Primitive polynomials and irreducible polynomials? This question may seem annoying to some scholars as it is a very silly question but as I am new on this topic, it seems quite confusing. 


  All minimal polynomials or primitive polynomials are considered as irreducible polynomials but all irreducible polynomials are not considered as minimal/primitive polynomials. 


Can anyone please clarify this statement?
The definition of a primitive polynomial $P$ is that it is an irreducible polynomial with an extra condition, which is that $X$ is a generator of the whole group of non-zero polynomials modulo $P$.

To clarify things:

We consider a finite field $K$ (for instance, $K$ may be the integers modulo a given prime $q$). $K[X]$ is the ring of polynomials with coefficients in $K$. Each polynomial has a degree, which is the largest index of non-zero coefficient; for instance, $X^2+1$ has degree $2$. By convention, we will say that the zero polynomial ($P = 0$) has degree $-1$; other constant polynomials ($P = a$ for some constant $a \neq 0$) have degree $0$.

We can define an Euclidean division of polynomials, in which, given two polynomials $A$ and $B$, such that $B$ is not zero, one can find two polynomials $Q$ and $R$ such that $A = BQ + R$, and the degree of $R$ is strictly less than the degree of $B$. Moreover, this decomposition is unique: for given $A$ and $B$, there is a unique pair $(Q,R)$ that works. The remainder $R$ is also known as "$A$ modulo $B$". If $R = 0$, then $A$ is said to be a multiple of $B$.

An irreducible polynomial $P$ is one such that it is the multiple of no other polynomial whose degree is less than the degree of $P$, but more than $0$. For instance, over field $\mathbb{Z}_2$, the polynomial $X^2+X+1$ is irreducible; but $X^2+X$ is not, since it is the product of $X$ with $X+1$. (Note that irreducibility depends on the base field; $X^2+1$ is irreducible in $\mathbb{R}[X]$, not in $\mathbb{C}[X]$.)

Given a polynomial $P$ of degree $n &gt; 0$, we can consider $K[X]/P$, which is the ring of polynomials modulo $P$: after all additions or multiplications, we reduce the result modulo $P$. This defines a ring structure, and contains all polynomials of $K[X]$ whose degree is strictly lower than the degree of $P$ (i.e. polynomials of degree $0$ to $n-1$, inclusive). If $K$ contains $q$ elements, then $K[X]/P$ contains $q^n$ elements.

It so happens that, when $P$ is irreducible, then $K[X]/P$ is itself a field. Non-zero polynomials in $K[X]/P$ are then a multiplicative group, of size $q^n-1$.

Consider, then, the subgroup generated by $X$ (the polynomial of degree $1$ whose coefficient of index $1$ is $1$, and coefficient of index $0$ is $0$). You take $X$, $X^2 \pmod P$, $X^3 \pmod P$, and so on. At one point, you will get back to $1$. The order of $X$ modulo $P$ is the smallest $k &gt; 0$ such that $X^k = 1 \pmod P$. Since all these $X^j$ form a multiplicative group, it follows that $k$ is a divisor of $q^n-1$.

When $k$ is equal to $q^n-1$, then the polynomial $P$ is said to be primitive. This means that by computing the various $X^j$ modulo $P$, you end up covering all non-zero polynomials modulo $P$.

All primitive polynomials are irreducible, because we simply defined them that way: to be primitive, a polynomial must first be irreducible. However, not all irreducible polynomials are primitive. For instance, if we work with $K = \mathbb{Z}_2$ (the field with two elements, $0$ and $1$), then polynomial $P = X^8+X^4+X^3+X+1$ is irreducible, but not primitive (there are $255 = 2^8-1$ elements modulo $P$; but the subgroup generated by $X$ only contains $51$ of them).

Primitive polynomials are mostly relevant for linear feedback shift registers. It can be shown that the contents of a LFSR of size $n$ bits can be interpreted as a polynomial in $\mathbb{Z}_2[X]/P$ for a given polynomial $P$ of degree $n$, that depends on the positions of the feedback bits of the LFSR. Shifting the LFSR by $1$ bit then corresponds to multiplying the contents by $X$. Therefore, the period of the LFSR (number of shifts until reaching back the starting state) is maximal ($2^n-1$) only when $P$ is primitive.

LFSR are "old crypto", back from a time when that notion of period was felt to be really important (when you hear somebody talk of "the cryptoperiod", you know that he is doing cryptography from the 1960s, and he missed out the last 50 years of research in cryptography). Nowadays, LFSR and primitive polynomials are less relevant. Irreducible polynomials, however, are necessary to define field extensions. The degree-8 polynomial shown above, which is irreducible but not primitive, is used in the definition of the AES S-Box.