Milestone Assignment #3 Choose one of the following assignments to complete: A.Lunar Meteorite Impact Risks B.Tales from the Cryptography Lunar Meteorite Impact Risks A December 4, 2006 CNN.Com news story, based on the research by Bill Cooke, head of NASA’s Meteoroid Environment Office suggests that one of the largest dangers to lunar explorers will be meteorite impacts. Between November 2005 and November 2006, Dr. Cooke’s observations of lunar flashes (see image) found 12 of these events in a single year. The flashes were caused primarily by Leonid Meteors about 3-inches across, impacting with the equivalent energy of 150–300 pounds of TNT. The diameter of the moon is 2158.8 miles. Problem 1: From the formula for the surface of a sphere, what is the area, in square miles, of the side of the moon facing Earth? Problem 2: Although an actual impact only affects the few square meters within its immediate vicinity, we can define an impact zone area as the total area of the surface being struck, by the number of objects striking it. What was the average impact zone area for a single event? Problem 3: Assuming the area is a square with a side length ‘ᡅ’, A) what is the length of the side of the impact area? B) What is the average distance between the centers of each impact area? (in miles)Problem 4: If the impacts happen randomly and uniformly in time, about what would be the time interval between impacts? Problem 5: From the vantage point of an astronaut standing on the Moon, the horizon is about 1.9 miles away. How long would the lunar colony have to wait before it was likely to see an impact within its horizon area? Tales from the Crypt(ography) Cryptography means “hidden,” as in a hidden message. It is the process of encoding a message, or encrypting it into an unreadable one, or a cipher. Only the person with the secret code can decipher, or decrypt, the message into the original message, or plaintext. From ATM cards, to computer passwords and email messages, to our nation’s secrets—all involve cryptography. One of the simplest examples is the Caesar Cipher, which is said to have been used by Julius Caesar to communicate with his army. Caesar is considered to be the one of the first people to use encryption methods. Other, more complicated, ciphers require the use of matrices. For example, Caesar shifted the alphabet 6 letters to the left, so A would become T, B would become U, C would become V, etc. Therefore, his plaintext “Return to Rome” would be written “Uhwxua wr Urph” as the ciphertext. Plain TextABCDEFGHIJKLMNOPQRSTUVWXYZCiphertextTUVWXYZABCDEFGHIJKLMNOPQRSEXAMPLE 1: Practice decrypting the message: “QEXL MW IZIVCALIVI!” using the key I = M. Plain TextABCDEFGHIJKLMNOPQRSTUVWXYZCiphertextMDid you get: “Math is everywhere!”? Example 2: Sometimes the message writer will change the spacing so that it is more difficult to determine the message. For example, practice decrypting the message using the key below: “ENAG QKXE JWRYZ KRGAL” Plain TextABCDEFGHIJKLMNOPQRSTUVWXYZCiphertextEDCBAZYXWVUTSRQPONMLKJIHGFDid you get: “Are you having fun yet?” It is really easy to decode a message (like the examples above) if you have the key, but what if you don’t? Let’s say you were trying to decode a message from an opposing country during wartime. In this case, you would have to use what you know about the English language and the way words and phrases are constructed. For example, single-letter words will be either I or a. Common words and phrases like the, is, of, and the suffix ing often repeat in a message. Also, notice smaller words within bigger words such as he in the. Certain consonants occur together, such as why, th, sh, ch, ed, tion, ent, and ant. Example e: Try to figure out this cipher. “FD VTC’H XQLID WZQELDKX ES RXNCY HMD XTKD PNCO QU HMNCPNCY FD RXDO FMDC FD VZDTHDO HMDK” Did you figure out the phrase? It was a quote by Albert Einstein: “We can’t solve problems by using the same kind of thinking we used when we created them.” It was more difficult because there was no pattern to how the message was encoded, as there was in the previous examples. Plain TextABCDEFGHIJKLMNOPQRSTUVWXYZCiphertextTEVODUYMNAPLKCQWBZXHRIFGSJIn order to encode example 1, you would use the linear equation: ᡷ = ᡶ + 4, which means that you would take whatever letter you wanted (ᡶ) and add 4letters to get the new letter (ᡷ). The person decoding the message would need to find the inverse of the equation. The inverse of ᡷ = ᡶ + 4 is ᡷ = ᡶ − 4 because: ᡷ = ᡶ + 4start with original equation ᡶ = ᡷ + 4switch ᡶ and ᡷ ᡶ − 4 = ᡷsolve for ᡷIn order to encode example 2, you would use a more complicated linear equation, and to encode example 3, you wouldn’t be able to use an equation because the letters were randomly placed. This is when it is beneficial to use matrices to encode messages. In order to use matrices to encode and decode messages, first start by assigning each letter in the alphabet to a number, starting with 0 for a space, 1 for A, 2 for B, …, 26 for Z. To encode STUDY MATHEMATICS, you would write: 19 20 21 4 25 0 13 1 20 8 5 13 1 20 9 3 19 0. Next, you would determine what size matrix you want to use. Your matrix can be 1 x 3, 1 x 2, 1 x 4, 1 x 5, etc. In this example, I am going to use 1 x 3 matrices. Notice that I put a 0 at the end so the last matrix would be a 1 x 3. [ 19 20 21] [ 4 25 0] [ 13 1 20] [ 8 5 13] [ 1 20 9] [ 3 19 0] S T U D Y M A T H E M A T I C S Then choose a key matrix to decode the message. It can be any matrix that has an inverse. In this example, I am going to use matrix A: 㐩1 0 −12 1 −20 1 1㐳. Now, multiply each of your 1 x 3 matrices by A to decode your message. 䙰19 20 21䙱㐩1 0 −12 1 −20 1 1㐳=䙰59 41 −38䙱䙰4 25 0䙱㐩1 0 −12 1 −20 1 1㐳=䙰54 25 −54䙱䙰13 1 20䙱㐩1 0 −12 1 −20 1 1㐳=䙰15 21 5䙱䙰8 5 13䙱㐩1 0 −12 1 −20 1 1㐳=䙰18 18 −5䙱䙰1 20 9䙱㐩1 0 −12 1 −20 1 1㐳=䙰41 29 −32䙱䙰3 19 0䙱㐩1 0 −12 1 −20 1 1㐳=䙰41 19 −41䙱So the encoded message is 䙰59 41 −38䙱䙰54 25 −54䙱䙰15 21 5䙱䙰18 18 −5䙱䙰41 29 −32䙱䙰41 19 −41䙱To decode the message the recipient must have either the inverse matrix of ᠧ, which is ᠧ⡹⡩, or know ᠧ and know how to find the inverse of matrix ᠧ. Take a minute and determine the inverse of ᠧ. Did you get ᠧ⡹⡩=㐩3 −1 1−2 1 02 −1 1㐳? Now that you have the inverse, multiply the inverse matrix by each of the matrices from the encoded message. Once you do that, you will get the original message! Here is what your Tales From the Crypt(ography) project will consist of: •Cover page (name, date, teacher name, title) (1 point) •A brief history of cryptography (15 points) oResearch on your own o3–5 paragraphs, with 7–9 sentences in each paragraph oYou can include information about the history of cryptography, about how to solve shift ciphers, about how to solve matrix ciphers, about famous ciphers, etc. •Five ways cryptography is used today (2 points each, 10 points total) oInclude for each: A sentence explaining how cryptography is used (1 point) A picture (1 point) •Three examples using a shift cipher (10 points each, 30 points total) oInclude for each: The plaintext (2 points) The linear equation (the rule/key) the encoder would use (2 points) The inverse equation (the rule/key) the decoder would use (2 points) The cipher text (2 points) An explanation of how to solve the cipher text. In other words, you either need to show the math you used orexplain in words how someone would solve it. (2 points) •Two examples using a Matrix cipher (16 points each, 32 points total) oInclude for each: The plaintext (2 points) The plaintext as a matrix, and using numbers (2 points) The key matrix that the encoder would use (4 points) The inverse matrix that the decoder would use (4 points) The cipher text (4 points) An explanation of how to solve the cipher text. In other words, you either need to show the math you used orexplain in words how someone would solve it. (4 points) •Bibliography/Works Cited (5 points) •Be creative in your project! (7 points) Milestone Assignment Rubric – Tales from the Crypt(ography) Skills Being Assessed Exemplary Achieved Developing Cover Page The student’s name, date, teacher’s name, and title are done. (1 point) The cover page is incomplete or not done (0 points) Brief History of Cryptography The history includes at least three unique pieces of information about cryptography. (12–15 points) The history includes two unique pieces of information about cryptography. (8–10 points) The history includes one or no unique pieces of information about cryptography. (0–5 points) Five Ways Cryptography Is Used Today An explanation and picture both show the present use of cryptography. (2 points each—10 total) An explanation or picture shows the present use of cryptography. (1 point each) Neither an explanation nor picture shows the present use of cryptography. (0 points each) Three Examples Using a Shift Cypher The example includes the plaintext, linear equation, calculation of the inverse, cypher text, and clear explanation of how the cypher text works. (10 points each—30 total) The example is missing one or two of the plaintext, linear equation, calculation of the inverse, cypher text, and clear explanation of how the cypher text works. (7 points each) The example is missing three or more of the plaintext, linear equation, calculation of the inverse, cypher text, and clear explanation of how the cypher text works. (0–3 points each) Two Examples Using a Matrix Cypher The example includes the plaintext, key matrix, calculation of the inverse, cypher text, and clear explanation of how to solve the cypher text. (16 points each—32 total) The example is missing one or two of the plaintext, key matrix, calculation of the inverse, cypher text, and clear explanation of how to solve the cypher text. (12 points each) The example is missing three or more of the plaintext, key matrix, calculation of the inverse, cypher text, and clear explanation of how to solve the cypher text. (0–8 points each) Bibliography/ Works Cited The bibliography/works cited includes at least two sources. (5 points) The bibliography/works cited includes one source. (3 points) The bibliography/works cited includes no sources. (0 points) Creativity The examples show creative and original thought. (7 points) The examples are original and are well presented. (5 points) Theexamples are neat and show a good amount of thought. (0–3 points)Very simple examples (Do not copy)
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