Symbol  Meaning 

a \mathrm{\ mod\ } b  Modulus: it indicates the remainder of the division between a and b.
For example, 20 \mathrm{\ mod\ } 7 = 6 indicates that the division between 20 and 7 gives 6 as remainder. Modulus has priority over additions and subtractions, while multiplications, divisions and powers have priority over it. Then, given the following expression: a + (b + c) x \mathrm{\ mod\ } d^2 + e
execution order of operations is: a + (((b + c) x) \mathrm{\ mod\ } (d^2)) + e

a \mathrm{\ mod^{\star}\ } b  Modulus Star: it’s defined as
a \mathrm{\ mod^{\star}\ } b := \begin{cases} b & \text{if }a \mathrm{\ mod\ } b = 0 \\ a \mathrm{\ mod\ } b & \text{otherwise} \end{cases}
It behaves like a \mathrm{\ mod\ } b, but returns b in place of 0. For example, if b = 5: 1 \mathrm{mod} 5 = 1 \mathrm{mod^{\star}} 5 = 1; Like modulus, it has priority only over additions and subtractions. 
(C) 
Integer value of a proposition: it’s defined as
(C) := \begin{cases} 1 &\text{if }C\text{ is true}\\ 0 &\text{if }C\text{ is false}\end{cases}
where C is a proposition. For example, (a \gt 10) is 1 if a \gt 10, 0 otherwise. 
\sigma_i(x_1, x_2, ... x_j) 
Elementary symmetric polynomials: they are defined as
\sigma_0(x_1, x_2, x_3, x_4, \ldots, x_j) = 1 Generally speaking, in order to compute \sigma_i(x_1, x_2, ... x_j) you have to list all the combinations i by i if the j arguments, transform each into a product, and then sum up the so obtained products: for example, \sigma_2(3, 5, 7) = 3 \cdot 5 + 3 \cdot 7 + 5 \cdot 7. If i is zero, the arguments are irrelevant, since the result is always 1. Similarly, we also agree that the result is 1 also when the argument list is empty, i.e. \sigma_i() = 1 for all i. 
[a, b]  Range: it’s the set of the integer numbers between a and b: [a,b] := \{n \in \mathbb{Z} \mid a \leq n \leq b\}. The number b  a + 1, that is the range cardinality, is defined width of the range. It’s supposed that a \leq b. 
p, q  The symbols p and q are used for indicating prime numbers, except if differently specified. 
p_i  The symbol p_i indicates the ith prime number, with i \geq 1, so that p_1 = 2, p_2 = 3, p_3 = 5, etcetera. 
q_i  The symbol q_i, with i \geq 1, indicates a number taken from a list of prime numbers, not necessarily sorted or complete. For example if the list is 5, 17, 3, then q_1 := 5, q_2 := 17 and q_3 := 3. 