Encoding Multiple Slots in CKKS Using Conjugates

As cryptographers, we often encounter complex concepts that can be challenging to grasp. In this article, I will demystify one such concept – negligible functions – and explain its significance in the field of cryptography. By using everyday language and engaging metaphors, I hope to provide a comprehensive understanding of this crucial component without oversimplifying it.
What are Negligible Functions?
In cryptography, we often encounter functions that decrease rapidly as the input size grows. These functions are called negligible functions. Imagine you have a bucket filled with water, and you gradually add more water to it. If the amount of water added is very small compared to the original volume, then the bucket will continue to hold most of its original water. Similarly, negligible functions decrease rapidly as the input size grows, making them essential in quantifying the advantage an adversary could gain over random guessing.
The Role of Negligible Functions in Security Proofs
Negligible functions play a critical role in security proof verification within cryptographic systems. When we prove that an adversary’s probability of distinguishing between ciphertexts is bounded by a negligible function, we ensure the scheme’s IND-CPA security. Think of this as a game of chance where you need to show that the opponent’s chances of winning are extremely small, making it unlikely for them to beat you.
Precomputing Procedure and Asymptotic Complexity
The precomputing procedure is usually carried out at an offline stage, and the cost is not a significant concern in practice. The asymptotic complexity of the precomputing algorithm is relatively efficient, with a constant cost that doesn’t depend on the scale of the plaintexts. Compared to the overhead of Rache, which grows logarithmically with the maximum value of the plaintext space, the cost of using negligible functions is much more favorable.
Conclusion
In conclusion, negligible functions are a crucial component in cryptographic security proofs. By understanding their properties and how they relate to adversary probabilities, we can develop more robust and secure cryptographic systems. I hope this summary has provided a comprehensive introduction to the concept of negligible functions without oversimplifying it. If you have any questions or would like further clarification, please feel free to ask!