Understanding DIQKD protocol, a few questions I'm refering to this paper here "Fully Device-Independent Quantum Key Distribution" (Umesh Vazirani and Thomas Vidick) and unfortunately there are many things I don't understand.

1) Page 3: by the word round, what do the authors actually mean? Is it right to say it's basically how many times Alice and Bob use their own devices?

2) Page 3, Testing section: this equality aᵢ ⊕ bᵢ = xᵢ ∧ yᵢ I was told it describes the requirement of the CHSH game. Alice input can be anything belonging to the set {0,1,2} while Bob's input can only be an element of the following set{0,1}. Their output can be 0 or 1, respectively.
The paper says "They compute the fraction of inputs in B that satisfy the CHSH condition aᵢ ⊕ bᵢ = xᵢ ∧ yᵢ..". My understanding is: if ⊕ is exclusive OR and ∧ is logical AND then the following makes sense:  aᵢ=1 bᵢ=0    1 ⊕ 0 = 1 ∧ 1 Is that correct? If I'm right what does the following expression mean? "If this fraction is smaller than cos2 π=8 − η they abort the protocol"



Cross-posted on quantumcomputing.SE
All of these are defined in page 2 of that paper.

1) A round is a single interaction with the device: Alice gives it an input, Bob gives is  an input, and the device outputs a bit to Alice and a bit to Bob.

2) While Alice's input is from the set $\{0,1,2\}$, the output she gets from the device, $a_i$, is a single bit. Then, the equation $a_i \oplus b_i = x_i \wedge y_i$ is on bits (the case of $x_i=2$ is treated separately). This equation is indeed the CHSH condition. If the bits where classical, the expected CHSH term is bounded by $2$, while if the system is quantum, values of up to $2\sqrt2$ are possible. By measuring the amount of times the equation holds you can estimate the value of the expected CHSH term, and if it is large enough, this implies the generation of "true" randomness.

There is a longer and more detailed version of that paper in 
https://arxiv.org/abs/1210.1810