# CS276 Lecture 26: Quadratic Residuosity and Proofs of Knowledge

Scribed by Anindya De

Summary

In this lecture, we show that the protocol for quadratic residuosity discussed last week is indeed zero-knowledge. Next we move on to the formal definition of proof of knowledge, and we show that the quadratic residuosity protocol is also a proof of knowledge. We also start discussing the primitives required to prove that any language in ${NP}$ admits a zero-knowledge proof.

# CS276 Lecture 25: Quadratic Residuosity and Zero Knowledge

Scribed by Alexandra Constantin

Summary

Today we show that the graph isomorphism protocol we defined last time is indeed a zero-knowledge protocol. Then we discuss the quadratic residuosity problem modulo a composite, and define a protocol for proving quadratic residuosity. (We shall prove that the protocol is zero knowledge next time.)

# CS276 Lecture 19: Trapdoor Permutations

Scribed by Cynthia Sturton

Summary

Today we continue to discuss number-theoretic constructions of CPA-secure encryption schemes.

First, we return to the Decision Diffie Hellman assumption (the one under which we proved the security of the El Gamal encryption scheme) and we show that it fails for ${{\mathbb Z}^*_p}$, although it is conjectured to hold in the subgroup of quadratic residues of ${{\mathbb Z}^*_p}$.

Then we introduce the notion of trapdoor permutation and show how to construct CPA-secure public-key encryption from any family of trapdoor permutations. Since RSA is conjectured to provide a family of trapdoor permutations, this gives a way to achieve CPA-secure encryption from RSA.

# CS276 Lecture 26 (draft)

Summary

After showing that last week’s protocol for quadratic residuosity is indeed zero-knowledge, we move on to the formal definition of proof of knowledge, and we show that the quadratic residuosity protocol is also a proof of knowledge.

# CS276 Lecture 25 (draft)

Summary

Today we show that the graph isomorphism protocol we defined last time is indeed a zero-knowledge protocol. Then we discuss the quadratic residuosity problem modulo a composite, and define a protocol for proving quadratic residuosity. (We shall prove that the protocol is zero knowledge next time.)

# CS276 Lecture 19 (draft)

Summary

Today we continue to discuss number-theoretic constructions of CPA-secure encryption schemes.

First, we return to the Decision Diffie Hellman assumption (the one under which we proved the security of the El Gamal encryption scheme) and we show that it fails for ${{\mathbb Z}^*_p}$, although it is conjectured to hold in the subgroup of quadratic residues of ${{\mathbb Z}^*_p}$.

Then we introduce the notion of trapdoor permutation and show how to construct CPA-secure public-key encryption from any family of trapdoor permutations. Since RSA is conjectured to provide a family of trapdoor permutations, this gives a way to achieve CPA-secure encryption from RSA.