This paper tackles the problem of designing efficient binary-level verification for a subset of information flow properties encompassing constant-time and secret-erasure. These properties are crucial for cryptographic implementations, but are generally not preserved by compilers. Our proposal builds on relational symbolic execution enhanced with new optimizations dedicated to information flow and binary-level analysis, yielding a dramatic improvement over prior work based on symbolic execution. We implement a prototype, Binsec/Rel, for bug-finding and bounded-verification of constant-time and secret-erasure, and perform extensive experiments on a set of 338 cryptographic implementations, demonstrating the benefits of our approach. Using Binsec/Rel, we also automate two prior manual studies on preservation of constant-time and secret-erasure by compilers for a total of 4,148 and 1,156 binaries respectively. Interestingly, our analysis highlights incorrect usages of volatile data pointer for secret erasure and shows that scrubbing mechanisms based on volatile function pointers can introduce additional register spilling which might break secret-erasure. We also discovered that gcc -O$\varnothing$ and backend passes of clang introduce violations of constant-time in implementations that were previously deemed secure by a state-of-the-art constant-time verification tool operating at LLVM level, showing the importance of reasoning at binary-level.