the aim of the present invention is to propose an alternative scheme to the classical boneh - franklin scheme in order to simplify the generation and the use of the asymmetric keys . according to the present invention , it is proposed a method to generate an i - th private key in a public key encryption scheme with traceable private keys formed by a public component γ and a secret component θ i , according to a maximal coalition factor k , with all arithmetic operations performed within a multiplicative group z / qz where q is a prime number , said public component being defined as : γ pl = and said secret component being defined as : θ i = ∑ r j  α j ∑ r j  γ j ( i )  mod   q where r j and α j are random values in the group z / z .