integer multiplication
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In this modern era, data protection is very important. To achieve this, the data must be secured using either public-key or private-key cryptography (PKC). PKC eliminates the need of sharing key at the beginning of communication. PKC systems such as ECC and RSA is implemented for different security services such as key exchange between sender, receiver and key distribution between different network nodes and authentication protocols. PKC is based on computationally intensive finite field arithmetic operations. In the PKC schemes, modular multiplication (MM) is the most critical operation. Usually, this operation is performed by integer multiplication (IM) followed by a reduction modulo M. However, the reduction step involves a long division operation that is expensive in terms of area, time and resources. Montgomery multiplication algorithm facilitates faster MM operation without the division operation. In this paper, low latency hardware implementation of the Montgomery multiplier is proposed. Many interesting and novel optimization strategies are adopted in the proposed design. The proposed Montgomery multiplier is based on school-book multiplier, Karatsuba-Ofman algorithm and fast adders techniques. The Karatsuba-Ofman algorithm and school-book multiplier recommends cutting down the operands into smaller chunks while adders facilitate fast addition for large size operands. The proposed design is simulated, synthesized and implemented using Xilinx ISE Design Suite by targeting different Xilinx FPGA devices for different bit sizes (64-1024). The proposed design is evaluated on the basis of computational time, area consumption, and throughput. The implementation results show that the proposed design can easily outperform the state of the art

The background of this research is that generally students understand the concept of multiplication and division of integers by memorizing. Basically, the memorization method will be appropriate if you memorize it, you know and understand what you have memorized. So that students do not just memorize alone. However, students really understand what they have memorized, and the teacher in providing material on multiplication and division of integers is only glued to the textbook but does not use other solutions that are easier for students to understand. To improve students' understanding of the operations of multiplication and division of integers, learning uses an Open-Ended approach that designs a solution and answer in the operation of multiplication and division of integers with more than one solution and answer. The main objective of this research is to find out how big the Open-Ended learning approach is in improving the understanding of fifth grade students of Al Ankabut SDIT Al Fahmi on the operations of multiplication and division of integers. In this study, the researcher conducted classroom action research with six students as research subjects who were selected based on their ability level, consisting of six students, namely two students with low abilities, two students with moderate abilities, and two students with high abilities. The process of collecting data was done through tests, observations, interviews and field notes. The action was carried out four times, namely Action (1) was learning about all the elements in multiplication using Open-Ended learning through a structural approach. Action (2) is learning about integer multiplication operations with the application of Open-Ended learning through a structural approach. Action (3) is learning about all the elements in the division using Open-Ended learning through a structural approach. Action (4) is learning about the operation of dividing integers by applying Open-Ended learning through a structural approach. The results show that learning with an Open-Ended approach can improve the understanding of fifth grade students at Al Ankabut SDIT Al Fahmi Palu in the operations of multiplication and division of integers.

The interval discrete logarithm problem(IDLP) is to find a solution n such that gn = h in a finite cyclic group G = 〈g〉, where h ∈ G and n belongs to a given interval. To accelerate solving IDLP, a restricted jump method is given to speed up Pollard’s kangaroo algorithm in this paper. Since the Pollard’ kangaroo-like method need to compute the intermediate value during every iteration, the restricted jump method gives another way to reuse the intermediate value so that each iteration is speeded up at least 10 times. Actually, there are some variants of kangaroo method pre-compute the intermediate value and reuse the pre-computed value in each iteration. Different from the pre-compute method that reuse the pre-computed value, the restricted jump method reuse the value naturally arised in pervious iteration, so that the improved algorithm not only avoids precomputation, but also speeds up the efficiency of each iteration. So only two or three large integer multiplications are needed in each iteration of the restricted jump method. And the average large integer multiplication times is (1:633 + o(1)) N in restricted jump method, which is verified in the experiment.

This study aimed to determine the effect of lattice learning methods on mathematics learning outcomes on the integer multiplication operation for the grade VII students of SMPN 2 Bua. This is a pre-experimental research design that uses a one-group pretest-posttest research design type. The instrument used a description test consist of 3 questions. The samples were students of class VII A of SMPN 2 Bua. The analysis result of the two-tailed curve, Ha was accepted, and Ho was rejected. The conclusion was a significant influence on the lattice learning method on mathematics learning outcomes in the integer for grade VII students of SMPN 2 Bua. So, a successful student not only by learning theory but also by the right learning methods. The results of this study will be able to help teachers to improve student learning outcomes.