/**************************************************************************\ MODULE: zz_pEX SUMMARY: The class zz_pEX represents polynomials over zz_pE, and so can be used, for example, for arithmentic in GF(p^n)[X]. However, except where mathematically necessary (e.g., GCD computations), zz_pE need not be a field. \**************************************************************************/ #include #include class zz_pEX { public: zz_pEX(); // initial value 0 zz_pEX(const zz_pEX& a); // copy zz_pEX& operator=(const zz_pEX& a); // assignment zz_pEX& operator=(const zz_pE& a); zz_pEX& operator=(const zz_p& a); zz_pEX& operator=(long a); ~zz_pEX(); // destructor zz_pEX(long i, const zz_pE& c); // initilaize to X^i*c zz_pEX(long i, const zz_p& c); zz_pEX(long i, long c); }; /**************************************************************************\ Comparison \**************************************************************************/ long operator==(const zz_pEX& a, const zz_pEX& b); long operator!=(const zz_pEX& a, const zz_pEX& b); long IsZero(const zz_pEX& a); // test for 0 long IsOne(const zz_pEX& a); // test for 1 // PROMOTIONS: ==, != promote {long,zz_p,zz_pE} to zz_pEX on (a, b). /**************************************************************************\ Addition \**************************************************************************/ // operator notation: zz_pEX operator+(const zz_pEX& a, const zz_pEX& b); zz_pEX operator-(const zz_pEX& a, const zz_pEX& b); zz_pEX operator-(const zz_pEX& a); zz_pEX& operator+=(zz_pEX& x, const zz_pEX& a); zz_pEX& operator+=(zz_pEX& x, const zz_pE& a); zz_pEX& operator+=(zz_pEX& x, const zz_p& a); zz_pEX& operator+=(zz_pEX& x, long a); zz_pEX& operator++(zz_pEX& x); // prefix void operator++(zz_pEX& x, int); // postfix zz_pEX& operator-=(zz_pEX& x, const zz_pEX& a); zz_pEX& operator-=(zz_pEX& x, const zz_pE& a); zz_pEX& operator-=(zz_pEX& x, const zz_p& a); zz_pEX& operator-=(zz_pEX& x, long a); zz_pEX& operator--(zz_pEX& x); // prefix void operator--(zz_pEX& x, int); // postfix // procedural versions: void add(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a + b void sub(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a - b void negate(zz_pEX& x, const zz_pEX& a); // x = - a // PROMOTIONS: +, -, add, sub promote {long,zz_p,zz_pE} to zz_pEX on (a, b). /**************************************************************************\ Multiplication \**************************************************************************/ // operator notation: zz_pEX operator*(const zz_pEX& a, const zz_pEX& b); zz_pEX& operator*=(zz_pEX& x, const zz_pEX& a); zz_pEX& operator*=(zz_pEX& x, const zz_pE& a); zz_pEX& operator*=(zz_pEX& x, const zz_p& a); zz_pEX& operator*=(zz_pEX& x, long a); // procedural versions: void mul(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a * b void sqr(zz_pEX& x, const zz_pEX& a); // x = a^2 zz_pEX sqr(const zz_pEX& a); // PROMOTIONS: *, mul promote {long,zz_p,zz_pE} to zz_pEX on (a, b). void power(zz_pEX& x, const zz_pEX& a, long e); // x = a^e (e >= 0) zz_pEX power(const zz_pEX& 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_pEX operator<<(const zz_pEX& a, long n); zz_pEX operator>>(const zz_pEX& a, long n); zz_pEX& operator<<=(zz_pEX& x, long n); zz_pEX& operator>>=(zz_pEX& x, long n); // procedural versions: void LeftShift(zz_pEX& x, const zz_pEX& a, long n); zz_pEX LeftShift(const zz_pEX& a, long n); void RightShift(zz_pEX& x, const zz_pEX& a, long n); zz_pEX RightShift(const zz_pEX& a, long n); /**************************************************************************\ Division \**************************************************************************/ // operator notation: zz_pEX operator/(const zz_pEX& a, const zz_pEX& b); zz_pEX operator/(const zz_pEX& a, const zz_pE& b); zz_pEX operator/(const zz_pEX& a, const zz_p& b); zz_pEX operator/(const zz_pEX& a, long b); zz_pEX operator%(const zz_pEX& a, const zz_pEX& b); zz_pEX& operator/=(zz_pEX& x, const zz_pEX& a); zz_pEX& operator/=(zz_pEX& x, const zz_pE& a); zz_pEX& operator/=(zz_pEX& x, const zz_p& a); zz_pEX& operator/=(zz_pEX& x, long a); zz_pEX& operator%=(zz_pEX& x, const zz_pEX& a); // procedural versions: void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEX& b); // q = a/b, r = a%b void div(zz_pEX& q, const zz_pEX& a, const zz_pEX& b); void div(zz_pEX& q, const zz_pEX& a, const zz_pE& b); void div(zz_pEX& q, const zz_pEX& a, const zz_p& b); void div(zz_pEX& q, const zz_pEX& a, long b); // q = a/b void rem(zz_pEX& r, const zz_pEX& a, const zz_pEX& b); // r = a%b long divide(zz_pEX& q, const zz_pEX& a, const zz_pEX& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 long divide(const zz_pEX& a, const zz_pEX& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 /**************************************************************************\ GCD's These routines are intended for use when zz_pE is a field. \**************************************************************************/ void GCD(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); zz_pEX GCD(const zz_pEX& a, const zz_pEX& b); // x = GCD(a, b), x is always monic (or zero if a==b==0). void XGCD(zz_pEX& d, zz_pEX& s, zz_pEX& t, const zz_pEX& a, const zz_pEX& b); // d = gcd(a,b), a s + b t = d /**************************************************************************\ 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 polynomials of degree < zz_pE::degree() and a_n not zero (the zero polynomial is [ ]). On input, the coefficients are arbitrary polynomials which are reduced modulo zz_pE::modulus(), and leading zeros stripped. \**************************************************************************/ istream& operator>>(istream& s, zz_pEX& x); ostream& operator<<(ostream& s, const zz_pEX& a); /**************************************************************************\ Some utility routines \**************************************************************************/ long deg(const zz_pEX& a); // return deg(a); deg(0) == -1. const zz_pE& coeff(const zz_pEX& a, long i); // returns a read-only reference to the coefficient of X^i, or zero if // i not in range const zz_pE& LeadCoeff(const zz_pEX& a); // read-only reference to leading term of a, or zero if a == 0 const zz_pE& ConstTerm(const zz_pEX& a); // read-only reference to constant term of a, or zero if a == 0 void SetCoeff(zz_pEX& x, long i, const zz_pE& a); void SetCoeff(zz_pEX& x, long i, const zz_p& a); void SetCoeff(zz_pEX& x, long i, long a); // makes coefficient of X^i equal to a; error is raised if i < 0 void SetCoeff(zz_pEX& x, long i); // makes coefficient of X^i equal to 1; error is raised if i < 0 void SetX(zz_pEX& x); // x is set to the monomial X long IsX(const zz_pEX& a); // test if x = X void diff(zz_pEX& x, const zz_pEX& a); // x = derivative of a zz_pEX diff(const zz_pEX& a); void MakeMonic(zz_pEX& x); // if x != 0 makes x into its monic associate; LeadCoeff(x) must be // invertible in this case void reverse(zz_pEX& x, const zz_pEX& a, long hi); zz_pEX reverse(const zz_pEX& a, long hi); void reverse(zz_pEX& x, const zz_pEX& a); zz_pEX reverse(const zz_pEX& a); // x = reverse of a[0]..a[hi] (hi >= -1); // hi defaults to deg(a) in second version void VectorCopy(vec_zz_pE& x, const zz_pEX& a, long n); vec_zz_pE VectorCopy(const zz_pEX& 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_pEX& x, long n); zz_pEX random_zz_pEX(long n); // x = random polynomial of degree < n /**************************************************************************\ Polynomial Evaluation and related problems \**************************************************************************/ void BuildFromRoots(zz_pEX& x, const vec_zz_pE& a); zz_pEX BuildFromRoots(const vec_zz_pE& a); // computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length() void eval(zz_pE& b, const zz_pEX& f, const zz_pE& a); zz_pE eval(const zz_pEX& f, const zz_pE& a); // b = f(a) void eval(zz_pE& b, const zz_pX& f, const zz_pE& a); zz_pE eval(const zz_pEX& f, const zz_pE& a); // b = f(a); uses ModComp algorithm for zz_pX void eval(vec_zz_pE& b, const zz_pEX& f, const vec_zz_pE& a); vec_zz_pE eval(const zz_pEX& f, const vec_zz_pE& a); // b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length() void interpolate(zz_pEX& f, const vec_zz_pE& a, const vec_zz_pE& b); zz_pEX interpolate(const vec_zz_pE& a, const vec_zz_pE& b); // interpolates the polynomial f satisfying f(a[i]) = b[i]. /**************************************************************************\ Arithmetic mod X^n Required: n >= 0; otherwise, an error is raised. \**************************************************************************/ void trunc(zz_pEX& x, const zz_pEX& a, long n); // x = a % X^n zz_pEX trunc(const zz_pEX& a, long n); void MulTrunc(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, long n); zz_pEX MulTrunc(const zz_pEX& a, const zz_pEX& b, long n); // x = a * b % X^n void SqrTrunc(zz_pEX& x, const zz_pEX& a, long n); zz_pEX SqrTrunc(const zz_pEX& a, long n); // x = a^2 % X^n void InvTrunc(zz_pEX& x, const zz_pEX& a, long n); zz_pEX InvTrunc(zz_pEX& x, const zz_pEX& a, long n); // computes x = a^{-1} % X^m. 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_pEXModulus data structure and associated routines below for better performance. \**************************************************************************/ void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, const zz_pEX& f); zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEX& f); // x = (a * b) % f void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f); zz_pEX SqrMod(const zz_pEX& a, const zz_pEX& f); // x = a^2 % f void MulByXMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f); zz_pEX MulByXMod(const zz_pEX& a, const zz_pEX& f); // x = (a * X) mod f void InvMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f); zz_pEX InvMod(const zz_pEX& a, const zz_pEX& f); // x = a^{-1} % f, error is a is not invertible long InvModStatus(zz_pEX& x, const zz_pEX& a, const zz_pEX& f); // if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise, // returns 1 and sets x = (a, f) /**************************************************************************\ Modular Arithmetic with Pre-Conditioning If you need to do a lot of arithmetic modulo a fixed f, build zz_pEXModulus F for f. This pre-computes information about f that speeds up subsequent computations. As an example, the following routine the product modulo f of a vector of polynomials. #include void product(zz_pEX& x, const vec_zz_pEX& v, const zz_pEX& f) { zz_pEXModulus F(f); zz_pEX res; res = 1; long i; for (i = 0; i < v.length(); i++) MulMod(res, res, v[i], F); x = res; } NOTE: A zz_pEX may be used wherever a zz_pEXModulus is required, and a zz_pEXModulus may be used wherever a zz_pEX is required. \**************************************************************************/ class zz_pEXModulus { public: zz_pEXModulus(); // initially in an unusable state zz_pEXModulus(const zz_pEX& f); // initialize with f, deg(f) > 0 zz_pEXModulus(const zz_pEXModulus&); // copy zz_pEXModulus& operator=(const zz_pEXModulus&); // assignment ~zz_pEXModulus(); // destructor operator const zz_pEX& () const; // implicit read-only access to f const zz_pEX& val() const; // explicit read-only access to f }; void build(zz_pEXModulus& F, const zz_pEX& f); // pre-computes information about f and stores it in F. Must have // deg(f) > 0. Note that the declaration zz_pEXModulus F(f) is // equivalent to zz_pEXModulus 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_pEXModulus& F); // return n=deg(f) void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, const zz_pEXModulus& F); zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEXModulus& F); // x = (a * b) % f; deg(a), deg(b) < n void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F); zz_pEX SqrMod(const zz_pEX& a, const zz_pEXModulus& F); // x = a^2 % f; deg(a) < n void PowerMod(zz_pEX& x, const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F); zz_pEX PowerMod(const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F); void PowerMod(zz_pEX& x, const zz_pEX& a, long e, const zz_pEXModulus& F); zz_pEX PowerMod(const zz_pEX& a, long e, const zz_pEXModulus& F); // x = a^e % f; e >= 0, deg(a) < n. Uses a sliding window algorithm. // (e may be negative) void PowerXMod(zz_pEX& x, const ZZ& e, const zz_pEXModulus& F); zz_pEX PowerXMod(const ZZ& e, const zz_pEXModulus& F); void PowerXMod(zz_pEX& x, long e, const zz_pEXModulus& F); zz_pEX PowerXMod(long e, const zz_pEXModulus& F); // x = X^e % f (e may be negative) void rem(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F); // x = a % f void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEXModulus& F); // q = a/f, r = a%f void div(zz_pEX& q, const zz_pEX& a, const zz_pEXModulus& F); // q = a/f // operator notation: zz_pEX operator/(const zz_pEX& a, const zz_pEXModulus& F); zz_pEX operator%(const zz_pEX& a, const zz_pEXModulus& F); zz_pEX& operator/=(zz_pEX& x, const zz_pEXModulus& F); zz_pEX& operator%=(zz_pEX& x, const zz_pEXModulus& F); /**************************************************************************\ vectors of zz_pEX's \**************************************************************************/ NTL_vector_decl(zz_pEX,vec_zz_pEX) // vec_zz_pEX NTL_eq_vector_decl(zz_pEX,vec_zz_pEX) // == and != NTL_io_vector_decl(zz_pEX,vec_zz_pEX) // 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_pEX& x, const zz_pEX& g, const zz_pEX& h, const zz_pEXModulus& F); zz_pEX CompMod(const zz_pEX& g, const zz_pEX& h, const zz_pEXModulus& F); // x = g(h) mod f; deg(h) < n void Comp2Mod(zz_pEX& x1, zz_pEX& x2, const zz_pEX& g1, const zz_pEX& g2, const zz_pEX& h, const zz_pEXModulus& F); // xi = gi(h) mod f (i=1,2); deg(h) < n. void Comp3Mod(zz_pEX& x1, zz_pEX& x2, zz_pEX& x3, const zz_pEX& g1, const zz_pEX& g2, const zz_pEX& g3, const zz_pEX& h, const zz_pEXModulus& 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_pEXArgument 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_pEXArgument { vec_zz_pEX H; }; void build(zz_pEXArgument& H, const zz_pEX& h, const zz_pEXModulus& F, long m); // Pre-Computes information about h. m > 0, deg(h) < n. void CompMod(zz_pEX& x, const zz_pEX& g, const zz_pEXArgument& H, const zz_pEXModulus& F); zz_pEX CompMod(const zz_pEX& g, const zz_pEXArgument& H, const zz_pEXModulus& F); extern long zz_pEXArgBound; // Initially 0. If this is set to a value greater than zero, then // composition routines will allocate a table of no than about // zz_pEXArgBound KB. Setting this value affects all compose routines // and the power projection and minimal polynomial routines below, // and indirectly affects many routines in zz_pEXFactoring. /**************************************************************************\ power projection routines \**************************************************************************/ void project(zz_pE& x, const zz_pEVector& a, const zz_pEX& b); zz_pE project(const zz_pEVector& a, const zz_pEX& b); // x = inner product of a with coefficient vector of b void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k, const zz_pEX& h, const zz_pEXModulus& F); vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k, const zz_pEX& h, const zz_pEXModulus& 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. void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k, const zz_pEXArgument& H, const zz_pEXModulus& F); vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k, const zz_pEXArgument& H, const zz_pEXModulus& F); // same as above, but uses a pre-computed zz_pEXArgument class zz_pEXTransMultiplier { /* ... */ }; void build(zz_pEXTransMultiplier& B, const zz_pEX& b, const zz_pEXModulus& F); void UpdateMap(vec_zz_pE& x, const vec_zz_pE& a, const zz_pEXMultiplier& B, const zz_pEXModulus& F); vec_zz_pE UpdateMap(const vec_zz_pE& a, const zz_pEXMultiplier& B, const zz_pEXModulus& F); // Computes the vector // project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f) // Required: a.length() <= deg(F), deg(b) < deg(F). // This is "transposed" MulMod by B. // Input may have "high order" zeroes stripped. // Output always has high order zeroes stripped. /**************************************************************************\ Minimum Polynomials These routines should be used only when zz_pE is a field. 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_pEX& h, const vec_zz_pE& a, long m); zz_pEX MinPolySeq(const vec_zz_pE& 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_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m); void ProbMinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& 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/2^{zz_pE::degree()}. void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m); void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F); // same as above, but guarantees that result is correct void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m); void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& 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). /**************************************************************************\ Composition and Minimal Polynomials in towers These are implementations of algorithms that will be described and analyzed in a forthcoming paper. The routines require that p is prime, but zz_pE need not be a field. \**************************************************************************/ void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEXArgument& h, const zz_pEXModulus& F); zz_pEX CompTower(const zz_pX& g, const zz_pEXArgument& h, const zz_pEXModulus& F); void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEX& h, const zz_pEXModulus& F); zz_pEX CompTower(const zz_pX& g, const zz_pEX& h, const zz_pEXModulus& F); // x = g(h) mod f void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m); void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F); // Uses a probabilistic algorithm to compute the minimal // polynomial of (g mod f) over zz_p. // The parameter m is a bound on the degree of the minimal polynomial // (default = deg(f)*zz_pE::degree()). // In general, the result will be a divisor of the true minimimal // polynomial. For correct results, use the MinPoly routines below. void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m); void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F); // Same as above, but result is always correct. void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m); zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m); void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F); zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F); // Same as above, but assumes the minimal polynomial is // irreducible, and uses a slightly faster, deterministic algorithm. /**************************************************************************\ Traces, norms, resultants \**************************************************************************/ void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEXModulus& F); zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& F); void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f); zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& f); // x = Trace(a mod f); deg(a) < deg(f) void TraceVec(vec_zz_pE& S, const zz_pEX& f); vec_zz_pE TraceVec(const zz_pEX& f); // S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f) // The above trace routines implement the asymptotically fast trace // algorithm from [von zur Gathen and Shoup, Computational Complexity, // 1992]. void NormMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f); zz_pE NormMod(const zz_pEX& a, const zz_pEX& f); // x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f) void resultant(zz_pE& x, const zz_pEX& a, const zz_pEX& b); zz_pE resultant(const zz_pEX& a, const zz_pEX& b); // x = resultant(a, b) // NormMod and resultant require that zz_pE is a field. /**************************************************************************\ Miscellany A zz_pEX f is represented as a vec_zz_pE, 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_pEX& x) // x = 0 void set(zz_pEX& x); // x = 1 void zz_pEX::normalize(); // f.normalize() strips leading zeros from f.rep. void zz_pEX::SetMaxLength(long n); // f.SetMaxLength(n) pre-allocate spaces for n coefficients. The // polynomial that f represents is unchanged. void zz_pEX::kill(); // f.kill() sets f to 0 and frees all memory held by f. Equivalent to // f.rep.kill(). zz_pEX::zz_pEX(INIT_SIZE_TYPE, long n); // zz_pEX(INIT_SIZE, n) initializes to zero, but space is pre-allocated // for n coefficients static const zz_pEX& zero(); // zz_pEX::zero() is a read-only reference to 0 void swap(zz_pEX& x, zz_pEX& y); // swap x and y (via "pointer swapping")