Welcome to Introduction to Applied Cryptography. Cryptography is an essential component of cybersecurity. The need to protect sensitive information and ensure the integrity of industrial control processes has placed a premium on cybersecurity skills in today’s information technology market. Demand for cybersecurity jobs is expected to rise 6 million globally by 2019, with a projected shortfall of 1.5 million, according to Symantec, the world’s largest security software vendor. According to Forbes, the cybersecurity market is expected to grow from $75 billion in 2015 to $170 billion by 2020. In this specialization, you will learn basic security issues in computer communications, classical cryptographic algorithms, symmetric-key cryptography, public-key cryptography, authentication, and digital signatures. These topics should prove especially useful to you if you are new to cybersecurity Course 1, Classical Cryptosystems, introduces you to basic concepts and terminology related to cryptography and cryptanalysis. It is recommended that you have a basic knowledge of computer science and basic math skills such as algebra and probability.
How are Cryptographic Hash Function Attacked? Part II
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Из курса от партнера University of Colorado System
Classical Cryptosystems and Core Concepts
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Курс 1 из 4 — Specialization Introduction to Applied Cryptography
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Hash Functions
Continuing on our exploration of the fundamental concept of cryptography, this module will explain the Hash Function, its purpose and application, potential attack vectors, and the importance of hash functions on cryptographic design. Upon completion you will be able to understand the role that hash functions play in cryptography and how cryptographic hash functions differ from other types of hash functions.
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William BahnLecturer
Computer Science
Richard WhiteAssistant Research Professor
Computer Science
Sang-Yoon ChangAssistant Professor
Computer Science
In part one of this lesson, we looked at the preimage and second preimage attacks,
which are both order 2 to the n when carried out via brute force,
which is the only way they can be carried out against a random oracle.
Now we get to a very interesting attack that is quite non-intuitive.
In a hash collision attack, all we are interested in doing is finding
two messages that happen to have the same digest.
And we don't care what the specific digest is.
Intuitively, we probably expect this to be little different from a second
preimage attack.
After all, it would appear to be essentially the same thing.
We want two messages that have the same digest.
But appearances can be deceiving because the seemingly minor distinction that
we don't care what the actual digest is,
makes all the difference due to what is known as the birthday paradox.
And which is why this attack is usually referred to as a birthday attack.
Many practical breaks in real cryptosystems,
such as the breaking of WEP keys and
the recently demonstrated break in SHA-1, involve some kind of birthday attack.
So let's spend some time getting familiar with this.
The Birthday Paradox asks a very simple question that most people,
the first time they see it, can confidently guess an answer that
is almost always not even close to being correct.
How many randomly chosen people do I need to have in a room
before there is a 50/50 chance that at least two people have the same birthday?
Let's simplify the problem by ignoring leap years, so we have 365 possible
birthdays, and also assuming that the distribution of birthdays is uniform.
Before answering this question, let's look at the analog to a preimage attack
that would be how many people I have to gather in a room before there's
a 50/50 chance of there being someone that has a specific birthday?
I can make this a second preimage attack by just specifying that I want
to find someone that has the same birthday as me.
Most people guess, fairly reasonably, that finding someone with a specific
birthday requires that we consider 183 people on average,
because that's half of the number of possible birthdays.
We already talked through the solution to this problem, which for small d,
the number of days or the number of digests, is given exactly by
this expression, which requires about 253 people on average.
This may or may not seem too surprising to you,
since it might not seem that different from 183 people.
But it's important to understand why it is significantly more than half.
Remember that we are asking how many people we need to consider before finding
someone that has a specific birthday.
If you think about it, you'll realize that you we're actually never guaranteed of
finding someone that has that birthday no mater how many people we look at.
One way to think of this is that there's a finite,
albeit vanishingly small, possibility that no one else on this planet
happens to have been born on the same day of the year as me.
Now let's consider the analog to the hash collision attack.
How many people do I need to consider before there is a 50/50 chance of finding
two people that happen to have the same birthday?
The first major difference is that because of the pigeonhole principle,
if I put 366 people in a room,
I am absolutely guaranteed that there are two people with the same birthday.
So as long as there are more messages than digests, I will find a collision.
But how many people do I need to be in that room before I have
a 50/50 chance of succeeding?
We can get at this by calculating the probability that
none of the k people currently in the room have the same birthday,
which we can do by adding people to the room one at a time.
If only one person is in the room,
then the likelihood that no two people in the room have the same birthday is 100%.
Or another way of saying this is that the one person can have any of
the 365 birthdays out of the 365 available birthdays.
When I put the second person into the room, the chance that they don't share
a birthday with the person already there is 364 divided by 365,
because they can have any birthday except one.
When I add the third person,
they can have any birthday except two, or 363 out of 365.
The next person can have any of 362 birthdays.
This pattern continues all the way down to the kth person.
We can now see mathematically why the 366th person
guarantees a solution, since the term associated with them is identically zero.
Meaning that the probability is zero
that no two people in the room have the same birthday.
We can write this using product notation.
We can also express it using factorial notation.
While there is no closed form solution for k, computing this probability using
one of these expressions is actually quite easy if D and
k are pretty small, especially if we use a simple spreadsheet.
We can work up incrementally from one person until the probability drops to less
than 50% since that means that the probability that at least two people do
have the same birthday has just risen above 50%.
Doing this, we find that the number of people needed to reach the 50/50
threshold is a surprisingly low 23 people.
But what about when D becomes very large, like 2 to the 256th power?
We can get a closed form approximation that gets better as D gets larger.
We can do this a couple of ways.
One is to use Stirling's approximation for the factorial function.
But a simpler way is to use an exponential approximation for
the factor in our product formula.
We can start by writing our factor as the difference between 1 and
something very small.
Then we can use the Taylor series approximation for
the exponential function valid for small exponents.
This lets us write our product formula as the product of exponentials.
By taking the natural log of both sides, we turn this into a summation which can be
reduced to one of the classic arithmetic series having a simple closed form.
Since we know k is going to be much larger than 1,
we can also ignore the minus 1 here.
This leaves us with a very simple expression that we can solve
directly for k.
We see that we just need a little more than the square root of the number
of digests.
As a sanity check for D equals 365 days,
we get 22.5 people, which is in excellent agreement with our previous results.
So we can have very good confidence that we haven't done anything wrong.
For a hash function, we can write D in terms of the size of the digest.
The key observation is that the number of hashes that must be computed is on
the order of the square root of the number of possible digests.
It also means that while the difficulty of a hash collision attack is still
exponential in the size of the hash, the effective number of bits is
only half compared to the difficulty of a preimage or second preimage attack.
This is extremely significant.
An example in a prior module described the related key attack against the original
WIFI WEP encryption which had a 64-bit key, but
only 24 bits of it actually changed with each message.
This meant that the effective number of keys were about 16 million.
But since it was only necessary to find two packets that used the same key,
a birthday attack only required about 5,000 packets in order to recover the key.
An attack that could be carried out in real time.
Recall from another example, that if we calculate a million digests a second,
then it would take half a million years to carry out an exhaustive preimage or
second preimage attack against a 64-bit hash function,
though we would have an even chance of finding a solution in about 400,000 years.
A birthday attack, on the other hand,
would have an even chance of finding a collision in less than an hour and a half.
This is why adversaries go to great lengths to figure out ways to perform
birthday attacks against cryptosystems, and
why some form of birthday attack is often the first successful crack into a system.
The first hash collision involving SHA-1 was published in early 2017, and
was the result of finding weaknesses that reduced the expected number of digests
that needed to be calculated from the 2 to the 80th needed against a random oracle,
down to about 2 to the 63rd,
which still cost about $100,000 worth of computer time to carry out.
While expensive, however, this is well within the reach
of large criminal organizations, not to mention national intelligence agencies.
Whether this capability is worth the cost for either is debatable.
And for the foreseeable future,
it is still highly unlikely that anyone will actually be able to benefit from it.
But security specialists don't think in these terms if they don't have to.
And thus, the use of SHA-1 is very quickly on the way out.
For instance, most major Internet browsers announced several years ago,
well before the first and so far only actual break,
that they will no longer accept SHA-1 signed certificates by the end of 2017.
Present best practices recommend that a 160-bit hash function
is the smallest that should be used if any kind of birthday attack is conceivable,
but 256-bit is required for most classified cryptosystems.
In the two parts of this lesson,
we've looked at the three attacks against the Random Oracle, and
seen that both preimage and second preimage attacks are ordered 2 to the n.
But the hash collision attack is order square root of 2 to the n,
which makes it extremely powerful.
Nearly all other attacks against real hash functions exploit
weaknesses in the algorithms in an attempt to reduce the amount of computational
effort to agree where some kind of brute force attack becomes feasible.
We are now at the point where we can discuss the properties that a real
cryptographic hash function should have.
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