congruence modulo and equality

Question

why in cryptography most of the equalities written in the form of

$$a:=b$$

why not we write $a=b$

why in congruence modulo $a \equiv c \pmod b$ that bracket is put. Is it refers the priority.

can some one help me in the basic doubt please

Answer 1

Usually $:=$ is used to denote assignment, not equality. You'd write $a := b$ if you want to express "set $a$ to $b$'s value", and $a = b$ to express the assertion "$a$ and $b$ have the same value".

For congruences, the modulus applies to the equivalence operator, not to one side of the equation. $a \equiv b \,(\textrm{mod} n)$ is read as "$a$ is congruent modulo n to $b$". This is different from saying "$a$ equals $b \textrm{ mod } n$, where you state the $a$ has the same value as the expression $b \textrm{ mod } n$. The brackets around the $\textrm{mod } n$ in a congruence serve to make this distinction more clear, and to avoid ambiguities.

BTW, some people write $a \equiv_{n} b$ instead of $a \equiv b \,(\textrm{mod } n)$, to emphasize that it's the equivalence relation that is affected by the modulus, not the individual sinces of the equation.