File Name: cryptography multiplication questions and answers .zip
Since that time, this paper has taken on a life of its own
This is a big multi-part lesson that introduces the concept of public key cryptography which is an answer to the crucial question: How can two people send encrypted messages back and forth over insecure channels the Internet without meeting ahead of time to agree on a secret key? In a nutshell, there are two main principles we want students to understand:. The lesson gets at these two core ideas through a deliberate chain of thought experiments, demonstrations, activities and widgets.
All parts are building blocks that lead to deeper understanding of how it works. This is a fairly hefty lesson because the underlying ideas are subtly quite sophisticated. It's worth noting that much of the material here - all but the highest level takeaways - are beyond the scope of what's covered on the AP exam. Students need to know the basic public key encryption process, and what asymmetric encryption is.
For programming they need to know how the modulo operation works. Our purpose here is to reveal some of the magic that happens every day on the Internet to enable secure transactions. To many the fact that encrypted messages can be sent between parties who have never met before is both taken for granted and opaque.
Our belief is that understanding how it works with some depth - getting to experiment with the mathematical principles that make asymmetric keys possible, and the resulting encryption hard to crack - is deeply satisfying. The widget in the lesson mimics the RSA encryption algorithm with smaller numbers and slightly easier mathematics.
This lesson will likely take two days to complete. Preparing for these activities the first time will take some time. Once you've been through it once, the activities actually go quicker than you might expect. Differentiation and other adjustments to this lesson. Prompt: "How can two people send encrypted messages to each other if they can't communicate, or agree on an encryption key ahead of time, and the only way they have to communicate is over the Internet?
Goal : Realize the difficulty of the problem. No form of symmetric encryption will work. There is no way for parties to establish a shared key without agreeing ahead of time in a way that secures it from an observer.
Hopefully some students will recall from the video in the last lesson the ideas of using different keys - one to encrypt data and one to decrypt it. Possible Responses : Students may come up with some fantastic ideas, but most will amount to some secret ahead-of-time agreement about a key, or simply some strategy that obscures the key "security by obscurity". Today we're going to dig in a little bit deeper to how this idea of using different keys actually works.
The ideas behind how it works are sophisticated, and so to get a deeper understanding we're going to do a series of short activities that stringing together several different ideas, bringing them all together in the end. Remind students - we're still a ways from the real thing but we're taking baby steps to string ideas together.
Option 1 preferred : Teacher Demo. We recommend doing this activity as a teacher demonstration in the interest of time. Instructions and teacher guide below. Option 2: Groups of 3 Students. You can have students work through an activity guide that explains it as well. It will take more time. Materials : Cups and Beans - enough for a demonstration or for groups of 3, if running as student activity. Display : You may want to display a picture of a jar full of candies to give a visual for the analogy you're about to explain.
The lock box analogy from the video is a good start, but our first step to seeing how public key cryptography works requires us to look at the same process of using public and private keys but with an analogy that goes a step further.
Discuss : Relate this process using cups and beans to the lockbox analogy from the video. What's similar? What's different? What took place of the public key? The message? The private key? Okay so that's one step. We now have a clearer idea of the public key encryption process. If we can keep extending this we'll have a solution to the problem of how two people can encrypt messages without meeting ahead of time.
The modulo operation is a math operation that returns the remainder from dividing two numbers. The mod operation gives the remainder portion. There is a well known visual analogy for modular arithmetic using clocks since modulo is often thought to "wrap" the number system. If, for example, you use 12 as a modulus then any result must be in the range since those are the only possible remainders.
Similarly, no matter how many hours you count off on a traditional analog clock, there is a limited number of hours that the hour hand can be pointing to. It's even called "Clock Arithmetic" in some places wikipedia: modular arithmetic. The modulo operation is important for cryptography because it can act as a one-way function - the output obscures the input. The purpose of this thought experiement is to understand the clock analogy for modulo.
It is a setup for the next step. Students should understand the concept of numbers that "wrap" around the clock and that the "size" of the clock could be arbitrary - it doesn't have to be The same principle would apply for a "clock" of any size.
Materials : two pictures of analog clocks - one with hour hand at and another at Display : picture of clock at You can use this interactive clock rather than pictures if you like. Modulo is important for cryptography as a one way function - you can't tell based on the remainder what went into the clock. To understand how it's used in cryptography, we're going to investigate what happens when we use simple multiplication to produce the number we input into the clock.
There are certain properties that are useful when we combine simple multiplication with modulo. This is not on the AP exam. Students do not need to memorize or be facile with these mathematics for the AP Exam. The modulo operation is part of the AP pseudocode and there might be simple programming questions on the exam that use it. However, the mathematics for Public Key Cryptography is beyond the scope of the course.
You cannot solve it like a typical equation in math class because there are many equations. Student do the activity: students should work with a partner to work through the problems on the activity guide.
Circulate as students work. Make sure that they are trying out the problems given which ask them to try to guess numbers. They should also be using the Mod Clock to check their results. Discuss : "Why is it hard to guess which numbers multiplied together produce the result?
These points are made at the bottom of the activity guide. After students have worked on the problems for a bit they should be able to give a few responses here such as:. Okay, now to finally bring everything together. This is last and final step in which we'll see how we can use the math we just learned about to create public and private keys.
Real Public Key Cryptography? You put the public key somewhere that someone can grab, like your personal web page there are other ways too. If you would like to try or demonstrate for your students, you can. Just google "RSA Keygen" and follow instructions for your type of computer. Answers to some FAQs about the widget. Clock size is chosen randomly by Alice but there is a set list of values to choose from.
The clock sizes in the list provided are prime numbers between 1 and 10, This ensures certain properties of the encryption. Alice simply has to pick one. Bob is sending a secret number to Alice , not vice-versa. In public key cryptography for Bob to send a secret to Alice, alice has to act first, producing a public key for Bob to use. Bob can send any number to Alice - as long as the number is between 0 and clockSize - 1. The clock size limits the range of values - the secret numbers that Bob and Alice use are confined to the output range of the mod clock.
For example: if the clock size is 13, then Bob can only send a secret number in the range If the clock size is then the secret values can be Introduce the Public Key Crypto widget providing the background and instructions given on the Instructions page in code studio.
Make sure to point out the similarities and differences between using this widget and cups and beans. Demonstrate the first step of using the widget.
Click past the the instructions page to get to the widget if necessary. After pairs have gotten the hang of playing Bob and Alice, regroup to review how Eve works. Display Eve's screen in the widget. Pick 2 students on opposite sides of the room to play Alice and Bob and demonstrate intercepting their spoken broadcasts and entering the info in Eve's screen.
Option 2: Small group experimentation - Have previous Alice-and-Bob pairs get together in groups of 4. One pair plays Bob and Alice, the other pair plays Eve as a team of 2 on one computer or two.
Students exchange numbers a few more times, trying to make it hard for Eve to crack.
This is a big multi-part lesson that introduces the concept of public key cryptography which is an answer to the crucial question: How can two people send encrypted messages back and forth over insecure channels the Internet without meeting ahead of time to agree on a secret key? In a nutshell, there are two main principles we want students to understand:. The lesson gets at these two core ideas through a deliberate chain of thought experiments, demonstrations, activities and widgets. All parts are building blocks that lead to deeper understanding of how it works. This is a fairly hefty lesson because the underlying ideas are subtly quite sophisticated.
Edit Reply. Cryptarithmetic questions are most commonly asked in the Infosys recruitment and eLitmus exam. The most repeated Cryptarithmetic questions with answers are discussed here. Cryptarithmetic is a mathematical puzzle which involves the replacement of digits with alphabets, symbols and letters. Only through certain practice, one can become an expert in solving the cryptarithmetic questions. The digits are distinct and positive. From this, we can tell that the largest value of A can be 2.
Since that time, this paper has taken on a life of its own Does increased security provide comfort to paranoid people? Or does security provide some very basic protections that we are naive to believe that we don't need? During this time when the Internet provides essential communication between literally billions of people and is used as a tool for commerce, social interaction, and the exchange of an increasing amount of personal information, security has become a tremendously important issue for every user to deal with. There are many aspects to security and many applications, ranging from secure commerce and payments to private communications and protecting health care information.
In this section, we will learn to find the inverse of a matrix, if it exists. Later, we will use matrix inverses to solve linear systems. In this section you will learn to. Encryption dates back approximately years.
Teachers can now assign pages to students digitally, and students can fill in their answers on the digital overlay! I have created answer boxes for all of the pages. These pages are designed to be self checking in that if the students have correctly solved the arithmetic problems, they will be able to finish the clues and discover the mystery animal! But teachers can also to check students' work digitally when it is completed. Note: the answer boxes cannot be formatted, so the division problems might be a little wonky for students to fill in as the boxes left justify right justify would work better And please let me know if there are any issues with these digital overlays! If your students like Crypto-Riddles , they're going to love Crypto-Animals!
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