Abstract
In order to understand cryptography, beyond the ancient ciphers, a certain level of mathematics is required. This involves number theory, abstract algebra, and linear algebra. One might think that algebra is a topic normally taught in secondary school. Algebra is much more than what is typically taught in secondary school. In Chap. 4, the essentials of number theory and discrete mathematics were covered. In this chapter, we will explore the essential facts of abstract algebra and linear algebra. You will find elements of abstract algebra used in many cryptographic algorithms including symmetric algorithms such as AES and asymmetric algorithms such as RSA. We will also explore linear algebra which is widely used in some cryptographic algorithms such as NTRU. Finally, we will discuss algorithm analysis, in a basic manner. The concepts in this chapter provide a foundation for understanding these algorithms.
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Test Your Knowledge
Test Your Knowledge
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1.
A __________ is any equation for which you are interested only in the integer solutions to the equation.
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2.
A matrix which has all 1s in its main diagonal and the rest of the elements zero is a what?
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(a)
Inverse matrix
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(b)
Diagonal matrix
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(c)
Identity matrix
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(d)
Revers matrix
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(a)
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3.
Omega notation is ____?
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(a)
The asymptotic upper bound of an algorithm
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(b)
The asymptotic lower bound of an algorithm
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(c)
The average performance of an algorithm
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(d)
Notation for the row and column of a matrix
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(a)
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4.
A ____ is an algebraic system consisting of a set, an identity element, two operations, and the inverse operation of the first operation.
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(a)
Ring
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(b)
Field
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(c)
Group
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(d)
Galois field
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(a)
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5.
A ____ is an algebraic system consisting of a set, an identity element for each operation, two operations, and their respective inverse operations.
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(a)
Ring
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(b)
Field
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(c)
Group
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(d)
Galois field
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(a)
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6.
Is the set of integers a group with reference to multiplication? ____
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7.
Is the set of natural numbers a subgroup of the set of integers with reference to addition? ____
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8.
If a | b and a | c, then ____.
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9.
A group that also has the commutative property is an abelian group.
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10.
___________ is a process used to measure the performance of computer algorithm.
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11.
Solve this determinant |A|\( \left[\begin{array}{cc}2& 1\\ {}1& 3.\end{array}\right] \)
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(a)
8
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(b)
6
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(c)
0
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(d)
5
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(a)
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Easttom, C. (2022). Essential Algebra. In: Modern Cryptography. Springer, Cham. https://doi.org/10.1007/978-3-031-12304-7_5
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DOI: https://doi.org/10.1007/978-3-031-12304-7_5
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