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Define a GPS-type problem. Give the goal condition, initial conditions, and operations. For each operation, define the action, preconditions, “add” conditions, and “delete” conditions. At least five operators must be defined.

Then, list a sequence of actions (in the correct order) that solve the problem. At least three operators must be used.

Give the sequence of states visited (states evaluated) in a hill-climbing search applied to the Goodale routing problem. starting at Woodruff & Tuttle and ending at Goodale parking lot. You’ll find the goodale-distances.txt file useful.

For each “choice” that the algorithm faces, list the options, the computed scores, and indicate the best choice.

Note, give the sequence of states visited, not just the states that make up the final route. There may (or may not) be some backtracking so the states visited may differ from the route.

Assume hill-climbing does not reconsider previously-visited states.

Your answer should look like this:

Repeat task 2 but for A* search.

Like before, for each “choice” that the algorithm faces, list the options, the computed scores, and indicate the best choice. Since A* may “jump around,” indicate the prior (connecting) state for each choice. Also give the current path cost for each state when that state is visited.

Assume A* does not reconsider previously-visited states.

Your answer should look like this:

The Goodale routing problem used road intersection distance to find the routes (when informed search was used). However, this distance (“as the crow flies”) is only one aspect of what makes one route better than another. What other information should we use to find the truly minimal-time route between these two locations?

What are some *admissible* heuristics for the 8-puzzle problem? Give at least two.

Define breadth-first search in terms of the cost function \(f(s) = g(s) + h(s)\) where \(g(s)\) is the cost of arriving at state \(s\) and \(h(s)\) is the (estimated) cost of getting from \(s\) to the goal state. That is to say, give simple definitions for \(g(s)\) and \(h(s)\) so that the A search algorithm acts like breadth-first search.

Also define depth-first search in terms of \(f(s) = g(s) + h(s)\).

Show a graph of minimax operation for this tic-tac-toe board, where ‘x’ is to make a move (and ‘o’ is the opponent). The graph should look similar to those in the adversarial search notes.

1, 2. job, assignment. Task, chore, job, assignment refer to a definite and specific instance or act of work. Task and chore and, to a lesser extent, job often imply work that is tiresome, arduous, or otherwise unpleasant. Task usually refers to a clearly defined piece of work, sometimes of short or limited duration, assigned to or expected of a person: the task of pacifying angry customers; a difficult, time-consuming task. A chore is a minor task, usually one of several performed as part of a routine, as in farming, and often more tedious than difficult: the daily chore of taking out the garbage; early morning chores of feeding the livestock. Job is the most general of these terms, referring to almost any work or responsibility, including a person's means of earning a living: the job of washing the windows; a well-paying job in advertising. Assignment refers to a specific task allocated to a person by someone in a position of authority: a homework assignment; a reporter's assignment to cover international news.

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Based on the Random House Dictionary, © Random House, Inc. 2017.

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Examples for task Expand

Some *task* Force Ranger vets left the military relatively soon after Black Hawk Down.

In a world of many squeaky wheels, Obama glided smoothly from *task* to *task*.

With that *task* accomplished, the SAMI team can get to work on the *task* of galactic demographics in earnest.

Sabrine is a trained lawyer, likely a helpful quality when your *task* is to push politicians.

Of course, recognizing our common humanity is only the beginning of our *task*.

It was charged with the *task* of cutting a way through to relieve Przemysl.

For what to shun will no great knowledge need; But what to follow is a *task* indeed.

For ourselves, men of Lacedaemon and of the allied states, our *task* is completed.

And so Clif put every ounce of muscle he had into that *task*.

He saw instantly that her dread was for him, and it made his *task* the harder.

*Homework 5 (16 points, due Wednesday, April 23)*

*Option for programmers (16 points + bonus points)*

Develop an efficient implementation of "mini-DSA" that is DSA with the length of the modulus *p* equal to the length of the maximum long integer supported by the language and the computer system you use.

Your implementation should consist of the separate procedures for the system setup, key generation, signing and verifying. All procedures should be fully tested and optimized for speed.

*Task 1 (bonus 8 points):*

Implement fixed-base windowing method of exponentiation (basic handbook, algorithm 14.109) and use it for your implementation of "mini-DSA".

*Task 2 (bonus 4 points):*

Replace the SHA-1 hash function used together with DSA by a simplified hash function based on modular arithmetic and defined as follows: H(0) = 0, H(i) = (H(i-1) xor M(i))^2 xor H(i-1) mod N, h(M=M(1)..M(t)) = H(t), where N is a product of two primes, and the bitlength of N is equal to the bitlength of *q* in "mini-DSA".

*Option for analysts (16 points required / 20 points total)*

*Problem 1 (4 points)*

Doing a signature with RSA on a long message would be too slow (presumably using cipher block chaining). Would it be reasonable to compute an RSA signature on a long message by first finding what a message equals mod N, and signing this?

*Problem 2 (4 points)*

Message digests are reasonably fast, but here's a much faster function to compute. Take your message, divide it into 160-bit chunks M1, M2. Mn, XOR all chunks with odd indices together to compute B1 = M1 xor M3 xor M5 xor. and all chunks with even indices together to compute B2 = M2 xor M4 xor M6 xor.

B1 and B2 are both 160-bit long. Use these two blocks as an input for SHA-1. Is this a good collision-resistant hash function?

*Problem 3 (5 points)*

Find a collision in the following construction for a collision-resistant hash function: H0 = IV1, G0 = IV2, H(i)=DES(M(i), H(i-1)), G(i)=DES(M(i), G(i-1)). Hash-value for the message M=M(1), M(2). M(N) is a concatenation of H(N) and G(N). DES(K, M) denotes the result of the DES encryption of the message M with the key K.*Hint:**Use your knowledge about properties of weak keys in DES.*

*Problem 4 (7 points)*

Find a collision in the following construction for a one-way hash function: H0 = IV, H(i)=DES(M(i) xor H(i-1), M(i-1)) xor M(i-1). Hash-value for the message M=M1, M2. Mn is equal to Hn. Prove that the collision you have found holds. DES(K, M) denotes the result of the DES encryption of the message M with the key K.*Hint:**Use your knowledge about the complementation property of DES.*

You are developing an efficient implementation of RSA for the sizes of the modulus *N* equal to 768 bits and 1024 bits. You use Chinese Remainder Theorem to speed up the decryption. You consider using the following algorithms for modular multiplication:

a) Montgomery multiplication;

b) Paper-and-pencil (classical) algorithm of multiplication followed by the Barrett's reduction.

For both given above cases and for both sizes of the modulus length, determine the optimum length *k-opt* of the exponent block processed in a single iteration of the extended classical algorithm of exponentiation (left-to-right *k* -ary exponentiation, basic handbook, algorithm 14.82). For all cases, estimate the speed-up obtained by using extended algorithm of exponentiation (left-to-right *k* -ary exponentiation) with the optimal value of the parameter *k* as compared to the classical algorithm of exponentiation (left-to-right binary exponentiation). For all four cases, estimate the size of the additional memory required to store precomputed numbers in all algorithms in use.

*Hint:**For the given assumptions concerning the algorithms in use, compute the ratio of times taken by the modular multiplication of two different numbers and for the modular squaring. Use the knowledge of this ratio, and the knowledge of the exponent length to minimize the exponentiation time.*

*Option for programmers (16 points + bonus points)*

Develop an efficient implementation of "mini-RSA" that is RSA with the length of the modulus N equal to the length of the maximum long integer supported by the language and the computer system you use.

Your implementation should consist of the separate procedures for the key generation, encryption, and decryption. All procedures should be fully tested and optimized for speed.

The following requirements must be fulfilled:

- Choose e=F4 = 2^16+1.
- Prime numbers P and Q should be found using an incremental Rabin- Miller search with an appropriate number of bases.
- Modulus N should have the full length of a long integer.
*Hint:*

Set two most significant bits of P and Q to one. - The number of messages not concealed by RSA should be minimal.
- Compute d=e^-1 mod phi(N) and d'=e^-1 mod lcm(p-1, q-1). Decide whether it is more efficient to use d or d' for decipherment.

- Implement modular exponentiation using both versions of the square-and-multiply algorithm (left-to-right and right-to-left). Compare the speed of both procedures and choose the faster one for your implementation.
- Use Chinese Remainder Theorem to speed up the decryption.

*Answer the following questions related to your implementation:*

- What is the probability that the numbers P and Q you have chosen using the Miller-Rabin test are composite?
- Is there any difference between the speed of left-to-right and right-to- left versions of square-and-multiply modular exponentiation? Explain why?
- Measure the ratio of encryption and decryption times. Find this ratio analytically. Would this ratio change if you performed operations on large integers (in the range of 200 decimal digits)?

*Extra task for bonus points:*

Implement one of the more sophisticated algorithms for exponentiation described in the basic handbook in sections 14.6 and 14.7.

Measure the ratio of times necessary for decryption of messages using the square-and-multiply algorithm of exponentiation and the more complex algorithm of your choice. *The total number of points you will get for this homework (option for programmers) will be multiplied by this ratio.*

*Send the source code of the program to me by e-mail together with your test vectors and the answers to the given above questions. Provide the information about the computer system and the tools you have used to compile or interpret the program.*

*Option for analysts (16 points required / 25 points total)*

*Problem 1 (3 points)*

Suppose Bob has an RSA cryptosystem with a very large modulus N for which the factorization cannot be found in a reasonable amount of time. Suppose Alice sends a message to Bob by representing each alphabetic character as an integer between 0 and 25 (A->0. Z->25), and then encrypting each number separately using RSA with large e and large N. Is this method secure? Explain your answer.

*Problem 2 (4 points)*

Check if the encipherment and decipherment procedures work well for the following choice of key components: e=865, P=97, Q=109. Have you noticed something unusual? How would you change the key components to obtain a better key. Verify you idea.

*Problem 3 (3 points)*

What is the probability that a randomly chosen number would not be relatively prime to some particular RSA modulus N? What threat would finding such a number pose?

*Problem 4 (5 points)*

Assume that you generate an authenticated and encrypted message by first applying the RSA transformation determined by your private key, and then enciphering the message using recipient's public key (note that you do NOT use hash function before the first transformation). Will this scheme work correctly, i.e. give the possibility to reconstruct the original message at the recipient's side, for all possible relations between the sender's modulus NS and the recipient's modulus NR (NS>NR, NS<NR, NS=NR). Explain your answer. In case your answer is "No", how would you make this scheme effective?

*Problem 5 (3 points)*

Prove that if

c = cH*2^16 + cL.

than

c = cL + /cH + 1 (mod 2^16+1),

where

/cH denotes a bit complement of cH.

- FFFE FFFF
- 00FF FF00

*Problem 6 (7 points)*

In the IDEA cryptosystem, the key K used to encipher the message M has the following value (in the hexadecimal notation):

K = 0020 00FE FFFF 0000 0010 0001 FFE1 5678

The message M in the hexadecimal notation is equal to:

M = 0020 0002 0002 0000

What are the values of the four internal keys used during the output transformation (the last, 9th round) of the IDEA during the decipherment?

What is the output of the first full round of the IDEA (including the effect of the mengler function) during the encipherment of the message M?

*Hint: Perform all operations in the hexadecimal notation. Do not convert numbers to the decimal notation.*

*Homework 2 (16 points, due Wednesday, March 5)*

*Please solve four out of the following five problems. If you are an undergraduate: solving version for graduates instead of version for graduates gives 1 bonus point; solving both versions gives bonus 2 points. If you are a graduate: solving version for undergraduates instead of version for graduates gives -1 point; solving both versions gives additional 1 point.*

*Problem 1 (4 points)*

Suppose the DES mangler function mapped every 32-bit value R, regardless of the value of the K input, to*a)* 32-bit string of ones,*b1) (undergraduates)* R,*b2) (graduates)* bitwise complement of R.

What function would DES then compute? Would it still be a non-affine function? How would the decryption look like?

*Problem 2 (4 points)*

How many 64-bit blocks of the corresponding plaintext and ciphertext, on average, do you need to find a right key using an exhaustive key search attack against DES, with the probability of error smaller than 0.01%? What is the expected number of encryptions/decryptions you need to perform?

How many 64-bit blocks of the ciphertext will you need to find the right key with the probability of error smaller than 0.01%, using exhaustive- key-search ciphertext-only attack assuming that you know that the message consists of 8-bit ASCII characters with the most significant bit equal to zero. What is the expected number of decryptions you need to perform?

*Hint:* Consider how many DES keys, on average, encrypt a particular plaintext block to a particular ciphertext block.

*Problem 3 (4 points)*

Prove that the following pairs of keys are DES semi-weak keys, that is the following equation holds for an arbitrary choice of message M

DES(K, DES(K*, M)) = M.

*Problem 4 (4 points)*

*(undergraduates and graduates)*

The pseudorandom keystream generated by 64-bit OFB (k=64), k[0] = IV, k[i+1] = DES(K, k[i]) must eventually repeat (since at most 2^64 different blocks are generated). Will the initial vector IV necessarily be the first block to be repeated?

Does the same hold if we use OFB with k EE392 Challenge

(solutions accepted till the end of the semester)

Try to recover messages encrypted using one of the remaining three ciphers. Describe shortly the most important steps of your algorithm. For each successfully mounted attack (meaningfully different than exhaustive key search) you obtain a 10 point bonus.

*Option 2 (for programmers)*

Write a program that solves equation:

a*x = b (mod n)

for arbitrary values of a, b, and n provided by the user.

*Send the source code of the program to me by e-mail, together with the output for three different test inputs. If your program is not written in C/C++, please provide the information about the tools I can use to compile or interpret the program.*

*Homework 0 (only for volunteers, 10 bonus points, equivalent to two regular homework problems)*

The following ciphertext has been obtained by applying the affine cipher to the text in English:

VYHJQ JWDCG YJMLW BLDGH OBZDJ CZJBJ JNBPO GJWDX XJEDP GJMWP ZGDZP SLGDS DGFXP FTJCJ WPSSF WJBGP BBLWJ EGYPG YJVDS SBJJN DCQPDC

Find the key that was used to encipher this message and the message itself. Describe shortly the most important steps that permitted you to recover the key. The solutions based on the exhaustive key search do not count.

- You are welcome to use the attached tables of frequencies for single letters and digrams in English.
- You are assumed to be familiar with the basics of the number theory covered in the basic textbook in sections 2.4.1 (points 2.79-2.93), 2.4.2, 2.4.3 (points 2.110-2.119) and 2.4.4 (point 2.142) (we will review this material later in class).
- Useful reading:
- basic textbook - sec. 7.3, and
- Sinkov, "Elementary Cryptanalysis," Mathematical Association of America, 1966.

Year 7 Music HomeworkTopic 2: African Music

Each slide in this presentation contains one of the 4 pieces of homework that you will be set during the topic.

They will be set fortnightly by your teacher, but you can begin any of the homework tasks whenever you want.

Homework Task 1:

Here are the main music keywords for this topic. They are in the first column. In the second column are the definitions for each one but they are jumbled up!

Task: Complete this activity in your book. Write out the keyword and match it up to the correct definition.

Homework Task 2:

- Research the following musical instruments commonly used in African Music.

Homework Task 3:

This homework task is designed for you to extend your learning and give your own opinions. You can also do research to really extend your learning and achieve the highest levels in Music.

Write a paragraph to answer the following questions. Focus carefully on your spelling and grammar:

Q: Why is African Music never written down?

Q: How do you think African musicians learn music?

Homework Task 4:

African Vocal Music

Music is a way of life for A______ people. Music, rhythms, dance and singing go hand in hand. S_____ have many uses in African culture. They are not just used for entertainment.

African musicians often sing ____________. This means there are no i__________. They make the music more interesting by singing in h______ and creating rich sounds with layers of voices. Songs are taught o________. This means they are not w______ down. People learn songs and music by singing and performing with others and this is how songs are learned and passed from generation to generation.

There are many different types of songs. There are work songs, teaching songs, songs for c__________ and songs for worship. Story telling songs are a way of preserving tribal history. In some cultures there are even special families of musicians whose job it is to teach, through the songs, the history and legends surrounding the tribe.

African singing and songs has influenced more popular singing styles too such as S_________ and G_____Music.

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