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@@ -7,40 +7,37 @@ is_finished: False
# Password cracking
-With the development of the Internet and long-distance communication went hand in hand the need to verify whether the person on the other side of the monitor is actually the person with whom we are communicating or just someone pretending to be an acquaintance. Similar to the situation, when introducing two friendly spies in a foreign territory, the possibility of using a password is offered. Today, one encounters passwords in cyberspace on a daily basis, when logging into email, school or work accounts, or online banking.
+With the development of the internet and remote communication went hand in hand the need to verify whether the person on the other side of the screen is genuinely the individual with whom we are communicating or simply an imposter. Similar to the way two friendly spies verify each other’s identity in a foreign territory, the option of using a password has emerged. Nowadays, one encounters passwords in cyberspace on a daily basis, whether it’s logging into email, school or work accounts, or online banking.
+But does the mere existence of passwords guarantee secure user authentication? Ongoing reports of new hacks and stolen accounts tell us otherwise. The methods by which attackers get to a user's password can be broadly categorized into two groups, depending on whether it is stolen or guessed. Since the following task deals with the second case, let's take a closer look at it.
+*Brute Force Attack*, as introduced in the task, involves trying out all possible passwords. Depending on the computer’s processing power and the software used, the speed of password testing can range from thousands to several hundred billion passwords per second. Thus, very short passwords can be guessed by a computer in a relatively short time (i.e., instantly or within hours).
-But does the mere existence of passwords guarantee secure user authentication? Ongoing reports of new hacks and stolen accounts tell us not. The methods by which attackers get to a user's password can basically be divided into two groups, depending on whether it is stolen or guessed. Since the following problem deals with the second case, let's take a closer look at it.
+A more sophisticated form of brute force attack is the *Dictionary Attack*, where the computer does not randomly try out passwords but selects them from a dictionary of prepared words. In addition to actual words, this dictionary contains commonly used passwords such as `password1234` or `password`. If the victim's password is in the attacker's dictionary, the cracking time is significantly reduced compared to a conventional brute force attack.
-*The brute force attack*, which we will learn about in the task, consists of trying all possible passwords. Depending on the computing power of the computer and the software used, the speed of testing can range from a few thousand to several hundred billion passwords per second. Thus, very short passwords can be guessed by the computer in a relatively short time (i.e., instantly or within hours).
-
-A more sophisticated form of brute force attack is the *dictionary attack*, where the computer does not try passwords at random, but selects them from a dictionary of prepared words. In addition to actual words, this contains commonly used passwords such as `password1234` or `password`. If the victim's password is in the attacker's dictionary, the cracking time is significantly reduced compared to a conventional brute force attack.
-
-An essential protection against both types of attacks is the use of sufficiently long passwords (at least 12 characters) made up of upper and lower case letters, numbers and other special characters.
+Essential protection against both types of attacks is the use of sufficiently long passwords (at least 12 characters) made up of upper- and lowercase letters, digits, and other special characters.
## Assignment
-The hacker program, in a brute force attack, is guaranteed to crack an eight-character password made up of upper and lower case letters of the English alphabet in about 22 minutes. (Assume that the set of keyboard characters to be tested can be set in the program settings.)
+The hacking program in a brute force attack is guaranteed to crack an eight-character password made up of upper- and lowercase letters of the English alphabet in at most about 22 minutes (for a certain specified computational power of a computer). Also, let’s assume that the set of keyboard characters to be tested can be set in the program’s settings.
-> **Exercise 1.** How many passwords does the program try in 1 second?
+> **Exercise 1.** How many passwords can the program try out in one second?
-*Solution.* Since the English alphabet has 26 characters, there can be 52 possibilities in each position of an eight-character password (upper and lower case letters). Using the combinatorial product rule, we can deduce that the total number of possible passwords is equal to $52^{8}$.
+*Solution.* Since the English alphabet has 26 characters, there are 52 possibilities at each position of an eight-character password (upper- and lowercase letters). Using the combinatorial product rule, we can deduce that the total number of possible passwords is $52^{8}$.
-The number of passwords the program tries in one second is the total
+Then, the number of passwords the program can test in one second is approximately
$$\frac{52^8}{22\cdot 60} \doteq 40\ 500\ 000\ 000.$$
-> **Exercise 2.** How long would it take the program to crack an eight-character password if we also allow using digits?
-
-*Solution.* By adding ten new characters, there can be 62 different characters in each position. According to the combinatorial product rule, the number of possible passwords is $62^8$; using the result of the previous problem, we get the time $t$ in which the program tries all the passwords as
+> **Exercise 2.** How long at most would it take for the program to guaranteedly crack an eight-character password if we also allowed the use of digits?
+*Solution.* By adding ten new characters, there are 62 different characters possible at each position. According to the combinatorial product rule, the number of possible passwords is $62^8$; using the result of the previous exercise, the time $t$ for the program to test all passwords is given by
$$
t=\frac{62^8}{40\ 500\ 000\ 000} \doteq 5\ 391\ \text{s} \doteq 90\ \text{min}.
$$
-> **Exercise 3.** How many characters would a password consisting of numbers and lowercase or uppercase letters of the English alphabet have to be strong enough, i.e. guaranteed to take at least 100 years to crack? How does the result change if we allow for the possible use of another 40 special keyboard characters?
+> **Exercise 3.** How many characters should a password consisting of digits and lowercase or uppercase letters of the English alphabet have to be strong enough, i.e., guaranteed time for cracking would be at least 100 years? How does the result change if we allow for the possible use of an additional 40 special keyboard characters?
-*Solution.* We assume that each year has 365 days, i.e. 31 536 000 seconds. Let's denote $n$ the required number of characters and add them as in the previous problem. But now we get an exponential equation with unknown $n$, which we solve:
+*Solution.* We assume that each year has 365 days, i.e., 31 536 000 seconds. Let's denote by $n$ the required number of characters, and substitute similarly to the previous exercise. However, this time, we get an exponential equation with the unknown $n$, which we solve:
$$
\begin{aligned}
@@ -51,15 +48,15 @@ n &= \frac{\log (40\ 500\ 000\ 000 \cdot 3\ 153\ 600\ 000)}{\log 62} \doteq 11{,
\end{aligned}
$$
-A password with the required security would have to be at least 12 characters long.
+Therefore, a password with the required security would have to be at least 12 characters long.
-If we allow an additional 40 characters on the keyboard, we obtain by similar calculation a result of the form
+If we allow an additional 40 characters of the keyboard, a similar calculation gives the result:
$$
n' = \frac{\log (40\ 500\ 000\ 000 \cdot 3\ 153\ 600\ 000)}{\log 102} \doteq 10{,}01.
$$
-A password with the required security must now have at least 11 characters.
+Now, a password with the required security must have at least 11 characters.
## References and literature