EmbeddingGemma 2: an open, lightweight multimodal embedding model
deepmind.google·6d ago
AI Engineering
Lobsters·Aug 13·10 min read
A long division story
TL;DR
A software engineer discovered a bug in Knuth's long division Algorithm D, leading to a new theorem on its correctness and revealing a flaw in LLVM's implementation. This oversight persisted for decades, highlighting the complexities of division in multiprecision arithmetic.
✦ Why It Matters
Review your division algorithms to ensure they align with the latest findings and avoid potential pitfalls.
Key Takeaways
- Examine division algorithms for hidden flaws.
- Consider alternative methods to avoid division in cryptographic applications.
- Stay updated on algorithmic proofs and their implications.
- Test edge cases to uncover potential errors in established algorithms.
How It Works
The implementation of long division in multiprecision arithmetic involves breaking down the division process into manageable parts, using various division algorithms. The engineer utilized a top-down approach to compute the quotient and remainder, ensuring that the highest limbs of the dividend were smaller than the divisor.
This method allowed for a structured way to handle the complexities of division, particularly in cryptographic contexts where performance is critical.
⚠ The Catch
The flaw in Algorithm D arises when the trial quotient exceeds the expected single limb size after corrections, leading to incorrect results. This specific case, which was overlooked for decades, highlights the importance of precise definitions and assumptions in algorithm design.