Oded Goldreich Weizmann Institute of Science Demystifying the Master Thesis and Research in General: The Story of Some Master Theses Oded Goldreich Weizmann Institute of Science

My own thesis (1981) :                A permutation group over D is represented by a set of generators S. The group is denoted <S>. <S> = {g1○g2 ○ ∙ ∙ ∙ ○gt : g1,g2,…,gtS} Given S and a permutation π, does p belong to <S>? Given S, π, and t, can π be expressed by a sequence of up to t elements of S?

My first MSc student: Ronen Vainish (1988) Background: A general construction of secure multi-party protocols by reduction to the two-party case. Suffices to compute the inner product mod 2 of two input vectors held by the two parties. 1st 2nd Inputs: x1,…,xn y1,…,yn Outputs: r r+∑ixiyi Study it 1st 2nd Inputs: x,z y Outputs: - z+xy The $i$th invocation uses inputs $(x_i,r_i)$ and $y_i$, where $r_i\rnd\bitset$. The final output is the sum of $r_i$’s and sum of $r_i+x_iy_i$’s. Sender Receiver Inputs: s0,s1 c Outputs: - sc

Eyal Kushilevitz (1989) Background: Few sets known to have perfect zero-knowledge proof systems. E.g., Graph-Iso, Quad-Res. Can we provide stronger evidence to PZK not in BPP? Solve it YES: A promise problem based on DLP.

Ran Canetti (1992) Invent your own... (inspired by a course) Background: communication complexity, gap between the complexity of randomized and deterministic protocols. Is there a randomness-communication trade-off? YES: Presents a trade-off. The ID function: two parties, each holds an n-bit long string. Deterministic lower bound: need n bits of communication. Randomized protocols: (1) via error-correcting codes: send a random position. (2) via the CRT: send integer modulo a random prime

Iftach Haitner (2004) Sender Receiver Inputs: s0,s1 c Outputs: - sc Background: assuming a collection of TDP {fi:Di→Di} Sender Receiver Inputs: s0,s1 c desired outputs: - sc selects an index i yc=fi(xc) , y1-c find the fi-preimages of both: z0 , z1 b(z0)+s0 , b(z1)+s1 The problem: what is assumed about sampling Di? Can we relax?

Lidor Avigad (2009) Background: property testing, the dense graph model, lowest level of query complexity. Specifically, c-CC is in that low level. Extend this result The work: Testing Graph blow-up in minimum query complexity (i.e., linear in 1/proximity, non-adaptively)

The End The slides of this talk are available at http://www.wisdom.weizmann.ac.il/~oded/T/de-mysti.ppt