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From Byzantine Consensus to Blockchain Consensus
Published in Kuan-Ching Li, Xiaofeng Chen, Hai Jiang, Elisa Bertino, Essentials of Blockchain Technology, 2019
There is a line or research on useful puzzles, i.e., on alternatives to PoW that spend energy solving useful tasks. The challenge is to find puzzles that are hard to compute, but with solutions that that can be verified efficiently. Primecoin seems to be the first cryptocurrency based on such an puzzle [145]. Its puzzles involve finding chains of prime numbers that are large enough to be hard to find, but not to verify. These prime numbers might be useful as cryptographic keys. Permacoin substitutes PoW by proofs-of-retrievability (PoR) to support distributed storage of data [146]. The purpose of a PoR is to certify that a node is using storage space to store a file. Therefore, Permacoin is a peer-to-peer file storage system in which nodes have an incentive to provide storage space, instead of mere altruism.
Numbers and Elementary Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A prime number is a positive integer greater than 1 with no positive, integral divisors other than 1 and itself. There are infinitely many prime numbers, 2, 3, 5, 7, . . .. The sum of the reciprocals of the prime numbers diverges: ∑n1pn=13+15+17+…=∞. $ \mathop \sum \limits_{n}^{{}} \frac{1}{{p_{n} }} = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \ldots = \infty . $
Spooky Action at a Distance
Published in Ted G. Lewis, The Signal, 2019
Problem number two for Diffie–Hellman–Merkle is the growing shortage of prime numbers. As far as we know, there are an infinite number of prime numbers if an infinite number of digits are allowed. Practically speaking, there is only a finite supply of prime numbers. With billions of public–private key pairs being used every day to support secure transactions over the global Internet, the finite number of primes is quickly used up. When a prime number has been used too often, it becomes a target for brute-force methods of discovery.
Fostering collateral creativity through teaching school mathematics with technology: what do teachers need to know?
Published in International Journal of Mathematical Education in Science and Technology, 2022
What is a prime number? One of the definitions is as follows: Prime number is a natural number greater than 1 which is not a product of smaller natural numbers. How can one find such numbers? The first number satisfying the definition is 2. It is a product of two numbers, 1 and 2, but only 1 is smaller than 2. The same can be said about 3. It is a product of two numbers 1 and 3, but only 1 is smaller than 3. Yet the number 4 is not a prime as it can be written as a product of two 2’s. In the 3rd century B.C., a Greek scholar Eratosthenes devised a method (nowadays called the sieve of Eratosthenes; the term sieve used in the previous section was borrowed from this context) of separating primes from natural numbers by continuously eliminating all multiples of 2, then all multiples of 3, then all multiples of 5 (the third prime number), then all multiples of 7 (as 7 is the first number to survive elimination by 2, 3, and 5), and so on.
Leveraging the power of quantum computing for breaking RSA encryption
Published in Cyber-Physical Systems, 2021
Moolchand Sharma, Vikas Choudhary, R. S. Bhatia, Sahil Malik, Anshuman Raina, Harshit Khandelwal
Supercomputers can, therefore, study only the simplest molecules. Yet quantum computers work using the same quantum properties as modelled molecules. Even with the most sophisticated responses, they will have no trouble handling. Wherever there is a massive, uncertain, complicated structure that needs to be simulated, Quantum computers find use [14]. A lot of encryption systems, as of today are relying on the difficulty of breaking down large numbers into prime numbers. This is called factoring, and it is slow, costly and impractical for classical computers. The conventional machines are the existing systems, which are Silicon-based computing systems we use today. They perform well and very efficiently for everyday tasks, like sending emails or processing images, etc. But the RSA encryption was designed with these computing systems in mind and these classical systems are not great at calculating factors. They still have to use the factorisation loops to break the RSA encryption, which may take more than the age of universe to find the factors for fairly large numbers [15]. But this time can be reduced to mere seconds with the use of quantum computers as we will see in the proposed solution.
LFSS-KF: lightweight fast real-time security standards with key fusion for surveillance videos
Published in The Imaging Science Journal, 2022
Chandan Kumar, Shailendra Singh
A data encryption algorithm is an International Data Encryption Standard (IDEA) [24]. Its algorithms are based on XOR, additions, and modularity multiplication. All data operations are performed with unsigned 16-bit integers. Al-Kadei et al. [25] have designed the fast RSA-based encryption algorithm for image encryption and security. This encryption takes the prime numbers as the key generation process. The Rivet–Shamir–Adelman (RSA) method is an asymmetric key algorithm and uses both public and private keys for encryption. Usually, the major limitation of RSA-based encryption is its slow speed of execution. It is expected to design fast methods for eliminating the speed limitations. Suresh et al. [26] have proposed a block cipher-based RC5 algorithm for encrypting the images.