/**************************************************************************\ MODULE: GF2X SUMMARY: The class GF2X implements polynomial arithmetic modulo 2. Polynomial arithmetic is implemented using a combination of classical routines and Karatsuba. \**************************************************************************/ #include #include class GF2X { public: GF2X(); // initial value 0 GF2X(const GF2X& a); // copy explicit GF2X(long a); // promotion explicit GF2X(GF2 a); // promotion GF2X& operator=(const GF2X& a); // assignment GF2X& operator=(GF2 a); GF2X& operator=(long a); ~GF2X(); // destructor GF2X(GF2X&& a); // move constructor (C++11 only) // declared noexcept unless NTL_EXCEPTIONS flag is set #ifndef NTL_DISABLE_MOVE_ASSIGN GF2X& operator=(GF2X&& a); // move assignment (C++11 only) // declared noexcept unless NTL_EXCEPTIONS flag is set #endif GF2X(INIT_MONO_TYPE, long i, GF2 c); GF2X(INIT_MONO_TYPE, long i, long c); // initialize to c*X^i, invoke as GF2X(INIT_MONO, i, c) GF2X(INIT_MONO_TYPE, long i); // initialize to c*X^i, invoke as GF2X(INIT_MONO, i) // typedefs to aid in generic programming typedef GF2 coeff_type; typedef GF2E residue_type; typedef GF2XModulus 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: unlike other polynomial classes, the coefficient vector for GF2X has a special representation, packing coefficients into words. This has two consequences. First, when using the indexing notation on a non-const polynomial f, the return type is ref_GF2, rather than GF2&. For the most part, a ref_GF2 may be used like a GF2& --- see GF2.txt for more details. Second, when applying f.SetLength(n) to a polynomial f, this essentially has the effect of zeroing out the coefficients of X^i for i >= n. \**************************************************************************/ long deg(const GF2X& a); // return deg(a); deg(0) == -1. const GF2 coeff(const GF2X& a, long i); // returns the coefficient of X^i, or zero if i not in range const GF2 LeadCoeff(const GF2X& a); // returns leading term of a, or zero if a == 0 const GF2 ConstTerm(const GF2X& a); // returns constant term of a, or zero if a == 0 void SetCoeff(GF2X& x, long i, GF2 a); void SetCoeff(GF2X& x, long i, long a); // makes coefficient of X^i equal to a; error is raised if i < 0 void SetCoeff(GF2X& x, long i); // makes coefficient of X^i equal to 1; error is raised if i < 0 void SetX(GF2X& x); // x is set to the monomial X long IsX(const GF2X& a); // test if x = X ref_GF2 GF2X::operator[](long i); const GF2 GF2X::operator[](long i) const; // indexing operators: f[i] is the coefficient of X^i --- // i should satsify i >= 0 and i <= deg(f) void GF2X::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 GF2X::normalize(); // f.normalize() strips leading zeros from coefficient vector of f void GF2X::SetMaxLength(long n); // f.SetMaxLength(n) pre-allocate spaces for n coefficients. The // polynomial that f represents is unchanged. /**************************************************************************\ Comparison \**************************************************************************/ long operator==(const GF2X& a, const GF2X& b); long operator!=(const GF2X& a, const GF2X& b); long IsZero(const GF2X& a); // test for 0 long IsOne(const GF2X& a); // test for 1 // PROMOTIONS: operators ==, != promote {long, GF2} to GF2X on (a, b) /**************************************************************************\ Addition \**************************************************************************/ // operator notation: GF2X operator+(const GF2X& a, const GF2X& b); GF2X operator-(const GF2X& a, const GF2X& b); GF2X operator-(const GF2X& a); // unary - GF2X& operator+=(GF2X& x, const GF2X& a); GF2X& operator+=(GF2X& x, GF2 a); GF2X& operator+=(GF2X& x, long a); GF2X& operator-=(GF2X& x, const GF2X& a); GF2X& operator-=(GF2X& x, GF2 a); GF2X& operator-=(GF2X& x, long a); GF2X& operator++(GF2X& x); // prefix void operator++(GF2X& x, int); // postfix GF2X& operator--(GF2X& x); // prefix void operator--(GF2X& x, int); // postfix // procedural versions: void add(GF2X& x, const GF2X& a, const GF2X& b); // x = a + b void sub(GF2X& x, const GF2X& a, const GF2X& b); // x = a - b void negate(GF2X& x, const GF2X& a); // x = -a // PROMOTIONS: binary +, - and procedures add, sub promote {long, GF2} // to GF2X on (a, b). /**************************************************************************\ Multiplication \**************************************************************************/ // operator notation: GF2X operator*(const GF2X& a, const GF2X& b); GF2X& operator*=(GF2X& x, const GF2X& a); GF2X& operator*=(GF2X& x, GF2 a); GF2X& operator*=(GF2X& x, long a); // procedural versions: void mul(GF2X& x, const GF2X& a, const GF2X& b); // x = a * b void sqr(GF2X& x, const GF2X& a); // x = a^2 GF2X sqr(const GF2X& a); // PROMOTIONS: operator * and procedure mul promote {long, GF2} to GF2X // on (a, b). /**************************************************************************\ 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: GF2X operator<<(const GF2X& a, long n); GF2X operator>>(const GF2X& a, long n); GF2X& operator<<=(GF2X& x, long n); GF2X& operator>>=(GF2X& x, long n); // procedural versions: void LeftShift(GF2X& x, const GF2X& a, long n); GF2X LeftShift(const GF2X& a, long n); void RightShift(GF2X& x, const GF2X& a, long n); GF2X RightShift(const GF2X& a, long n); void MulByX(GF2X& x, const GF2X& a); GF2X MulByX(const GF2X& a); /**************************************************************************\ Division \**************************************************************************/ // operator notation: GF2X operator/(const GF2X& a, const GF2X& b); GF2X operator%(const GF2X& a, const GF2X& b); GF2X& operator/=(GF2X& x, const GF2X& a); GF2X& operator/=(GF2X& x, GF2 a); GF2X& operator/=(GF2X& x, long a); GF2X& operator%=(GF2X& x, const GF2X& b); // procedural versions: void DivRem(GF2X& q, GF2X& r, const GF2X& a, const GF2X& b); // q = a/b, r = a%b void div(GF2X& q, const GF2X& a, const GF2X& b); // q = a/b void rem(GF2X& r, const GF2X& a, const GF2X& b); // r = a%b long divide(GF2X& q, const GF2X& a, const GF2X& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 long divide(const GF2X& a, const GF2X& b); // if b | a, sets q = a/b and returns 1; otherwise returns 0 // PROMOTIONS: operator / and procedure div promote {long, GF2} to GF2X // on (a, b). /**************************************************************************\ GCD's \**************************************************************************/ void GCD(GF2X& x, const GF2X& a, const GF2X& b); GF2X GCD(const GF2X& a, const GF2X& b); // x = GCD(a, b) (zero if a==b==0). void XGCD(GF2X& d, GF2X& s, GF2X& t, const GF2X& a, const GF2X& 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 0 or 1, and a_n not zero (the zero polynomial is [ ]). On input, the coefficients may be arbitrary integers which are reduced modulo 2, and leading zeros stripped. There is also a more compact hex I/O format. To output in this format, set GF2X::HexOutput to a nonzero value. On input, if the first non-blank character read is 'x' or 'X', then a hex format is assumed. \**************************************************************************/ istream& operator>>(istream& s, GF2X& x); ostream& operator<<(ostream& s, const GF2X& a); /**************************************************************************\ Some utility routines \**************************************************************************/ void diff(GF2X& x, const GF2X& a); GF2X diff(const GF2X& a); // x = derivative of a void reverse(GF2X& x, const GF2X& a, long hi); GF2X reverse(const GF2X& a, long hi); void reverse(GF2X& x, const GF2X& a); GF2X reverse(const GF2X& a); // x = reverse of a[0]..a[hi] (hi >= -1); // hi defaults to deg(a) in second version void VectorCopy(vec_GF2& x, const GF2X& a, long n); vec_GF2 VectorCopy(const GF2X& a, long n); // x = copy of coefficient vector of a of length exactly n. // input is truncated or padded with zeroes as appropriate. // Note that there is also a conversion routine from GF2X to vec_GF2 // that makes the length of the vector match the number of coefficients // of the polynomial. long weight(const GF2X& a); // returns the # of nonzero coefficients in a void GF2XFromBytes(GF2X& x, const unsigned char *p, long n); GF2X GF2XFromBytes(const unsigned char *p, long n); // conversion from byte vector to polynomial. // x = sum(p[i]*X^(8*i), i = 0..n-1), where the bits of p[i] are interpretted // as a polynomial in the natural way (i.e., p[i] = 1 is interpretted as 1, // p[i] = 2 is interpretted as X, p[i] = 3 is interpretted as X+1, etc.). // In the unusual event that characters are wider than 8 bits, // only the low-order 8 bits of p[i] are used. void BytesFromGF2X(unsigned char *p, const GF2X& a, long n); // conversion from polynomial to byte vector. // p[0..n-1] are computed so that // a = sum(p[i]*X^(8*i), i = 0..n-1) mod X^(8*n), // where the values p[i] are interpretted as polynomials as in GF2XFromBytes // above. long NumBits(const GF2X& a); // returns number of bits of a, i.e., deg(a) + 1. long NumBytes(const GF2X& a); // returns number of bytes of a, i.e., floor((NumBits(a)+7)/8) /**************************************************************************\ Random Polynomials \**************************************************************************/ void random(GF2X& x, long n); GF2X random_GF2X(long n); // x = random polynomial of degree < n /**************************************************************************\ Arithmetic mod X^n Required: n >= 0; otherwise, an error is raised. \**************************************************************************/ void trunc(GF2X& x, const GF2X& a, long n); // x = a % X^n GF2X trunc(const GF2X& a, long n); void MulTrunc(GF2X& x, const GF2X& a, const GF2X& b, long n); GF2X MulTrunc(const GF2X& a, const GF2X& b, long n); // x = a * b % X^n void SqrTrunc(GF2X& x, const GF2X& a, long n); GF2X SqrTrunc(const GF2X& a, long n); // x = a^2 % X^n void InvTrunc(GF2X& x, const GF2X& a, long n); GF2X InvTrunc(const GF2X& 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 GF2XModulus data structure and associated routines below for better performance. \**************************************************************************/ void MulMod(GF2X& x, const GF2X& a, const GF2X& b, const GF2X& f); GF2X MulMod(const GF2X& a, const GF2X& b, const GF2X& f); // x = (a * b) % f void SqrMod(GF2X& x, const GF2X& a, const GF2X& f); GF2X SqrMod(const GF2X& a, const GF2X& f); // x = a^2 % f void MulByXMod(GF2X& x, const GF2X& a, const GF2X& f); GF2X MulByXMod(const GF2X& a, const GF2X& f); // x = (a * X) mod f void InvMod(GF2X& x, const GF2X& a, const GF2X& f); GF2X InvMod(const GF2X& a, const GF2X& f); // x = a^{-1} % f, error is a is not invertible long InvModStatus(GF2X& x, const GF2X& a, const GF2X& 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 GF2XModulus F for f. This pre-computes information about f that speeds up subsequent computations. As an example, the following routine computes the product modulo f of a vector of polynomials. #include void product(GF2X& x, const vec_GF2X& v, const GF2X& f) { GF2XModulus F(f); GF2X 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 GF2X can be used wherever a GF2XModulus is required, and a GF2XModulus can be used wherever a GF2X is required. The GF2XModulus routines optimize several important special cases: - f = X^n + X^k + 1, where k <= min((n+1)/2, n-NTL_BITS_PER_LONG) - f = X^n + X^{k_3} + X^{k_2} + X^{k_1} + 1, where k_3 <= min((n+1)/2, n-NTL_BITS_PER_LONG) - f = X^n + g, where deg(g) is small \**************************************************************************/ class GF2XModulus { public: GF2XModulus(); // initially in an unusable state ~GF2XModulus(); GF2XModulus(const GF2XModulus&); // copy GF2XModulus& operator=(const GF2XModulus&); // assignment GF2XModulus(const GF2X& f); // initialize with f, deg(f) > 0 operator const GF2X& () const; // read-only access to f, implicit conversion operator const GF2X& val() const; // read-only access to f, explicit notation long WordLength() const; // returns word-length of resisues }; void build(GF2XModulus& F, const GF2X& f); // pre-computes information about f and stores it in F; deg(f) > 0. // Note that the declaration GF2XModulus F(f) is equivalent to // GF2XModulus 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 GF2XModulus& F); // return deg(f) void MulMod(GF2X& x, const GF2X& a, const GF2X& b, const GF2XModulus& F); GF2X MulMod(const GF2X& a, const GF2X& b, const GF2XModulus& F); // x = (a * b) % f; deg(a), deg(b) < n void SqrMod(GF2X& x, const GF2X& a, const GF2XModulus& F); GF2X SqrMod(const GF2X& a, const GF2XModulus& F); // x = a^2 % f; deg(a) < n void MulByXMod(GF2X& x, const GF2X& a, const GF2XModulus& F); GF2X MulByXMod(const GF2X& a, const GF2XModulus& F); // x = (a * X) mod F void PowerMod(GF2X& x, const GF2X& a, const ZZ& e, const GF2XModulus& F); GF2X PowerMod(const GF2X& a, const ZZ& e, const GF2XModulus& F); void PowerMod(GF2X& x, const GF2X& a, long e, const GF2XModulus& F); GF2X PowerMod(const GF2X& a, long e, const GF2XModulus& F); // x = a^e % f; deg(a) < n (e may be negative) void PowerXMod(GF2X& x, const ZZ& e, const GF2XModulus& F); GF2X PowerXMod(const ZZ& e, const GF2XModulus& F); void PowerXMod(GF2X& x, long e, const GF2XModulus& F); GF2X PowerXMod(long e, const GF2XModulus& F); // x = X^e % f (e may be negative) void rem(GF2X& x, const GF2X& a, const GF2XModulus& F); // x = a % f void DivRem(GF2X& q, GF2X& r, const GF2X& a, const GF2XModulus& F); // q = a/f, r = a%f void div(GF2X& q, const GF2X& a, const GF2XModulus& F); // q = a/f // operator notation: GF2X operator/(const GF2X& a, const GF2XModulus& F); GF2X operator%(const GF2X& a, const GF2XModulus& F); GF2X& operator/=(GF2X& x, const GF2XModulus& F); GF2X& operator%=(GF2X& x, const GF2XModulus& F); /**************************************************************************\ vectors of GF2X's \**************************************************************************/ typedef Vec vec_GF2X; // 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(GF2X& x, const GF2X& g, const GF2X& h, const GF2XModulus& F); GF2X CompMod(const GF2X& g, const GF2X& h, const GF2XModulus& F); // x = g(h) mod f; deg(h) < n void Comp2Mod(GF2X& x1, GF2X& x2, const GF2X& g1, const GF2X& g2, const GF2X& h, const GF2XModulus& F); // xi = gi(h) mod f (i=1,2), deg(h) < n. void CompMod3(GF2X& x1, GF2X& x2, GF2X& x3, const GF2X& g1, const GF2X& g2, const GF2X& g3, const GF2X& h, const GF2XModulus& 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 GF2XArgument 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 GF2XArgument { vec_GF2X H; }; void build(GF2XArgument& H, const GF2X& h, const GF2XModulus& F, long m); // Pre-Computes information about h. m > 0, deg(h) < n void CompMod(GF2X& x, const GF2X& g, const GF2XArgument& H, const GF2XModulus& F); GF2X CompMod(const GF2X& g, const GF2XArgument& H, const GF2XModulus& F); extern thread_local long GF2XArgBound; // Initially 0. If this is set to a value greater than zero, then // composition routines will allocate a table of no than about // GF2XArgBound KB. Setting this value affects all compose routines // and the power projection and minimal polynomial routines below, // and indirectly affects many routines in GF2XFactoring. /**************************************************************************\ Power Projection routines \**************************************************************************/ void project(GF2& x, const vec_GF2& a, const GF2X& b); GF2 project(const vec_GF2& a, const GF2X& b); // x = inner product of a with coefficient vector of b void ProjectPowers(vec_GF2& x, const vec_GF2& a, long k, const GF2X& h, const GF2XModulus& F); vec_GF2 ProjectPowers(const vec_GF2& a, long k, const GF2X& h, const GF2XModulus& F); // Computes the vector // (project(a, 1), project(a, h), ..., project(a, h^{k-1} % f). // Restriction: must have a.length <= deg(F) and deg(h) < deg(F). // This operation is really the "transpose" of the modular composition // operation. void ProjectPowers(vec_GF2& x, const vec_GF2& a, long k, const GF2XArgument& H, const GF2XModulus& F); vec_GF2 ProjectPowers(const vec_GF2& a, long k, const GF2XArgument& H, const GF2XModulus& F); // same as above, but uses a pre-computed GF2XArgument // lower-level routines for transposed modular multiplication: class GF2XTransMultiplier { /* ... */ }; void build(GF2XTransMultiplier& B, const GF2X& b, const GF2XModulus& F); // build a GF2XTransMultiplier to use in the following routine: void UpdateMap(vec_GF2& x, const vec_GF2& a, const GF2XTransMultiplier& B, const GF2XModulus& F); vec_GF2 UpdateMap(const vec_GF2& a, const GF2XTransMultiplier& B, const GF2XModulus& 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) and deg(b) < deg(F). // This is really the transpose of modular multiplication. // Input may have "high order" zeroes stripped. // Output always has high order zeroes stripped. /**************************************************************************\ Minimum Polynomials 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(GF2X& h, const vec_GF2& 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(GF2X& h, const GF2X& g, const GF2XModulus& F, long m); GF2X ProbMinPolyMod(const GF2X& g, const GF2XModulus& F, long m); void ProbMinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F); GF2X ProbMinPolyMod(const GF2X& g, const GF2XModulus& 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; it always // returns a divisor of the minimal polynomial, possibly a proper divisor. void MinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F, long m); GF2X MinPolyMod(const GF2X& g, const GF2XModulus& F, long m); void MinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F); GF2X MinPolyMod(const GF2X& g, const GF2XModulus& F); // same as above, but guarantees that result is correct void IrredPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F, long m); GF2X IrredPolyMod(const GF2X& g, const GF2XModulus& F, long m); void IrredPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F); GF2X IrredPolyMod(const GF2X& g, const GF2XModulus& 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 \**************************************************************************/ void TraceMod(GF2& x, const GF2X& a, const GF2XModulus& F); GF2 TraceMod(const GF2X& a, const GF2XModulus& F); void TraceMod(GF2& x, const GF2X& a, const GF2X& f); GF2 TraceMod(const GF2X& a, const GF2X& f); // x = Trace(a mod f); deg(a) < deg(f) void TraceVec(vec_GF2& S, const GF2X& f); vec_GF2 TraceVec(const GF2X& 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]. /**************************************************************************\ Miscellany \**************************************************************************/ void clear(GF2X& x) // x = 0 void set(GF2X& x); // x = 1 void GF2X::kill(); // f.kill() sets f to 0 and frees all memory held by f. GF2X::GF2X(INIT_SIZE_TYPE, long n); // GF2X(INIT_SIZE, n) initializes to zero, but space is pre-allocated // for n coefficients static const GF2X& zero(); // GF2X::zero() is a read-only reference to 0 void GF2X::swap(GF2X& x); void swap(GF2X& x, GF2X& y); // swap (via "pointer swapping" -- if possible) GF2X::GF2X(long i, GF2 c); GF2X::GF2X(long i, long c); // initialize to c*X^i, provided for backward compatibility // SIZE INVARIANT: for any f in GF2X, deg(f)+1 < 2^(NTL_BITS_PER_LONG-4).