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2. You are making a new account on your favorite social media website. This great new website offers the most fun way to steal all of your personal information! Long story short, you need a new password. ords can include upper-case letters, lower-case letters, numbers and the symbols you can r. Characters may be repeated. Please do not simplify your answers. get by holding Shift + a numbe (a) Suppose you want to make a 7, 8 or 9-character long password. How many such passwords are there? (b) Now suppose you want to generate a password of length n that includes exactly one symbol and one n umber. Find an expression in terms of n for the number of such passwords. (c) Suppose you want to make a 9-character password, and passwords must contain upper- case letter and at least one number. How many such passwords are there? (d) A passwords entropy can be calculated as the log2 of the number of characters in the character set used, multiplied by the number of characters in the password itself. How many bits of entropy does a 9-character password have, if it may be chosen from the set of upper-case letters, lower-case letters, numbers and symbols (Shift +number)?
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Answer #1

With all the upper-case and lower-case alphabets, numbers, and characters with shift+number, there are 26+26+10+10 = 72 choices for each character. This will be important for calculations.

(a) There are 72^7, 72^8, 72^9 passwords for length 7,8,9 respectively, as any character can be used and repetitions are allowed.

(b) First choose the position for the symbol, and then the number. For the other positions, there are 52 choices, for these two positions, there are 10 choices each. Thus, the answer is 100 * 52^{n-2} .

(c) Count the passwords with no upper-case letter and no number. As there are 36 choices for each positions, there are 36^9 many of them. Now count the ones with no upper-case letters. This will be 46^9 . Now the ones with no numbers. This will be 62^9 . Now by the exclusion inclusion principle, there are 46^9 + 62^9 - 36^9 passwords which either don't have a number or an upper-case letter. Thus, there are 72^9 - 46^9 - 62^9 + 36^9 passwords with at least one upper-case letter and one number, by taking the complement.

(d) The set of characters has size 72 thus the entropy for a 9 character password is 9 * \log_272 = 55.53 .

Comment in case of any doubts.

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