Question

Find an item related to mathematics that was recently in the news, and write a short, 500 word essay. Again, it's okay if it's a bit longer.

Answer 1

The Mathematics of Hacking password .

Here is the logic behind setting hack-resistant passwords. When you are asked to create a password of a certain length and combination of elements, your choice will fit into the realm of all unique options that conform to that rule—into the “space” of possibilities. For example, if you were told to use six lowercase letters—such as, afzjxd, auntie, secret, wwwwww—the space would contain 26⁶, or 308,915,776, possibilities. In other words, there are 26 possible choices for the first letter, 26 possible choices for the second, and so forth. These choices are independent: you do not have to use different letters, so the size of the password space is the product of the possibilities, or 26 x 26 x 26 x 26 x 26 x 26 = 26⁶.

If you are told to select a 12-character password that can include uppercase and lowercase letters, the 10 digits and 10 symbols (say, !, @, #, $, %, ^, &, ?, / and +), you would have 72 possibilities for each of the 12 characters of the password. The size of the possibility space would then be 72¹² (19,408,409,961,765,342,806,016, or close to 19 x 10²¹).

That is more than 62 trillion times the size of the first space. A computer running through all the possibilities for your 12-character password one by one would take 62 trillion times longer. If your computer spent a second visiting the six-character space, it would have to devote two million years to examining each of the passwords in the 12-character space. The multitude of possibilities makes it impractical for a hacker to carry out a plan of attack that might have been feasible for the six-character space.

Calculating the size of these spaces by computer usually involves counting the number of binary digits in the number of possibilities. That number, N, is derived from this formula: 1 + integer(log₂(N)). In the formula, the value of log₂(N) is a real number with many decimal places, such as log₂(26⁶) = 28.202638…. The “integer” in the formula indicates that the decimal portion of that log value is omitted, rounding down to a whole number—as in integer(28.202638… 28). For the example of six lowercase letters above, the computation results in 29 bits; for the more complex, 12-character example, it is 75 bits. (Mathematicians refer to the possibility spaces as having entropy of 29 and 75 bits, respectively.) The French National Cybersecurity Agency (ANSSI) recommends spaces having a minimum of 100 bits when it comes to passwords or secret keys for encryption systems that absolutely must be secure. Encryption involves representing data in a way that ensures it cannot be retrieved unless a recipient has a secret code-breaking key. In fact, the agency recommends a possibility space of 128 bits to guarantee security for several years. It considers 64 bits to be very small (very weak); 64 to 80 bits to be small; and 80 to 100 bits to be medium (moderately strong).