What Does This Symbol Mean? (Hardcore Predicates for One-Way Functions)

Question

I am studying Pseudorandom Number Generators and when reading the discussion on One-Way Functions and Hardcore Predicates, I came upon this equation.

$$b(x,r)=\displaystyle \bigoplus_jx_jr_j$$

I understand that the $b(x,r)$ is the hardcore predicate (either a $1$ or $0$) but I do not understand the operation being performed here. Specifically, I have never before seen the XOR symbol ($\oplus$) used in this way and do not understand what it signifies. Is it the sum of the values found by XORing $x_j$ and $r_j$? $\bmod 2$?

Answer 1

Assume $x$ and $r$ are bit strings of same length and $x_i$ and $r_i$ denotes the $i$-th bit of $x$ and $r$ respectively. The operation performed between $r_j,x_j$ is multiplication which is equivalent to AND ($\wedge$) as they can both only take the values $1$ or $0$.

Generally any large symbol like $\displaystyle\sum_jf(j)$, $\displaystyle\bigwedge_jf(j)$ and $\displaystyle\bigoplus_jf(j)$ applies the operation denoted by the large sign "between" each iterated value of $f(j)$, resulting in $\displaystyle\sum_jf(j)=f(j_1)+f(j_2)+\dots$, $\displaystyle\bigwedge_jf(j)=f(j_1)\wedge f(j_2)\wedge\dots$ and $\displaystyle\bigoplus_jf(j)=f(j_1)\oplus f(j_2)\oplus\dots$.

So the formula you gave can be read as follows:
$$b(x,r)=\displaystyle\bigoplus_{j}x_jr_j=\displaystyle\bigoplus_{j}x_j\cdot r_j=x_0r_0\oplus x_1r_1\oplus \dots=(x_0\wedge r_0)\oplus(x_1 \wedge r_1)\oplus\dots$$.