This thesis focuses on establishing efficient parallel repetition theorems for computationally sound protocols, which assert that under parallel repetition, the computational soundness error of interactive protocols decreases at an exponential rate, and ideally, behaves as if the repetitions are completely independent. For example, suppose a protocol ⟨P, V⟩ has soundness error δ, then its n-fold parallel repetition ⟨Pⁿ, Vⁿ⟩, where V ⁿ (called direct-product verifier) accepts iff all n subverifiers accept, should have soundness error δ ⁿ.

The soundness error captures the probability of breaking a cryptographic protocol and/or the probability of convincing a party of a false assertion. Parallel repetition is a simple and desirable way to amplify soundness since it preserves the round complexity. However, existing negative examples show that this does not hold for all interactive protocols. Therefore, the question is, for what classes of protocols do parallel repetition theorems hold?

We prove new parallel repetition theorems for several classes of protocols such as public-coin protocols, three-message protocols, and a more general class of "simulatable" protocols. For some settings such as public-coin protocols with direct product verifiers, we obtain tight results that match information-theoretic bounds. In addition, we will discuss strength and limitations of different reduction ideas. We hope that the discussion can make the current progress more transparent, and lead to better understanding of parallel repetition.

The reductions used for proving parallel repetition theorems have several applications, in particular, to security amplification. We will also present our work on improving the efficient of security amplification for cryptographic primitives such as commitment schemes, signature schemes, message authentication codes, CAPTCHAs, etc.