Multiplication and squaring the binary polynomials I have tried to calculate $trace$ of a coordinate $X$ of EC in binary representation. Before that I tried to pre-calculate traces of the various bits of $X$ using formula: 
$$Tr(X) = Tr(\sum_{i = 0}^{m-1} c_ix_i)= \sum_{i = 0}^{m-1}(c_i Tr(x_i) )$$ 

For the field $F_{2^{163}}$ that i use  we have only 2 cases $x_i$ that yield  $Tr(x_i) = 1$ to common $trace$. This cases  when $i$ - 0 and 157. With 0 is ok 
but for others cases squaring the polynomials and reduction by modulus lead to the performance issues.

I would be appreciated if anyone proposes a more effective approach to such calculations. Or any references to ready implementations or services which can be used to calculate traces
What do you mean by trace of "bits" of $X$? How do you define $x_i$? 

Since $263$ is prime, the field $\mathbb{F}_{2^{163}}$ has no subfields other than the prime field $\mathbb{F}_2.$ This means that the trace function $$Tr:\mathbb{F}_{2^{163}}\rightarrow \mathbb{F}_2$$ is the only trace which exists for this field. However, this means that the function $Tr(u)$ is equidistributed on $\mathbb{F}_{2}$, which means it is $0$ half the time and $1$ half the time as $u$ ranges over $\mathbb{F}_{2^{163}}.$ 

This property would also apply to any "bit" of $X$, reasonably defined as the coordinate of $X$ with respect to some basis element. 

Edit:  If as in my comment $u=\alpha^i$, squaring gives $u^2=\alpha^{2i}.$
So square of $e_i$ is $e_{2i}$ but if $2i\geq n=163$  it needs to be reduced mod the field defining polynomial.

In $\mathbb{F}_{2^4}$. defined by $\alpha^4+\alpha+1=0,$ 
$\alpha=e_1=0010,$ $\alpha^2=e_2=0100,$ but $\alpha^4=\alpha+1=e_1+e_0=0011.$