Spreading Rumors Quietly and the Subgroup Escape Problem

Spreading Rumors Quietly and the Subgroup Escape Problem Aleksandr Yampolskiy Joint work with James Aspnes, Zoë Diamadi, Kristian Gjøsteen, and René Peralta

Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Our model ,[object Object],[object Object],The British are coming!

Our model (cont.) ,[object Object],[object Object],The British are coming! The British are coming! The British are coming! The British are coming!

The adversary… ,[object Object],[object Object],You started a rumor!

The goal ,[object Object],[object Object],[object Object],[object Object]

Blind coupon mechanism ,[object Object],[object Object],[object Object],[object Object]

Blind coupon mechanism (cont.) ,[object Object],[object Object],[object Object]

Blind coupon mechanism (cont.) ,[object Object],[object Object],[object Object],[object Object],¼ ,[object Object],¼ ,[object Object],¼ 0 1 1 ,[object Object],0 0 0 c 0 1 1 c , 1 0 , 1 1

Simple inefficient construction ,[object Object],[object Object],[object Object],[object Object],[object Object]

Abstract group structure (U, G, D) ,[object Object],[object Object],[object Object],U G D invalid dummy signal

Hardness assumptions ,[object Object],[object Object],G D G ??? ¼

Hardness assumptions (cont.) ,[object Object],[object Object],G G ¼ ??? D

Generic security of subgroup escape problem ,[object Object],[object Object],[object Object],[object Object],negligible if |G|/|U| is small

The BCM on abstract group structure (U, G, D) ,[object Object],[object Object],[object Object],[object Object],[object Object],{

The BCM on abstract group structure (U, G, D) (cont.) ,[object Object]

The BCM on abstract group structure (U, G, D) (cont.) ,[object Object],[object Object],[object Object],[object Object]

Elliptic Curves over Z n ,[object Object],[object Object],[object Object],[object Object],P 1 P 2 P 1 + P 2

Elliptic Curves over Z n (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Elliptic Curves over Z n (cont.) ,[object Object],[object Object],[object Object],[object Object],U G D invalid signal dummy

Elliptic Curves over Z n (cont.) ,[object Object],[object Object],[object Object]

Bilinear groups ,[object Object],[object Object],[object Object],U G D invalid signal dummy

Bilinear groups (cont.) ,[object Object],[object Object],[object Object]

Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Yay! Almost there!

Spreading rumors with the BCM ,[object Object],[object Object],1 0 0 0

Spreading rumors with the BCM ,[object Object],$#!@ 1 1 0 0 0 1

Spreading rumors with the BCM ,[object Object],[object Object],$#!@ ,[object Object],[object Object],[object Object],1 0 0 0 1 1

Spreading rumors with the BCM (cont.) ,[object Object],[object Object]

Conclusion ,[object Object],[object Object],[object Object],[object Object],[object Object]

Open problems ,[object Object],[object Object],?

Spreading Rumors Quietly and the Subgroup Escape Problem

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Spreading Rumors Quietly and the Subgroup Escape Problem