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

    MODULE: vec_ZZ_pE

    SUMMARY:

    Provides vectors over ZZ_pE, along with some related operations.

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

    #include <NTL/ZZ_pE.h>
    #include <NTL/vec_ZZ.h>
    #include <NTL/vector.h>

    typedef Vec<ZZ_pE> vec_ZZ_pE; // backward compatibility

    void mul(vec_ZZ_pE& x, const vec_ZZ_pE& a, const ZZ_pE& b);
    void mul(vec_ZZ_pE& x, const vec_ZZ_pE& a, const ZZ_p& b);
    void mul(vec_ZZ_pE& x, const vec_ZZ_pE& a, long b);

    void mul(vec_ZZ_pE& x, const ZZ_pE& a, const vec_ZZ_pE& b);
    void mul(vec_ZZ_pE& x, const ZZ_p& a, const vec_ZZ_pE& b);
    void mul(vec_ZZ_pE& x, long a, const vec_ZZ_pE& b);
    // x = a * b

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

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

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

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

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

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

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

    void VectorCopy(vec_ZZ_pE& x, const vec_ZZ_pE& a, long n);
    vec_ZZ_pE VectorCopy(const vec_ZZ_pE& 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_pE& x, long n);  // x = random vector of length n
    vec_ZZ_pE random_vec_ZZ_pE(long n);

    // operator notation:

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

    vec_ZZ_pE operator-(const vec_ZZ_pE& a);

    // vector/scalar multiplication:

    vec_ZZ_pE operator*(const vec_ZZ_pE& a, const ZZ_pE& b);
    vec_ZZ_pE operator*(const vec_ZZ_pE& a, const ZZ_p& b);
    vec_ZZ_pE operator*(const vec_ZZ_pE& a, long b);

    vec_ZZ_pE operator*(const ZZ_pE& a, const vec_ZZ_pE& b);
    vec_ZZ_pE operator*(const ZZ_p& a, const vec_ZZ_pE& b);
    vec_ZZ_pE operator*(long a, const vec_ZZ_pE& b);

    // inner product:

    ZZ_pE operator*(const vec_ZZ_pE& a, const vec_ZZ_pE& b);

    // assignment operator notation:

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

    vec_ZZ_pE& operator*=(vec_ZZ_pE& x, const ZZ_pE& a);
    vec_ZZ_pE& operator*=(vec_ZZ_pE& x, const ZZ_p& a);
    vec_ZZ_pE& operator*=(vec_ZZ_pE& x, long a);