Clarifying NOTE added on May 21, 2004
Carefully note that what is meant by "...this falsification of the kingdom to come under the name of millenarianism..." is not the real thousand years spoken of in Revelation 20:1-10.
What the Catechism is warning the faithful about is the "falsification of the kingdom", that is, when the False Christ, about whom we have written and warned the faithful about, will try to convince people to work with him to achieve the real thousand years of peace and prosperity. This "falsification of the kingdom" would be "the 'intrinsically perverse' political form of a secular messianism" that you may read below in Paragraph No. 676 and that we continually warn one and all about!
Before Christ's second coming the Church must pass through a final trial that will shake the faith of many believers. The persecution that accompanies her pilgrimage on earth will unveil the "mystery of iniquity" in the form of a religious deception offering men an apparent solution to their problems at the price of apostasy from the truth. The supreme religious deception is that of the Antichrist, a pseudo-messianism by which man glorifies himself in place of God and of his Messiah come in the flesh.
Paragraph No. 676
The Antichrist's deception already begins to take shape in the world every time the claim is made to realize within history that messianic hope which can only be realized beyond history through the eschatological judgment. The Church has rejected even modified forms o this falsification of the kingdom to come under the name of millenarianism, especially the "intrinsically perverse" political form of a secular messianism.
Paragraph No. 677
The Church will enter the glory of the kingdom only through this final Passover, when she will follow her Lord in his death and Resurrection. The kingdom will be fulfilled, then, not by a historic triumph of the Church through a progressive ascendancy, but only by God's victory over the final unleashing of evil, which will cause his Bride to come down from heaven. God's triumph over the revolt of evil will take the form of the Last Judgment after the final cosmic upheaval of this passing world.
Catechism of the
Catholic Church, pp 176-177, Liguori
Publications, ISBN 0-89243-566-6