/**************************************************************************\ MODULE: ZZ_pX SUMMARY: The class ZZ_pX implements polynomial arithmetic modulo p. Polynomial arithmetic is implemented using the FFT, combined with the Chinese Remainder Theorem. A more detailed description of the techniques used here can be found in [Shoup, J. Symbolic Comp. 20:363-397, 1995]. Small degree polynomials are multiplied either with classical or Karatsuba algorithms. \**************************************************************************/ #include #include class ZZ_pX { public: ZZ_pX(); // initialize to 0 ZZ_pX(const ZZ_pX& a); // copy constructor ZZ_pX& operator=(const ZZ_pX& a); // assignment ZZ_pX& operator=(const ZZ_p& a); // assignment ZZ_pX& operator=(const long a); // assignment ZZ_pX(long i, const ZZ_p& c); // initialize to X^i*c ZZ_pX(long i, long c); ~ZZ_pX(); // destructor }; /**************************************************************************\ Comparison \**************************************************************************/ long operator==(const ZZ_pX& a, const ZZ_pX& b); long operator!=(const ZZ_pX& a, const ZZ_pX& b); // PROMOTIONS: operators ==, != promote {long, ZZ_p} to ZZ_pX on (a, b). long IsZero(const ZZ_pX& a); // test for 0 long IsOne(const ZZ_pX& a); // test for 1 /**************************************************************************\ 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, const 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, const 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, const 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_p& b); ZZ_pX operator/(const ZZ_pX& a, long b); ZZ_pX& operator/=(ZZ_pX& x, const ZZ_pX& b); ZZ_pX& operator/=(ZZ_pX& x, const ZZ_p& b); ZZ_pX& operator/=(ZZ_pX& x, long b); ZZ_pX operator%(const ZZ_pX& a, const ZZ_pX& b); 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); void div(ZZ_pX& q, const ZZ_pX& a, const ZZ_p& b); void div(ZZ_pX& q, const ZZ_pX& a, long 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 /**************************************************************************\ 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, and 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. const ZZ_p& coeff(const ZZ_pX& a, long i); // returns a read-only reference to the coefficient of X^i, or zero if // i not in range const ZZ_p& LeadCoeff(const ZZ_pX& a); // read-only reference to leading term of a, or zero if a == 0 const ZZ_p& ConstTerm(const ZZ_pX& a); // read-only reference to constant term of a, or zero if a == 0 void SetCoeff(ZZ_pX& x, long i, const 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); // x = derivative of a ZZ_pX diff(const ZZ_pX& 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); // generate a 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, const ZZ_p& a); ZZ_p eval(const ZZ_pX& f, const 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 All routines require 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^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_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 a ZZ_pXModulus F for f. This pre-computes information about f that speeds up subsequent computations. It is required that deg(f) > 0 and that LeadCoeff(f) is invertible. As an example, the following routine computes the product modulo f of a vector of polynomials. #include 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(const ZZ_pXModulus&); // copy ZZ_pXModulus& operator=(const ZZ_pXModulus&); // assignment ~ZZ_pXModulus(); 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 n=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 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); } \**************************************************************************/ 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&); // copy ZZ_pXMultiplier& operator=(const ZZ_pXMultiplier&); // assignment ~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); // 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 Comp3Mod(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. 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: must have a.length() <= deg(F). // This is "transposed" MulMod by B. // Input may have "high order" zeroes stripped. // 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); ZZ_pX MinPolySeq(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 IrredPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F, long m); ZZ_pX IrredPolyMod(const ZZ_pX& g, const ZZ_pXModulus& F, long m); void IrredPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F); ZZ_pX IrredPolyMod(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 trace 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_p resultant(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; if f is irreducible, // it is faster to use the IrredPolyMod routine, and then exponentiate // if necessary (since in this case the CharPoly is just a power of // the IrredPoly). /**************************************************************************\ 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& 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")