Implementing-cryptography-using-python

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CHAPTER4Cryptographic Mathand Frequency AnalysisBy now, you have a basic understanding of historical ciphers and their cryptanalysisusing Python. In this chapter, you’ll turn your attention to the mathematicsthat is essential to understanding cryptography through the remainder of thisbook. This chapter introduces basic group theory and the Chinese remaindertheorem, shows how to solve systems of linear equations, and gives a morein-depth look at modular arithmetic. You’ll learn the proper importance ofsecure pseudorandom number generators and their use in cryptography. Thischapter is the basis for the remainder of this book as you explore public-keycryptography. Finally, you’ll start to construct a frequency analysis (FA) modulethat has a wide range of cryptanalysis functions. Through this chapter, you’llgain cryptographic knowledge as you do the following:■■Gain an understanding of modular arithmetic■■Understand the importance of the greatest common divisor (GCD)■■Gain an understanding of group theory■■Gain an understanding of pseudorandom numbers■■Create a Python script for frequency analysisImplementing Cryptography Using Python®, First Edition. Shannon W. Bray.© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.95
96 Chapter 4 ■ Cryptographic Math and Frequency AnalysisModular Arithmetic and the Greatest Common DevisorYou witnessed briefly in the previous chapter how Python uses modular arithmeticin cryptography. In this chapter, you’ll gain a greater understanding of it. Thelogic of modular arithmetic began with the quotient-remainder theorem. Thequotient-remainder theorem states that for every integer A and positive B thereexist different integers Q and R such that: A = B * Q + R, 0 = < r = < b. Whena = 95 and b = 10, what is the unique value of q (quotient) and r (remainder)?You find that the quotient equals 9 and the remainder equals 5.Once you understand the quotient-remainder theorem, it is easier to understandour first bit of cryptographic math: modular arithmetic.Here is an example: 23 ≡ 2(mod7), which reads as “23 is equivalent to 2 mod7.” You can also type it into a search engine as 23 mod 7 to see the answer. Youcan further examine the modulo by stating that a ≡ b (mod q) when a minusb is a multiple of q. Another way to state it numerically would be: 123 ≡ 13(mod 11) because 123 – 13 = 110 = 11 * 10. An alternative way to think about it is toexamine 53 (mod 13), which would be to say that 53 is equivalent to 53 − 13 = 40,which is equivalent to 27, which is equivalent to 14, which is equivalent to1, which is equivalent to −12, which is equivalent to −25, and so on. In fact,53 ≡ {53 + k⋅13|∀ k ∈Z} (you can read that as “is equivalent to the set of all numbersof form 53 plus an integer multiple of 13”). As shown in the previous chapter,we take the modulus by using the % sign.To illustrate this example in Python, type the following:>>> 53 % 131>>> 40 % 131>>> 27 % 131>>> 14 % 131>>> -12 % 131Now that you understand modular arithmetic, we turn our attention to thegreatest common divisor (GCD). The GCD is the largest number that perfectlydivides two integers: a and b. For example, the GCD of 12 and 18 is 6. This isan excellent opportunity to introduce Euclid’s algorithm, which is a techniquefor finding the GCD of two integers with negligible effort. To find the GCD oftwo integers A and B, use the following rules:If A = 0 then GCD(A, B) = BIf B = 0 then GCD(A, B) = A
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ImplementingCryptography UsingPytho
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To Stephanie, Eden, Hayden, and Ken
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viAbout the Authorbook, Shannon is
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Contents at a GlanceIntroductionxvi
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xiiContentsUsing Conditionals 16Usi
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xivContentsPseudorandomness115Break
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xviContentsInstalling and Testing W
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xviiiIntroductionprivacy advocates.
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CHAPTER1Introduction to Cryptograph
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Chapter 1 ■ Introduction to Crypt
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CHAPTER2Cryptographic Protocolsand
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Chapter 2 ■ Cryptographic Protoco
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CHAPTER6Using Cryptography with Ima
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Chapter 6 ■ Using Cryptography wi
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200 Chapter 7 ■ Message Integrity
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224 Chapter 8 ■ Cryptographic App
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CHAPTER9Mastering CryptographyUsing
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Chapter 9 ■ Mastering Cryptograph
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IndexSYMBOLS\ (backslash), 10- oper
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Index ■ D-D 279CIA Triad, 35-36ci
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Index ■ I-M 281hashlib module, 28
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Index ■ Q-S 283Preneel, Bart, 145
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