Systems and methods for finding a value in a combined list of private values

A system for computing a value at a position N in a combined list of values, the system comprising:

• at least one processor;
• and
• a computer-readable storage device storing instructions which, when executed by the processor, cause the processor to perform operations comprising:
• (1) providing a value Ri to a group of members;
• (2) computing, by each member of the group of members, how many numbers are smaller than Ri in a respective list of values for each respective member of the group of members to yield a respective number of values smaller than Ri;
• (3) computing, via the at least one processor, by each member of the group of members and using a secure multi-party computation protocol and the respective number of values smaller than Ri, a total number of smaller values (Pi);
• (4) identifying a position of Ri in a combined list of values comprising each respective list of values, the position of Ri equaling Pi+1;
• (5) when N=Pi+1, returning Ri;
• (6) when N is greater than Pi+1, removing, by each member, all values smaller than Ri in their respective list of values and sets N=N−(Pi+1);
• (7) when N is less than Pi+1, removing, by each member, all numbers bigger than Ri in their respective list of value;
• (8) setting an iteration value i=i+1;
• and
• (9) returning to step (5).