Cryptographic algorithms are fundamental to modern security, yet their implementations frequently harbor subtle logic flaws that are hard to detect. We introduce CryptoScope, a novel framework for automated cryptographic vulnerability detection powered by Large Language Models (LLMs). CryptoScope combines Chain-of-Thought (CoT) prompting with Retrieval-Augmented Generation (RAG), guided by a curated cryptographic knowledge base containing over 12,000 entries. We evaluate CryptoScope on LLM-CLVA, a benchmark of 92 cases primarily derived from real-world CVE vulnerabilities, complemented by cryptographic challenges from major Capture The Flag (CTF) competitions and synthetic examples across 11 programming languages. CryptoScope consistently improves performance over strong LLM baselines, boosting DeepSeek-V3 by 11.62%, GPT-4o-mini by 20.28%, and GLM-4-Flash by 28.69%. Additionally, it identifies 9 previously undisclosed flaws in widely used open-source cryptographic projects.