Explore the critical issue of mathematical backdoors in encryption algorithms in this 50-minute Black Hat conference talk. Delve into the concept of by-design backdoors placed directly within the mathematical structure of encryption systems. Understand how these backdoors differ from implementation-level vulnerabilities and why they pose a significant challenge to detect and exploit. Learn about the historical context, regulatory landscape, and international agreements surrounding cryptographic control. Examine case studies, including the Swiss company saga and the evolution of random number generators. Analyze specific algorithmic components like design transformations, linear partitions, and secret S-boxes. Discover techniques for last-round attacks and methods to detect potential tampering in existing ciphers. Gain insights into the implications of these findings for trusting foreign encryption algorithms and the future of cryptographic security.