This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

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From the course by University of Maryland, College Park

Cryptography

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This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

From the lesson

Week 3

Private-Key Encryption

- Jonathan KatzProfessor, University of Maryland, and Director, Maryland Cybersecurity Center

Maryland Cybersecurity Center

[SOUND] In the last few lectures, we've been talking

Â about security against chosen ciphertext attacks.

Â And in our definition of security against chosen ciphertext attacks, we allow

Â the attacker to obtain the encryption of any cypher text of its choice.

Â Of course, besides the challenge cypher text.

Â And while this makes sense from a definitional point of view because we

Â want to capture any possible sort of chosen cyphertext attack,

Â it's still an unrealistic sort of a model, because in practice,

Â the attacker will not be given the full decryption of arbitrary ciphertext.

Â So what I want to show here is a real-world scenario where only one bit

Â about decrypted ciphertext is leaked to the attacker, but nevertheless,

Â this information can be exploited by an attacker to learn the entire plaintext.

Â And the point of this is to demonstrate the importance of

Â protecting yourself against chosen ciphertext attacks.

Â Because a scheme that was secure against chosen ciphertext attacks would in

Â particular be secure against the attack we're going to show here.

Â Now to describe the attack, I just need to remind you about CBC mode encryption.

Â So remember that in CBC mode encryption,

Â we have a message consisting of say t blocks of plain text.

Â And what we do to encrypt that, is we choose a random initialization vector, IV.

Â Output the IV as the first block of the ciphertext, and then chain

Â preceding blocks of cyphertext XORing them with the next block of the plain text.

Â And then feeding that into our block cipher to produce the next block of

Â ciphertext output.

Â Now in our description of CBC mode encryption,

Â we've been assuming until now that the plain text

Â is an integral multiple of the block length of the block cipher.

Â But in practice, that may not be the case and what we'd like to do is allow for

Â more flexibility and allow the sender to encrypt messages whose length is

Â not an integral multiple of the block length of the underlying cipher.

Â So in general,

Â what we can do to achieve this, is to take the message that the sender wants to send.

Â Encode it in some way, some non-cryptographic way,

Â that will allow efficient reconstruction of the original message, and

Â then encrypt the encoded data using CBC mode encryption to yield the ciphertext.

Â Now in the description I'm going to present here, we're going to assume for

Â simplicity that the message is an integral number of bytes.

Â But it will not necessarily be an integral multiple of the block length

Â of the cipher.

Â And one particular encoding scheme we're going to look at is

Â called PKCS #5 encoding.

Â And what you do in PKCS #5 encoding is the following.

Â So let's let L denote the block length in bytes of our block cipher.

Â So, if we look at the message we've been given, we can determine from the length of

Â the message, how many bytes of padding we need to append to the message

Â in order to get something which is an integral multiple of the block length.

Â And let's let B denote the number of bytes that we need to append to the message

Â in order to get something, whose length is a multiple of L, the block link.

Â I'll note here that B, the number of bytes of padding we need,

Â is going to be strictly between 1 and L, and note that we cannot have B equal to 0.

Â That is we cannot have a case where we don't pad anything.

Â And that might seem a little bit strange,

Â because what if our message actually is an integral multiple of the block length?

Â Well the point is that we need the receiver to be able to

Â unambiguously recover the original message.

Â And if, and if there are cases in which we don't add any padding at all,

Â then that will be impossible.

Â So if it turns out that our message is indeed an integral multiple of the block

Â length, what we end up doing in that case is actually appending an additional L

Â bytes of padding to the end of that message before encrypting it.

Â So what we do then is we take b, the number of bytes that we need to append,

Â and we append that value and code it in one byte, a total of b times.

Â So we're appending indeed b bytes of encoding,

Â which consist of the value b repeated b times.

Â So for example, if we need to append three bytes of padding to our message, then what

Â we do is we append the three bytes, 030303, here written in hexadecimal.

Â To decrypt, the receiver will use CBC-mode decryption to obtain the encoded data.

Â And then they need to recover from the encoded data the original message.

Â And they do this in the following way.

Â So say the final byte of the encoded data has value b.

Â Well, if b is equal to 0, or if b is greater than L, then there's an error.

Â No properly encoded message can have a final byte.

Â Which is either 0 or greater than L.

Â So in that case, the receiver will return an error.

Â Otherwise, what the receiver does is to look at the final b

Â bytes of the encoded data.

Â And these should all be equal to b.

Â Again, any properly encoded message will be of that form.

Â So if the final b bytes of encoded data are not all equal to b,

Â then the receiver again returns an error.

Â And otherwise, what it does is simply strip off the final b bytes of

Â the encoded data and

Â output the initial portion of the message whatever's left as the message itself.

Â So here's just an example.

Â Here we have L equals 8, and

Â we have a 6 byte message that the sender wants to encrypt.

Â So what they'll do is they'll encode their message by appending 2 bytes.

Â And those will of course both have value 2.

Â It will then encrypt that and send it to the receiver.

Â And the receiver will decrypt to obtain the same encoded message.

Â The receiver looks at the final byte, which has value 2.

Â And then strips off the final 2 bytes of padding,

Â which were both indeed equal to 2, to recover the original 6 byte message.

Â Now, the attacker can potentially exploit the behavior of

Â the receiver to learn information about a ciphertext.

Â And this will be done in a sort of a chosen ciphertext attack, but

Â of a limited form.

Â So here we assume that the attacker has intercepted some cyphertext c,

Â which was generated by some honest sender in the appropriate way, i.e.,

Â by padding it appropriately and then encrypting it.

Â And what the attacker can then do, is to construct modified cyphertext,

Â say c prime, send that to the receiver as if it were coming from the honest sender.

Â The receiver will decrypt that to obtain some encoded data,

Â and then try to recover a message from that encoded data.

Â I would point the receiver will either return an error or not, i.e.,

Â either the encoded data was properly formatted, and then there's no error.

Â Or, as on the previous slide, there will be some error in the processing and

Â the receiver will return an error message.

Â We'll call this a padding oracle.

Â So that is we have imagine that the attacker will interact with this receiver

Â i.e this oracle, that takes cyphertext as input, decrypts them and

Â then checks the encoding and outputs a 0 or 1, depending on whether or

Â not the encoded data is correctly formatted.

Â Now this may seem like something that would never happen in real life.

Â However, padding oracles are frequently present in web applications.

Â And if you think about it, this makes sense.

Â Because when the receiver gets a ciphertext, and decrypts it and

Â gets some encoded data which is not properly formatted, in general,

Â it really does need to inform Defender that something went wrong.

Â Maybe the message, or maybe the ciphertext was garbled in transit.

Â Maybe there was a network level error.

Â But either way, there has to be some way for

Â the receiver to recover from that event.

Â And even if the receiver does not explicitly return an error

Â message in the case of improper encoding.

Â An attacker is still might be able to detect whether or

Â not an error occurred based on differences in the timing of what the receiver does or

Â in the behavior of the receiver overall, i.e if the receiver

Â terminates the connection, then that's indicative of an error.

Â And even any kind of change like that can be enough to effectively give

Â the attacker access to a padding oracle.

Â Now let me go through the main idea of the attack, again,

Â assuming that the attacker has access to this padding oracle.

Â For simplicity, let's assume we have a two-block ciphertext

Â throughout the node by IV, c.

Â So IV is the initial block of ciphertext, and c is the next block.

Â So this means we have a one block encoded data.

Â And therefore a, a, a plain text, which is strictly smaller than one block.

Â So, the encoded data is then equal to Fk inverse of c, XOR with IV.

Â This is simply the, the decryption operation when running CBC mode.

Â This is what the receiver will do to obtain the encoded data.

Â Now the attacker doesn't know k,

Â and the attacker of course doesn't know Fk inverse of c.

Â The point is that it does know that this is exactly how the receiver will be

Â computing the encoded data.

Â And the main observation here,

Â is that if the attacker modifies the ith byte of the IV,

Â this will cause a predictable change only to the ith byte of the encoded data.

Â Right, so if c remains unchanged and

Â the attacker modifies the ith byte of the IV, then Fk inverse remains unchanged.

Â And then when you're, when that gets XORed with IV,

Â the result will be effected only in the ith byte of the result.

Â So here I've tried to draw the picture from the point of view of the attacker.

Â We have Fk inverse of c,

Â which consists of a bunch of bytes which are unknown to the attacker.

Â And the attacker knows that those are going to be XORed with the IV,

Â which is known to the attacker, to give some encoded data,

Â which at this point in time is completely unknown to the attacker.

Â The attacker does know, however,

Â that the encoded data that results from this is properly formatted.

Â And that's because

Â the sender properly formatted the message before encrypting it.

Â So the attacker knows that if it forwards IV,c to the receiver,

Â the receiver will decrypt, the padding will be fine, and there will be no error.

Â What the attacker can now do, is try modifying the first byte of the IV.

Â This will result in a modification only to the first byte of the encoded data,

Â as we said before.

Â The attacker can forward the resulting ciphertext with this

Â modified first byte of IV.

Â And check whether or not this yields an error or not.

Â And let's assume that in this case there's no error, so

Â it does get successfully decrypted.

Â Or the attacker can then modify the second byte of the IV and

Â check whether it's successfully decrypted.

Â And then the third byte of the IV and so on.

Â Let's assume that after modifying the third byte of the IV,

Â the receiver returns an error.

Â So this means that at this point in time, when a third bit of the IV has been

Â modified, suddenly the encoded data is no longer properly formatted.

Â What does that tell the attacker?

Â Well, what that tells the attacker is that the receiver is checking the final 6

Â bytes of the encoded data to check whether the encoding was done properly, right?

Â And that's because when the attacker had only modified the second byte of the IV,

Â decryption succeeded, but when it modified the 3 byte, decryption suddenly failed.

Â That tells the attacker that exactly the final 6 bytes of

Â encoded data are being checked.

Â And that means in turn,

Â that the final 6 bytes of the original encoded data were all equal to 6.

Â So if we go back to the original situation with the original IV, the attacker has

Â now learned that the final 6 bytes of encoded data were all equal to 6.

Â But the remaining 2 bytes of encoded data,

Â which are the original message, are still unknown.

Â But let's see what the attacker can do next.

Â What they can do is they can modify, say the right-most byte of the IV,

Â in a particular way to cause a predictable change in the right-most byte of

Â the encoded data.

Â So the right most byte of the IV started out being equal to 9E.

Â If the attacker modified that to a value which is equal to 9E XORed with 6,

Â XORed with 7, then what it knows is that

Â that will result in a final byte of encoded data being equal to 7.

Â And it can do something similar with the second to right most byte of the IV

Â in order to cause the second to right most byte of the encoded data to come out

Â equal to 7.

Â And so on for the final 6 bytes of the IV and the 6 bytes of the encoded data.

Â Now if the attacker sends this to the padding oracle,

Â most likely decryption will fail because it's, it's unlikely in general that

Â the second byte of the encoded data will happen to be equal to 7.

Â But what the attacker can do now is simply try all possibilities for

Â the second byte of the IV.

Â So it can start out by setting the second byte of the IV to 0, and

Â check to see whether or not the resulting cyphertext decrypts correctly or not.

Â It can then try this with the byte having value 1, 2, et cetera, until

Â eventually it reaches a value for the second byte, at which decryption succeeds.

Â And when that happens, the attacker then knows that the second byte of

Â the encoded data, at that point, must be equal to the value 7.

Â Because that's the only way decryption will succeed.

Â And now we're done.

Â The attacker has learned the second byte of encoded data.

Â Why is that?

Â Because the attacker now knows that xx, that is the value of the second byte

Â of Fk inverse of c, XORed with 41 in this case, is equal to the value 7.

Â So now we can determine what xx XORed with 1 is equal to.

Â So remember 1 was the actual value in the cyphertext in

Â the second byte of the cyphertext that the attacker had intercepted.

Â And so we can figure out just through some basic math that the second byte

Â of the original encoded data transmitted by the sender was equal to 47.

Â That just comes out from doing the math.

Â And it can repeat the process now with the first byte of the IV,

Â in order to learn the first byte of the encoded data.

Â What's the complexity of this attack?

Â Well, it takes at most L tries for

Â the attacker to learn the number of padding bytes.

Â This is because it's just going through the bytes of the IV one by one.

Â For at most L bytes until decryption fails, at which point the attacker learns

Â exactly how much padding there is and what those padding bytes are.

Â And then for each byte of the original plaintext message,

Â it takes at most 256 tries to cycle through all possibilities.

Â All possible bytes of that position of the IV.

Â And at that point, the attacker was guaranteed to learn the value of

Â the original plaintext at that byte.

Â So it takes 256 tries per byte, so 256 times the number of bytes in

Â the original message, and the attacker learns the entire original message.

Â Now that might sound like a lot, but in fact if the receiver were not doing any

Â sort of throttling to prevent decrypting cyphertext, and, and, and

Â continuing to get errors like that, then it wouldn't actually take very long in

Â terms of time for the attacker to run 256 tries with the receiver.

Â This could be done on the order of seconds.

Â So what I've tried to show in this lecture,

Â is an example where chosen-cyphertext attacks are a significant

Â real-world threat on a deployed use of cryptography.

Â And I'll highlight again that these kind of attacks have been found to exist in

Â the real world.

Â And have, in fact, been shown to be exploitable in the real world.

Â So this is not just a theoretical sort of a threat.

Â But I think it also demonstrates what kind of things can be done with chosen

Â ciphertext attacks, and shows more generally why such

Â attacks are problematic, and why you would want to defend against them.

Â And for this reason, modern encryption schemes are generally designed to be

Â CCA-secure in order to rule out any such attacks,

Â in particular the padding oracle type attacks like we've shown here.

Â Now, we're not going to see an example of a CCA-secure scheme now.

Â However, we will see an example at the end of next week.

Â I'll see you next time.

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