Abstract: To explore the vertical vibration response of friction piles in saturated frozen soil foundations under the action of vertical P-waves, based on the theory of wave propagation in saturated frozen soil and considering the interaction among soil skeletons, ice and water, a theoretical model for the propagation of friction piles in saturated frozen soil under the action of vertical P-waves is established. By using the Helmholtz decomposition method and the separation variable method, combined with the boundary conditions and the propagation characteristics of P-waves, the analytical solution of saturated permafrost under the action of vertical P-waves is given. Considering the coupling effect between the friction pile and saturated frozen soil, the analytical solution of the pile's vibration in saturated frozen soil is obtained. Then, the motion response factor of the friction pile, the motion amplification factor of the friction pile, and the impedance of saturated frozen soil to the pile are given. The correctness of the calculation model were verified through numerical examples. Through parameter analysis, the influence laws of porosity, pile slenderness ratio, pile-soil modulus ratio, Poisson's ratio, soil saturation, soil damping and temperature on the vertical seismic characteristics of the pile body were studied. The research finds that the pile-soil modulus ratio, pile-slenderness ratio, soil saturation, soil damping and temperature have significant effects on the vibration of friction piles in saturated frozen soil, while the Poisson's ratio of the pile and the porosity ratio of saturated frozen soil have little influence on the motion response of the pile. The research results provide a theoretical basis for the seismic design of friction piles in saturated frozen soil in practical engineering, and offer key evidence for optimizing seismic parameters and improving the prediction accuracy of the interaction between frozen soil and piles.