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Network Reliability and Security
Published in Partha Pratim Sahu, Advances in Optical Networks and Components, 2020
We consider 3 and 11 which are multiplicative inverses mod 8 since 33 mod 8 = 1. This characteristic makes modular arithmetic very useful and appealing in cryptographic applications. When a and n are relatively prime, i.e., gcd (a, n) = 1, we can write x=aϕ(n)−1modn
Two Level Fractional Designs
Published in Thomas J. Lorenzen, Virgil L. Anderson, Design of Experiments, 2018
Thomas J. Lorenzen, Virgil L. Anderson
A few examples illustrate the use of modular arithmetic. If you have never seen modular arithmetic before, you should find it an easy concept to pick up. Modular arithmetic is the value of the remainder after dividing the original number by the modulus. For example, the value of 5 mod 2 is 1 since 5 divided by 2 is 2 with a remainder of 1. Similarly, 18 mod 2 = 0 since there is no remainder after dividing 18 by 2. Trivially, 0 mod 2 = 0 and 1 mod 2 = 1. The value of 5 mod 3 is 2 because 2 is the remainder after dividing 5 by 3, and the value of 18 mod 3 is 0 since there is no remainder after dividing 18 by 3. As a final example, 654 mod 23 = 10 since 654 divided by 23 is 28 with a remainder of 10.
Pseudo-Random Pixel Rearrangement Algorithm Based on Gaussian Integers for Image Watermarking
Published in Frank Y. Shih, Multimedia Security, 2017
Aleksey Koval, Frank Y. Shih, Boris S. Verkhovsky
There are many innovating watermarking algorithms and many more get published everyday (such as Al-Qaheri et al., 2010; Huang et al., 2010; Lin and Shiu, 2010; Yamamoto and Iwakiri, 2010). In many image watermarking algorithms, for example (Dawei et al., 2004; Wu and Shih, 2007; Yantao et al., 2008; Ye, 2010), it is required to rearrange the pixels as a part of watermarking process. Randomness is desired during this step. Modular arithmetic and, specifically, the integer exponentiation modulo prime numbers are widely used in modern cryptographic algorithms. One important property of integer exponentiation modulo prime is that it generates a sequence of integers that looks very much like a sequence of random numbers. This is a property that is desirable for pixel rearrangement algorithms. In this chapter, the rearrangement step of watermarking algorithms is revisited and a different universal method for doing it is described. It is easy to replace the rearrangement step in Dawei et al. (2004), Wu and Shih (2007), Yantao et al. (2008), and Ye (2010). Moreover, this method can be used with most image watermarking algorithms to enhance them.
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
Modular arithmetic is a part of mathematics that revolves around the remainders of integers. Shor’s algorithm for factorisation uses this as a basis for finding prime factors more efficiently. Shor introduced an algorithm for integer factorisation that is used for finding the prime factors of a number. When running on a quantum computer, the algorithm takes very little time to find the prime factors of the number – running in polynomial time. The efficiency of Shor’s algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by repeated squaring. The RSA and the algorithm are based on the same finite group theory. Shor’s algorithm gives us the best efficiency, but it relies on Size of Quantum Computers, which are not possible right now [20,21].