2 D Coordinates Calculating Absolute Difference Calculating Absolute

2 D Coordinates

Calculating Absolute Difference

Calculating Absolute Difference

Calculating Manhattan Distance

Circuit Size Manhattan Distance Calculation Block Total #XOR gates #AND gates

Minimum Value and Minimum Index [Kolesnikov ’ 09]

Minimum Value and Minimum Index [Kolesnikov ’ 09] Tree of MIN blocks can find both minimum value, minimum index

Circuit Size 1 Min Value/Index Block 1 Minimum Value/Index Block Total #XOR gates #AND gates

Circuit Size Tree of Min Blocks and Manhattan Distance to L locations Tree of Minimum Value/Index Blocks #XOR gates #AND gates Manhattan Distance to L locations #XOR gates #AND gates

Complete Circuit Size Closest Location and its Manhattan Distance #XOR gates #AND gates

Privacy Preserving Application: Closest Bank/ATM Location Example: Compute the location of closest Chase or Wells Fargo ATM in Salt Lake City in a privacy preserving manner Mobile client’s private input: Coordinates representing intersection of two streets (e. g. , 800 East 100 South) Question: What remains secret in our privacy preserving application? 1. Mobile client's location 2. The computed ATM location closest to the client 3. The computed distance to the closest ATM The secret is revealed only if evaluator colludes with all servers

ATM Locations Consider the area of Salt Lake City, UT that lies between: • • Main street (represents 0 East street) 1300 East street South Temple street (represents 0 South street) 800 South Street ATM Locations Bank Location Chase 201 South 0 East Chase 185 South 100 East Chase 376 East 400 South Chase 531 East 400 South Wells Fargo 299 South 0 East Wells Fargo 381 East 300 South Wells Fargo 79 South 0 East Wells Fargo 778 South 0 East Wells Fargo 570 South 700 East Wells Fargo 235 South 1300 East Source: www. chase. com, www. wellsfargo. com

Circuit Size for Closest ATM & Distance Closest Bank ATM Location, Distance #XOR gates #AND gates

Server-Side Cost to Construct Garbled Circuit for Closest ATM

Client-Side Cost to determine the Closest ATM client-side cost grows much slowly with number of parties, in comparison to server-side cost!

Backup Slides Alternative/Direct Method to Calculate Manhattan Distance

Maximum and Minimum Value [Huang’ 12] Minimum and Maximum Value Block Total #XOR gates #AND gates

Manhattan Distance Block Total #XOR gates #AND gates