To perform key updation which simultaneously rejects two designated terminals by making all greatest common divisors of secret keys of two arbitrary terminals different, making all greatest common divisors of secret keys of three optional terminals the same and making each terminal except the two rejected terminals calculate an updation key from the information undergoing multi-address communication and a secret key of a self-station.

First terminal information is an integer ei which satisfies a relation that becomes expressions I and II (where, 1 ≤ i, j and k ≤ N; i ≠j ≠ k ≠ i; d is a prescribed integer) and 2nd terminal information makes K information for producing an updating key, and is Xi of an expression III of an exponentiation residue value of the K which makes the ei power and an integer (n) a law. All terminals (i) except terminals (j) and (k) calculate integers (a), (b) and (c) which satisfy an expression IV, by using 1st terminal information ei, ej and ek of the terminals (i), (j) and (k) and all the terminals (i) except the terminals (j) and (k) calculate an expression V by using 2nd terminal information Xi, Xj and Xk of the terminals (i), (j) and (k) and calculate an updating key.