    /**************************************************************************\

    MODULE: vec_zz_p

    SUMMARY:

    Provides vectors over zz_p, along with some related operations.

    \**************************************************************************/

    #include "zz_p.h"
    #include "vec_zz.h"
    #include <NTL/vector.h>

    typedef Vec<zz_p> vec_zz_p; // backward compatibility

    void mul(vec_zz_p& x, const vec_zz_p& a, zz_p b);
    void mul(vec_zz_p& x, const vec_zz_p& a, long b);

    void mul(vec_zz_p& x, zz_p a, const vec_zz_p& b);
    void mul(vec_zz_p& x, long a, const vec_zz_p& b);
    // x = a * b

    void add(vec_zz_p& x, const vec_zz_p& a, const vec_zz_p& b);
    // x = a + b

    void sub(vec_zz_p& x, const vec_zz_p& a, const vec_zz_p& b);
    // x = a - b

    void clear(vec_zz_p& x);
    // x = 0 (length unchanged)

    void negate(vec_zz_p& x, const vec_zz_p& a);
    // x = -a

    long IsZero(const vec_zz_p& a);
    // test if a is the zero vector

    void VectorCopy(vec_zz_p& x, const vec_zz_p& a, long n);
    vec_zz_p VectorCopy(const vec_zz_p& a, long n);
    // x = a copy of a of length exactly n.
    // The input is truncated or padded with zeroes, as necessary.

    void random(vec_zz_p& x, long n);  // x = random vector of length n
    vec_zz_p random_vec_zz_p(long n);

    void InnerProduct(zz_p& x, const vec_zz_p& a, const vec_zz_p& b);
    // x = sum_{i=0}^{n-1} a[i]*b[i], where n = min(a.length(),
    // b.length())

    void InnerProduct(zz_p& x, const vec_zz_p& a, const vec_zz_p& b,
                      long offset);
    // x = sum_{i=offset}^{n-1} a[i]*b[i-offset], where n = min(a.length(),
    // b.length()+offset)

    long CRT(vec_ZZ& a, ZZ& prod, const vec_zz_p& A);
    // Incremental Chinese Remaindering: If p is the current zz_p modulus with
    // (p, prod) = 1; Computes a' such that a' = a mod prod and a' = A mod p,
    // with coefficients in the interval (-p*prod/2, p*prod/2];
    // Sets a := a', prod := p*prod, and returns 1 if a's value changed.

    // operator notation:

    vec_zz_p operator+(const vec_zz_p& a, const vec_zz_p& b);
    vec_zz_p operator-(const vec_zz_p& a, const vec_zz_p& b);

    vec_zz_p operator-(const vec_zz_p& a);

    // vector/scalar multiplication:

    vec_zz_p operator*(const vec_zz_p& a, zz_p b);
    vec_zz_p operator*(const vec_zz_p& a, long b);

    vec_zz_p operator*(zz_p a, const vec_zz_p& b);
    vec_zz_p operator*(long a, const vec_zz_p& b);

    // inner product:

    zz_p operator*(const vec_zz_p& a, const vec_zz_p& b);

    // assignment operator notation:

    vec_zz_p& operator+=(vec_zz_p& x, const vec_zz_p& a);
    vec_zz_p& operator-=(vec_zz_p& x, const vec_zz_p& a);

    vec_zz_p& operator*=(vec_zz_p& x, zz_p a);
    vec_zz_p& operator*=(vec_zz_p& x, long a);