The liturgical gesture is an ancient one, dating back to the early Church.
Here is a brief article on the ancient roots of the greeting at Mass. Before the reception of Holy Communion, the priest or deacon will say to the people; “Let us one another a sign of peace.” It is then customary in the United States to say to one another “Peace” or “Peace be with you,” and to shake the hand of those standing in close proximity. What’s interesting is that this liturgical gesture has ancient roots and has been part of the Mass since the very beginning. The spiritual symbolism of the act is found in the Gospel of Matthew, where Jesus said, “If you are offering your gift at the altar, and there remember that your brother has something against you, leave your gift there before the altar and go; first be reconciled to your brother, and then come and offer your gift” (Matthew 5:23-24).