/**************************************************************************\ 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 explicit ZZ_pEX(const ZZ_pE& a); // promotion explicit ZZ_pEX(const ZZ_p& a); explicit ZZ_pEX(long a); 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(ZZ_pEX&& a); // move constructor (C++11 only) // declared noexcept unless NTL_EXCEPTIONS flag is set #ifndef NTL_DISABLE_MOVE_ASSIGN ZZ_pEX& operator=(ZZ_pEX&& a); // move assignment (C++11 only) // declared noexcept unless NTL_EXCEPTIONS flag is set #endif ZZ_pEX(INIT_MONO_TYPE, long i, const ZZ_pE& c); ZZ_pEX(INIT_MONO_TYPE, long i, const ZZ_p& c); ZZ_pEX(INIT_MONO_TYPE, long i, long c); // initialize to c*X^i, invoke as ZZ_pEX(INIT_MONO, i, c) ZZ_pEX(INIT_MONO_TYPE, long i); // initialize to X^i, invoke as ZZ_pEX(INIT_MONO, i) // typedefs to aid in generic programming typedef ZZ_pE coeff_type; typedef ZZ_pEXModulus modulus_type; // ... }; /**************************************************************************\ Accessing coefficients The degree of a polynomial f is obtained as deg(f), where the zero polynomial, by definition, has degree -1. A polynomial f is represented as a coefficient vector. Coefficients may be accesses in one of two ways. The safe, high-level method is to call the function coeff(f, i) to get the coefficient of X^i in the polynomial f, and to call the function SetCoeff(f, i, a) to set the coefficient of X^i in f to the scalar a. One can also access the coefficients more directly via a lower level interface. The coefficient of X^i in f may be accessed using subscript notation f[i]. In addition, one may write f.SetLength(n) to set the length of the underlying coefficient vector to n, and f.SetMaxLength(n) to allocate space for n coefficients, without changing the coefficient vector itself. After setting coefficients using this low-level interface, one must ensure that leading zeros in the coefficient vector are stripped afterwards by calling the function f.normalize(). NOTE: the coefficient vector of f may also be accessed directly as f.rep; however, this is not recommended. Also, for a properly normalized polynomial f, we have f.rep.length() == deg(f)+1, and deg(f) >= 0 => f.rep[deg(f)] != 0. \**************************************************************************/ long deg(const ZZ_pEX& a); // return deg(a); deg(0) == -1. const ZZ_pE& coeff(const ZZ_pEX& a, long i); // returns the coefficient of X^i, or zero if i not in range const ZZ_pE& LeadCoeff(const ZZ_pEX& a); // returns leading term of a, or zero if a == 0 const ZZ_pE& ConstTerm(const ZZ_pEX& a); // returns 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 ZZ_pE& ZZ_pEX::operator[](long i); const ZZ_pE& ZZ_pEX::operator[](long i) const; // indexing operators: f[i] is the coefficient of X^i --- // i should satsify i >= 0 and i <= deg(f). // No range checking (unless NTL_RANGE_CHECK is defined). void ZZ_pEX::SetLength(long n); // f.SetLength(n) sets the length of the inderlying coefficient // vector to n --- after this call, indexing f[i] for i = 0..n-1 // is valid. void ZZ_pEX::normalize(); // f.normalize() strips leading zeros from coefficient vector of f void ZZ_pEX::SetMaxLength(long n); // f.SetMaxLength(n) pre-allocate spaces for n coefficients. The // polynomial that f represents is unchanged. /**************************************************************************\ 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 \**************************************************************************/ 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 \**************************************************************************/ typedef Vec vec_ZZ_pEX; // backward compatibility /**************************************************************************\ 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 thread_local 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 \**************************************************************************/ void clear(ZZ_pEX& x) // x = 0 void set(ZZ_pEX& x); // x = 1 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 ZZ_pEX::swap(ZZ_pEX& x); void swap(ZZ_pEX& x, ZZ_pEX& y); // swap (via "pointer swapping") ZZ_pEX::ZZ_pEX(long i, const ZZ_pE& c); ZZ_pEX::ZZ_pEX(long i, const ZZ_p& c); ZZ_pEX::ZZ_pEX(long i, long c); // initialize to c*X^i, provided for backward compatibility