The structure of invertible substitutions on a three-letter alphabet

AbstractWe study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw∘σ1∘σ2∘⋯∘σk, where Iw is the inner automorphism associated with w, and σj∈S for 1⩽j⩽k. As a consequence, M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices

Advances in the Bacteriophage-Based Precise Identification and Magnetic Relaxation Switch Sensor for Rapid Detection of Foodborne Pathogens

The development of novel and highly specific technologies for the rapid and sensitive detection of foodborne pathogens is very important for disease prevention and control. Bacteriophages can recognize viable and unviable bacteria, replacing antibodies as the recognition element in the immune response, which are currently being widely developed in novel precise identification biosensors. Magnetic relaxation switch sensors based on the magnetic relaxation signal has been used to construct a variety of background-free novel biosensors in recent years, which can realize rapid detection of foodborne pathogens. This chapter will mainly introduce the latest developments and future prospects of bacteriophages in the field of accurate identifications for foodborne pathogens. At the same time, it will introduce the research progress and development direction of novel magnetic relaxation switch sensors for detecting foodborne pathogens

PEGASUS: Bridging Polynomial and Non-polynomial Evaluations in Homomorphic Encryption

Homomorphic encryption (HE) is considered as one of the most important primitives for privacy-preserving applications. However, an efficient approach to evaluate both polynomial and non-polynomial functions on encrypted data is still absent, which hinders the deployment of HE to real-life applications. To address this issue, we propose a practical framework PEGASUS. PEGASUS can efficiently switch back and forth between a packed CKKS ciphertext and FHEW ciphertexts without decryption, allowing us to evaluate arithmetic functions efficiently on the CKKS side, and to evaluate look-up tables on FHEW ciphertexts. Our FHEW ! CKKS conversion algorithm is more practical than the existing methods. We improve the computational complexity from linear to sublinear. Moreover, the size of our conversion key is significantly smaller, e.g., reduced from 80 gigabytes to 12 megabytes. We present extensive benchmarks of PEGASUS, including sigmoid/ReLU/min/max/division, sorting and max-pooling. To further demonstrate the capability of PEGASUS, we developed two more applications. The first one is a private decision tree evaluation whose communication cost is about two orders of magnitude smaller than the previous HE-based approaches. The second one is a secure K-means clustering that is able to run on thousands of encrypted samples in minutes that outperforms the best existing system by 14 – 20. To the best of our knowledge, this is the first work that supports practical K-means clustering using HE in a single server setting