Explore faster algorithms for approximate common divisors in this 25-minute conference talk from Eurocrypt 2012. Delve into the security of Fully Homomorphic Encryption (FHE) schemes and the Approximate Common-Divisor Problem. Learn about classical tools like Product Trees and Evaluation, and discover applications to various cryptographic problems including noisy factoring, RSA-CRT with noisy messages, and low-exponent RSA. Gain insights into key recovery attacks and message recovery techniques. Conclude with an analysis of FHE schemes and their implications for cryptographic security.