXModulus& F); // x = g(h) mod f; deg(h) < n void Comp2Mod(zz_pX& x1, zz_pX& x2, const zz_pX& g1, const zz_pX& g2, const zz_pX& h, const zz_pXModulus& F); // xi = gi(h) mod f (i=1,2), deg(h) < n. void CompMod3(zz_pX& x1, zz_pX& x2, zz_pX& x3, const zz_pX& g1, const zz_pX& g2, const zz_pX& g3, const zz_pX& h, const zz_pXModulus& F); // xi = gi(h) mod f (i=1..3), deg(h) < n /**************************************************************************\ Composition with Pre-Conditioning If a single h is going to be used with many g's then you should build a zz_pXArgument for h, and then use the compose routine below. The routine build computes and stores h, h^2, ..., h^m mod f. After this pre-computation, composing a polynomial of degree roughly n with h takes n/m multiplies mod f, plus n^2 scalar multiplies. Thus, increasing m increases the space requirement and the pre-computation time, but reduces the composition time. \**************************************************************************/ struct zz_pXArgument { vec_zz_pX H; }; void build(zz_pXArgument& H, const zz_pX& h, const zz_pXModulus& F, long m); // Pre-Computes information about h. m > 0, deg(h) < n void CompMod(zz_pX& x, const zz_pX& g, const zz_pXArgument& H, const zz_pXModulus& F); zz_pX CompMod(const zz_pX& g, const zz_pXArgument& H, const zz_pXModulus& F); extern long zz_pXArgBound; // Initially 0. If this is set to a value greater than zero, then // composition routines will allocate a table of no than about // zz_pXArgBound KB. Setting this value affects all compose routines // and the power projection and minimal polynomial routines below, // and indirectly affects many routines in zz_pXFactoring. /**************************************************************************\ power projection routines \**************************************************************************/ void project(zz_p& x, const zz_pVector& a, const zz_pX& b); zz_p project(const zz_pVector& a, const zz_pX& b); // x = inner product of a with coefficient vector of b void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k, const zz_pX& h, const zz_pXModulus& F); vec_zz_p ProjectPowers(const vec_zz_p& a, long k, const zz_pX& h, const zz_pXModulus& F); // Computes the vector // project(a, 1), project(a, h), ..., project(a, h^{k-1} % f). // This operation is the "transpose" of the modular composition operation. // Input and output may have "high order" zeroes stripped. void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k, const zz_pXArgument& H, const zz_pXModulus& F); vec_zz_p ProjectPowers(const vec_zz_p& a, long k, const zz_pXArgument& H, const zz_pXModulus& F); // same as above, but uses a pre-computed zz_pXArgument void UpdateMap(vec_zz_p& x, const vec_zz_p& a, const zz_pXMultiplier& B, const zz_pXModulus& F); vec_zz_p UpdateMap(const vec_zz_p& a, const zz_pXMultiplier& B, const zz_pXModulus& F); // Computes the vector // project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f) // Restriction: a.length() <= deg(F). // This is "transposed" MulMod by B. // Input vector may have "high order" zeroes striped. // The output will always have high order zeroes stripped. /**************************************************************************\ Minimum Polynomials These routines should be used with prime p. All of these routines implement the algorithm from [Shoup, J. Symbolic Comp. 17:371-391, 1994] and [Shoup, J. Symbolic Comp. 20:363-397, 1995], based on transposed modular composition and the Berlekamp/Massey algorithm. \**************************************************************************/ void MinPolySeq(zz_pX& h, const vec_zz_p& a, long m); // computes the minimum polynomial of a linealy generated sequence; m // is a bound on the degree of the polynomial; required: a.length() >= // 2*m void ProbMinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m); zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m); void ProbMinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F); zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& F); // computes the monic minimal polynomial if (g mod f). m = a bound on // the degree of the minimal polynomial; in the second version, this // argument defaults to n. The algorithm is probabilistic, always // returns a divisor of the minimal polynomial, and returns a proper // divisor with probability at most m/p. void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m); zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m); void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F); zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F); // same as above, but guarantees that result is correct void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m); zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& F, long m); void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F); zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& F); // same as above, but assumes that f is irreducible, or at least that // the minimal poly of g is itself irreducible. The algorithm is // deterministic (and is always correct). /**************************************************************************\ Traces, norms, resultants These routines should be used with prime p. \**************************************************************************/ void TraceMod(zz_p& x, const zz_pX& a, const zz_pXModulus& F); zz_p TraceMod(const zz_pX& a, const zz_pXModulus& F); void TraceMod(zz_p& x, const zz_pX& a, const zz_pX& f); zz_p TraceMod(const zz_pX& a, const zz_pXModulus& f); // x = Trace(a mod f); deg(a) < deg(f) void TraceVec(vec_zz_p& S, const zz_pX& f); vec_zz_p TraceVec(const zz_pX& f); // S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f) // The above routines implement the asymptotically fast trace // algorithm from [von zur Gathen and Shoup, Computational Complexity, // 1992]. void NormMod(zz_p& x, const zz_pX& a, const zz_pX& f); zz_p NormMod(const zz_pX& a, const zz_pX& f); // x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f) void resultant(zz_p& x, const zz_pX& a, const zz_pX& b); zz_pX resultant(zz_p& x, const zz_pX& a, const zz_pX& b); // x = resultant(a, b) void CharPolyMod(zz_pX& g, const zz_pX& a, const zz_pX& f); zz_pX CharPolyMod(const zz_pX& a, const zz_pX& f); // g = charcteristic polynomial of (a mod f); 0 < deg(f), deg(g) < // deg(f). This routine works for arbitrary f. For irreducible f, // is it faster to use IrredPolyMod, and then exponentiate as // necessary, since in this case the characterstic polynomial // is a power of the minimal polynomial. /**************************************************************************\ Miscellany A zz_pX f is represented as a vec_zz_p, which can be accessed as f.rep. The constant term is f.rep[0] and the leading coefficient is f.rep[f.rep.length()-1], except if f is zero, in which case f.rep.length() == 0. Note that the leading coefficient is always nonzero (unless f is zero). One can freely access and modify f.rep, but one should always ensure that the leading coefficient is nonzero, which can be done by invoking f.normalize(). \**************************************************************************/ void clear(zz_pX& x) // x = 0 void set(zz_pX& x); // x = 1 void zz_pX::normalize(); // f.normalize() strips leading zeros from f.rep. void zz_pX::SetMaxLength(long n); // f.SetMaxLength(n) pre-allocate spaces for n coefficients. The // polynomial that f represents is unchanged. void zz_pX::kill(); // f.kill() sets f to 0 and frees all memory held by f. Equivalent to // f.rep.kill(). zz_pX::zz_pX(INIT_SIZE_TYPE, long n); // zz_pX(INIT_SIZE, n) initializes to zero, but space is pre-allocated // for n coefficients static const zz_pX& zero(); // zz_pX::zero() is a read-only reference to 0 void swap(zz_pX& x, zz_pX& y); // swap x and y (via "pointer swapping")