Large integer multiplication is becoming a major challenge for accelerating cryptographic applications on FPGAs. Existing techniques for decomposing and optimizing large integer multiplication bring about nontrivial trade-offs between different resource types as well as performance. In this work, we regard determining the level and order of multiplication decomposition as a phase ordering problem and leverage equality saturation to automatically produce a wide range of equivalent integer multiplication expressions with various hardware costs. We devise ILP-based extraction techniques with multiple objectives to automatically choose the optimal expressions based on the application requirements. Our framework offers significant control over resource utilization and balance, and it increases the maximum number of instances of cryptographic applications on FPGA.