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CAD REU Exercises
June 5, 2015
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Games
1.1. Consider the game G(M, S, A) where M is a positive integer, S is a set of
positive integers, and A is a subset of {0, 1, · · · , min(S)}. Determine, for the
following two examples, whether player ONE has a winning strategy:
(a) G(103, {6, 9}, {0, 2, 4})
(b) G(103, {7, 11}, {0, 2, 4, 6})
(c) G(103, {6, 9}, A), where A has only one element.
1.2.
In each case, describe an explicit winning strategy for the winning player.
1.3. Now consider the normal play versions of these three games, and answer
the above questions for these.
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Cryptography
1.1. Alice and Bob choose a key space K containing 25 6 keys. Eve build a
special computer that can check 10,000,000,000 per second.
(a) How much time does it take Eve to check half of the keys K?
(b) Alice and Bob replace the keyspace with a larger set containing 2n different
keys. How large should Alice and Bob choose n in order to force Eve’s
computer to spend 100 years checking half of the keys (assume that each
year has 365 days)?
For many years the US government used a symmetric key cipher (Data Encryption Standard) that used 56 bit keys. During the 1990s, cryptographers built
special computers and demonstrated that 56 bits keys do not provide enough
security.
1.2. Alice and Bob use a symmetric key cryptosystem in which their private key is a large prime number k and their plaintext and ciphertext are
integers. Bob encrypts a message by computing the product c = m × k.
Eve intercepts the following two ciphertexts: c1 = 12849217045006222 and
c2 = 6485880443666222. Find the shared private key k.
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1.3.
Alice and Bob created the following symmetric key crypto system:
• The messages (plain texts) will be integers with at most n digits, that is,
elements of Z10n = {0, 1, · · · , 10n − 1}.
• Their shared private key will be some large integer k.
• Encryption proceeds √
as follows: As illustration, assume that Alice is the
encryptor. Calculate k up to n decimal digits, and discard all digits preceding the decimal point, as well as the decimal point: Keep the remaining
n digits. Call this number κ. Then, encrypt the message m ∈ Z10n by
calculating e = m + κ mod 10n .
• Decryption proceeds by subtracting κ from e in Z10n .
Show that the number κ is given by the formula
√
√
κ = b10n · ( k − b kc)c
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1.4.
Number Theory
Please do the following problems.
(a) Show that ax ≡ ax mod φ(n) mod n
(b) Find all primitive roots modulo 8, 9, and 11. Does very number have a
primitive root? How many primitive roots each number has?
(c) Compute the value of 2(p−1)/2 mod p for every prime 2 < p < 20. Make
a conjecture for possible values of 2(p−1)/2 mod p when p is prime. Prove
your conjecture.
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