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In "On Defining Proofs of Knowledge" by Mihir Bellare and Oded Goldreich, the definition of knowledge soundness (KS) is (See Definition 3.1):

  • Validity (with error $\kappa$): Let $p(x)>\kappa(x)$ be the verifier accept probability. The expected number of steps of extractor bounded by: $$T1=\frac{|x|^c}{p(x)-\kappa(x)}$$

In "Incrementally Verifiable Computation" by Paul Valiant, Definition 6 (Noninteractive CS proof of knowledge), after rewrite the notation $\kappa(x)=\frac{1}{K'(|x|)}$:

  • (Kownledge extraction): Let $time_{P'}(M,x,r)\leq K'(|x|)$ and let $p(x)=\alpha>\kappa(x)$ be verifier accept probability, the expected running time is at most: $$T2=\frac{|x|^{c_2}}{p(x)}*\text{expected runtime of P'}$$

Since the expected runtime of P' is: $E_r[time_{P'}(M,x,r)]\leq K'$, we have: $$T2\leq \frac{|x|^{c_2}}{p(x)}*\frac{1}{\kappa(x)}$$

How do I prove these two definitions are equivalent?

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