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poly.h

POLY.H is a library that supports diverse polynomial calculation including linear computation, multiplication, inversion, square root, logarithm and exponent. Also, it supports polynomial with arbitrary modulus.

Usage

POLY.H is implemented in C++. You only need to download the header file poly.h.

There are two classes in POLY.H: poly and m_poly, both inside namespace fstdlib. You can call them by fstdlib::poly and fstdlib::m_poly, or simply use using namespace fstdlib.

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poly is a polynomial class with fixed modulus fstdlib::mod, whose value is $998,244,353$ by default. This varible is constant. If you are to modify it, you have to define macro VARIABLE_MODULO in advance and ensure that:

• The new modulus can be used for NTT. In other words, it has to be a prime and can be represented by $k \cdot 2^r + 1$ where $k$ and $r$ are integers and $2^r$ is big enough.
• Twice this modulus is within the range of 32b signed integer.

Besides, if $3$ is not a primitive root of your modulus, you have set fstdlib::grt to a primitive root of your modulus.

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m_poly is a polynomial class with arbitrary modulus mod, which is a 32b signed integer member of each single instance. You should ensure:

• Twice your modulus is within the range of 32b signed integer.
• Any instances that are calculated together have the same modulus.
• If the modulus is not a prime, no calculation except for linear computation and multiplication is applied.

Constructors

Method	Intro
poly(std::size_t n)	Create a polynomial of length $n$. All elements are initialized with $0$.
poly(std::vector<int> a)	Create a polynomial by a vector.
m_poly(std::size_t n, int p)	Create a polynomial of arbitrary modulus $p$ and length $n$. All elements are initialized with $0$.
m_poly(std::vector<int> a, int p)	Create a polynomial of arbitrary modulus $p$ by a vector.

Overloaded Operators

Method	Intro	Time Complexity
poly operator*(const poly &, const poly &)	Convolution of two polynomials.	$O(n\log n)$
poly &operator*=(poly &, const poly &)	Convolution of two polynomials.	$O(n\log n)$
poly operator*(const poly &, const int &	Convolution of a polynomial and a monomial.	$O(n)$
poly &operator*=(poly &, const int &)	Convolution of a polynomial and a monomial.	$O(n)$
poly operator*(const int &, const poly &)	Convolution of a polynomial and a monomial.	$O(n)$
poly operator>>(const poly &, const int &)	Right shift of a polynomial.	$O(n)$
poly &operator>>=(poly &, const int &)	Right shift of a polynomial.	$O(n)$
poly operator<<(const poly &, const int &)	Left shift of a polynomial.	$O(n)$
poly &operator<<=(poly &, const int &)	Left shift of a polynomial.	$O(n)$
poly operator+(const poly &, const poly &)	Sum of two polynomials.	$O(n)$
poly &operator+=(poly &, const poly &)	Sum of two polynomials.	$O(n)$
poly operator+(const poly &, const int &	Sum of a polynomial and a monomial.	$O(n)$
poly &operator+=(poly &, const int &)	Sum of a polynomial and a monomial.	$O(1)$
poly operator+(const int &, const poly &)	Sum of a monomial and a polynomial.	$O(n)$
poly operator-(const poly &, const poly &)	Substraction of two polynomials.	$O(n)$
poly &operator-=(poly &, const poly &)	Substraction of two polynomials.	$O(n)$
poly operator-(const poly &, const int &	Substraction of a polynomial and a monomial.	$O(n)$
poly &operator-=(poly &, const int &)	Substraction of a polynomial and a monomial.	$O(1)$
poly operator-(const int &, const poly &)	Substraction of a monomial and a polynomial.	$O(n)$
poly operator/(const poly &, const int &)	Division of a polynomial and a monomial.	$O(n)$
poly operator/=(const poly &, const int &)	Division of a polynomial and a monomial.	$O(n)$

Member Functions

Method	Intro	Time Complexity
poly poly::inv(void)const	Inversoin of a polynomial.	$O(n\log n)$
poly poly::inv(std::size_t)const	Inversion of a polynomial of a specified length.	$O(n\log n)$
poly poly::prefix(std::size_t)const	Prefix of a polynomial. The parameter can be greater than the polynomial's length.	$O(n)$

Other Functions

Method	Intro	Time Complexity
poly sqrt(const poly &)	Square root of a polynomial.	$O(n\log n)$
poly log(const poly &)	Exponent of a polynomial.	$O(n\log n)$
poly exp(const poly &)	Logarithm of a polynomial.	$O(n\log n)$

• The constant term of the polynomial sqrt applied to must be $1$.
• The constant term of the polynomial log applied to must be $1$.
• The constant term of the polynomial exp applied to must be $0$.