With the rising volume of transactions on blockchains, signature verification becomes a critical bottleneck of efficiency, hindering scalability and performance. This paper presents a general approach to batch verification of arbitrary signatures on blockchain. By leveraging the memory-friendliness of incremental verifiable computation (IVC) and optimizing for blockchain environments, the proposed scheme can enhance scalability, reduce memory consumption, and ensure compatibility with common devices while supporting an arbitrary number of signature verifications. This approach allows for the concurrent generation of IVC proofs while receiving signatures from other nodes, making it particularly well-suited for low-latency blockchain applications. As a concrete instantiation of our approach, we introduce BEATSBEATS (Batch ECDSA Transaction verification Scheme), where the underlying SNARK is instantiated by SpartanSpartan with Bulletproof commitment. Our implementation, evaluated on a virtual machine with 8 cores and 16 GB RAM, shows significant performance gains compared to SpartanBPSpartanBP, which is the direct construction using SpartanSpartan with BulletproofBulletproof commitment to verify a batch of ECDSA. The comparison shows that BEATSBEATS speeds up the prover by 3–7 times and the verifier by 48–240 times when handling up to 211211 ECDSA signatures, the maximum batch size supported by SpartanBPSpartanBP. For larger batches exceeding 210210, our scheme outperforms the baseline approach, which verifies ECDSA signatures one by one without any proof system. Our verifier achieved a speedup of 21–174 times compared to the baseline as the batch size grows to 220220. Furthermore, BEATSBEATS exhibits a remarkably low memory footprint, with peak memory usage remaining below 1 GB.

What happens when teachers from different educational and cultural backgrounds learn from one another over time? A study of mathematics teachers who participated in a seven-year China–Canada reciprocal learning project explores how cross-cultural professional exchange can move beyond sharing classroom practices to foster deeper mutual understanding and collaborative innovation. The findings identify three stages of reciprocal learning—getting to know each other, critically examining different practices, and learning through deeper understanding—and highlight the importance of cultural awareness, classroom observation, and sustained partnerships.

Dr. Daniel Hajovsky, associate professor in the Department of Educational Psychology at Texas A&M University, studies human intelligence and the cognitive processes that affect learning. In a study published in the Journal of Intelligence, Hajovsky and his co-authors examined the relationship between cognitive abilities and mathematics skills. The findings point to fluid reasoning and comprehension knowledge as two of the most consistent cognitive predictors of math skills.