Integrating　data　from　multiple　parties　to　achieve　cross-institutional　machine　learning　is　an　important　trend　in　Industry　4.0　era.　However,　the　privacy　risks　from　sharing　data　pose　a　significant　challenge　to　data　integration.　To　integrate　data　without　sharing　data　and　meet　large-scale　samples＇　modeling　needs,　we　propose　two　vertical　federation　learning　algorithms　for　ridge　regression　via　least-squares　solution　for　two-party　and　multi-party　scenarios,　respectively.　Compared　with　the　state-of-the-art　algorithms,　our　algorithms　only　need　one　round　of　calculation　for　the　optimization　instead　of　iteration.　Furthermore,　our　algorithms　can　effectively　handle　large-scale　samples　due　to　the　number　of　cryptographic　operations　in　our　algorithms　being　independent　of　the　number　of　samples.　Through　our　proposed　the　matrix　secure　agent　computing　theory　and　δ-data　indistinguishability　theory,　we　provide　quantitative　theoretical　guarantees　for　the　security　of　our　algorithms.　Our　algorithms　satisfy　complete　data　indistinguishability　under　the　＇semi-honest＇　assumption　and　the　quantitative　security　under　the　＇malicious＇　assumption.　The　experiments　show　that　our　proposed　algorithm　takes　only　about　400　seconds　to　handle　up　to　9.6　million　large-scale　samples,　while　the　state-of-the-art　algorithms　take　close　to　1000　seconds　to　handle　every　1000　samples,　which　embodies　the　advantage　of　our　algorithms　in　handling　large-scale　samples.　©　2013　IEEE.