"解密信息安全：数学基础手册"

信息安全

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信息安全的数学基础是一个十分重要的领域，它涵盖了广泛的数学原理和概念，这些对于保护和维护信息安全至关重要。Brendan W. Sullivan和John Mackey教授合著的《Everything You Always Wanted To Know About Mathematics*(*But didn’t even know to ask)》提供了一次深入抽象数学世界和证明写作的引导之旅，为信息安全的数学基础提供了宝贵的参考资料。在这本书中，作者首先介绍了学习数学的重要性，以及数学与现实生活的联系。他们强调数学思维的重要性，并指出数学的本质是如何帮助人们解决问题和应对挑战。这对于信息安全领域来说尤为重要，因为信息安全专业人士需要具备严谨的逻辑思维和解决问题的能力。书中还介绍了抽象数学的基本概念和原则，例如集合论、逻辑、函数和关系等。这些数学概念在密码学、网络安全和数据加密等领域起着至关重要的作用。同时，作者还深入探讨了证明的写作方法和技巧，这对于信息安全领域的研究和发展同样具有指导意义。除了书中介绍的内容，信息安全的数学基础还涉及到许多其他相关资料，例如数论、代数学、概率论和统计学等。这些数学领域在信息安全中均扮演着重要的角色，比如在密码算法设计、密钥管理和安全协议验证方面，数论和代数学有着重要的应用价值；而在风险评估、安全建模和攻击检测方面，概率论和统计学同样不可或缺。总的来说，信息安全的数学基础是一个广阔而复杂的领域，它涵盖了许多不同的数学原理和概念，需要信息安全专业人士具备坚实的数学基础和深厚的数学思维能力。《Everything You Always Wanted To Know About Mathematics*(*But didn’t even know to ask)》提供了一次深入抽象数学世界和证明写作的引导之旅，对于信息安全专业人士来说是一本宝贵的参考资料。同时，相关资料中的数论、代数学、概率论和统计学等领域也对信息安全的研究和发展起着至关重要的作用。希望未来能够有更多的学者和研究人员投入到信息安全的数学基础研究中，为保障信息安全做出更多的贡献。

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16 CHAPTER 1. WHAT IS MATHEMATICS?

Expanding the expression on the left and canceling a common term on both

sides yields



2ab + b

= c



2ab

Therefore, a

+ b

= c

is true.

Are you convinced? Did each step make sense? Maybe you’re not sure yet,

so let’s look at another “proof” of the theorem.

“Proof” 2. Suppose the Pythagorean Theorem is true and draw the right tri-

angle with the altitude from the vertex corresponding to the right angle. Label

the points and lengths as in the diagram below:

Since the Pythagorean Theorem is true, we can apply it to all three of the

right triangles in the diagram, namely ABC, BCD, ACD. This tells us (deﬁning

e = c − d)

= d

+ f

= f

+ e

= a

+ b

Adding the ﬁrst two equations together and replacing this sum in the third

equation, we get

= d

+ e

+ 2f

Notice that angles ∠ABC and ∠ACD are equal, because they are both comple-

mentary to angle ∠CAB, so we know triangles 4CDB and 4ADC are similar

triangles. (We are now assuming some familiarity with plane geometry.) This

tells us

, and thus f

= ed. We can use this to replace f

in the line

above and factor, as follows:

= d

+ e

+ 2de = (d + e)

18 CHAPTER 1. WHAT IS MATHEMATICS?

What’s nice about the truth requirement is that mathematics has been built up

so that we can read through an argument and verify each claim as True or False.

What’s diﬃcult to deﬁne is clear writing. In a way, it is much like Supreme

Court Justice Potter Stewart’s famous deﬁnition of obscenity: “I know it when

I see it”.

Given these four arguments for comparison, let’s assess them for clarity and

correctness:

Clarity:

• “Proofs” 1 and 2 are fairly well explained. There are clear statements

about what the writer is doing and why. They indicate where any equa-

tions come from, and even include some pictures to illustrate their ideas

for the reader.

Notice that “proof 1” does rely on some basic prior knowledge, like the

algebraic manipulation of variables and formulae for the area of a triangle

and square, but this is ﬁne.

Likewise, “Proof 2” relies on some understanding of similar triangles and

what this means about the lengths of their sides. At least the proof-writers

pointed this out, so an interested reader could look up some relevant ideas.

If they didn’t say this, a reader might be confused and have no idea how

to ﬁgure out what they’re missing!

• “Proof” 3 is very poorly worded! It oﬀers no explanation whatsoever.

This makes it quite diﬃcult to determine whether their claims are actually

correct. Yes, a picture is included, but there is no indication of why they

chose to draw a circle around the triangle, or why the stated equations

follow from the diagram.

• “Proof” 4 is a grammatically correct English sentence, but it doesn’t ex-

plain anything!

Already, we can see that “Proof” 4 is certainly ont a viable candidate for being

a good and proper proof. “Proofs” 1 and 2 are still in the running, since they

are at least written clearly. “Proof” 3, as it is written now, would probably

not be a good candidate; however, maybe it does contain correct ideas that just

require better explanations. Perhaps it could be rewritten as a good and proper

proof.

Let’s analayze the logical correctness of these four arguments:

Correctness:

• “Proof” 1 mostly good. The formulae for the areas of the square and

triangles are correctly applied, and the algebraic manipulation thereof is

correct. But how do we know that the process they described—putting

four copies of the given triangle inside a larger square—creates a square

with side length c on the inside? They merely say it does so without

1.1. TRUTHS AND PROOFS 19

really saying why. Other than this omission, though, this proof is both

well-written and correct.

(Can you prove that fact, that the shape inside is actually a square? Just

look at its angles: can you show why they are all right angles?)

• Unfortunately, “Proof” 2 is completely invalid! Every logical step that

it makes does follow from the previous one. For instance, assuming we

have the triangles set up this way, we can correctly deduce that ∆CDB

and ∆ADC are similar triangles. However, why is it that we can assume

the theorem is True right at the beginning? Isn’t that what we are trying

to accomplish in the proof, overall? This is a crucial ﬂaw. Assuming a

fact and deducing something True from it does not allow us to

conclude the original assumption was valid.

If this method were valid, we could “prove” just about anything we wanted!

Here’s an example: What do you think of the following “proof” that 0 = 1?

“Proof”. Assume 0 = 1. Then, by the symmetric property of =, it is also

true that 1 = 0. Adding these two equations tells us 1 = 1, which is True.

Therefore, 0 = 1 was a valid assumption, so it must be True.

Do you see the similarity between this and “Proof” 2 above? The same

sort of ﬂawed reasoning was used: we assumed one fact, did some work to

get to something else we know to be True, and then said that the assumed

fact must be True, as well.

• Regarding “Proof” 3, most mathematicians would say it is a “bad proof”,

despite the fact that everything it appears to claim is correct. We say

“appears” because, without any words to explain what’s going on, we

don’t actually know what the proof-writer is trying to say! However, we

will say that the kernel of a perfectly good proof is contained therein.

From the diagram, you can show that the stated equation,

c+b

c−b

must follow. (Hint: Use similar triangles!) From there, it is a simple

manipulation to deduce that a

+ b

= c

Can you write some sentences to go along with the diagram that would

turn this into a proper proof?

• Lastly, just about every reasonably logical person (we hope!) would say

that “Proof” 4 is not even close to being a proof, however convenient it

might be to make such statements.

This discussion shows that “Proof” 1 is actually a good proof. Amongst all

four, it is the most clearly-written, and the one that is logically correct. We can

refer to it now as a proof. “Proof” 2 is outright incorrect, despite how clearly

it is presented. “Proof” 3 contains correct ideas, but is not presented clearly.

“Proof” 4 is so far from a proof that we don’t even want to discuss it.

20 CHAPTER 1. WHAT IS MATHEMATICS?

Question

Before moving on to other topics, we’ll leave you with a question: if we give you

three positive numbers a, b, c that satisfy a

+ b

= c

, is it necessarily true that

there is a right triangle with side lengths a, b and hypotenuse length c? If so,

how could you go about constructing it? If not, why not?

1.1.2 Prime Time

While we’re on the topic of proofs, let’s look at another proof, for a diﬀerent

theorem. As a reminder (or brief introduction), let’s talk about prime numbers.

Deﬁnition, Examples, and Uses

Deﬁnition 1.1.2. A positive integer p that is larger than 1 is called a prime

number if the only positive divisors of p are 1 and p. A non-prime positive

integer is called a composite number.

Prime numbers have shown to be incredibly important in all branches of

mathematics, not just the study of integers and their properties, which is known

as number theory. One of the most famous conjectures (a guess at a theorem

that has been neither proven nor disproven thus far) in all of mathematics is

the Riemann Hypothesis. Its conclusion has been shown to be closely related to

the distribution of prime numbers throughout the integers. Many books have

been written on this topic. Also, most modern cryptography schemes are based

on multiplying huge prime numbers together, relying on the fact that it’s quite

diﬃcult to undo this process and ﬁgure out the two huge prime factors, given

their product. So now you know: every time you buy a song on iTunes with

your credit card, some computer just multiplied two large prime numbers!

The ﬁrst few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, . . . (remember, 1

does not ﬁt our deﬁnition). How many prime numbers are there? How far apart

are they? Is there a pattern? Answering questions like these can be interesting

and fun, but also diﬃcult (and sometimes, impossible!). Here, we’ll answer one

of the questions: are there an inﬁnite number of prime numbers?

Theorem and Proof

Theorem 1.1.3 (Inﬁnitude of the Primes). There are inﬁnitely-many prime

numbers.

“Proof”. Assume instead that there are only ﬁnitely-many prime numbers, and

list them in ascending order: p

, p

, . . . , p

, so that p

is the largest of these

prime numbers. Deﬁne the new number

N = (p

· p

· ··· · p

) + 1

It must be true that N is divisible by some prime number. However, it cannot

be divisible by p

or p

or . . . or p

, because that would leave a remainder of 1,

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