| CPC G06F 21/6245 (2013.01) [H04L 9/085 (2013.01); H04L 9/3066 (2013.01); H04L 2209/56 (2013.01)] | 20 Claims |

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1. A system, comprising:
a plurality of party electronic devices, each party electronic device associated with one of a plurality of parties; and
a privacy-preserving inventory matching electronic server comprising a computer processor that is configured to:
receive, from each of the plurality of party electronic devices, an axe submission, each axe submission comprising a direction and a quantity commitment to a quantity integer, wherein the quantity integer is not revealed in the quantity commitment;
arrange the parties into data structures based on the direction in the party's axe submission;
for a first party and a second party, the first party and the second party being in different data structures, send each party's quantity commitment to the party electronic device for the other party;
receive, from the party electronic device for the first party and the party electronic device for the second party, output secret-shares of an arithmetized comparison circuit;
verify that the output secret-shares of the arithmetized comparison circuit received from the party electronic device for the first party and the party electronic device for the second party match quantity commitments to the output secret-shares sent by the respective opposite party;
identify a minimal party based on the outputs of the arithmetized comparison circuit;
generate and send a minimal party identification proof of the minimal party identification to the party electronic device for the minimal party;
receive a minimal quantity integer from the party electronic device for the minimal party, wherein the minimal quantity integer is the quantity integer for the minimal party;
reveal the minimal quantity integer to the party electronic device for the first party and the party electronic device for the second party; and
execute a trade for the minimal quantity integer.
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