The New Alison App has just launched Download Now
We'll email you at these times to remind you to study
You can set up to 7 reminders per week
We'll email you at these times to remind you to study
Monday
Reminder set
7am
Tuesday
Reminder set
7am
Wednesday
Reminder set
7am
Thursday
Reminder set
7am
Friday
Reminder set
7am
Saturday
Reminder set
7am
Sunday
Reminder set
7am
Foundations of Cryptography Dr. Ashish Choudhury Department of Computer Science Indian Institute of Technology - Bangalore Lecture – 34 Composing CPA Secure Cipher with a Secure MAC Part I (Refer Slide Time: 00:34) Hello everyone, welcome to this lecture, just a quick recap. In the last lecture we had seen the definition of authenticated encryption scheme and the plan for this lecture is to construct one such authenticated encryption scheme namely, in this lecture we will see how to compose a CPA secure cipher and a secure MAC and obtain an authenticated encryption scheme. There are several approaches for composing a CPA secure cipher and a secure MAC. In this lecture, we will discuss one of those approaches of composing namely, the encrypt then authenticate approach. (Refer Slide Time: 01:02)So, the problem that we are interested to solve here is that we are given a CPA secure symmetric encryption and we are given a message authentication code and we want to combine them or compose them to obtain an authenticated encryption scheme. So, we have a secure ΠCPA = (GenCPA ,EncCPA , DecCPA) over its own plain text space, key space and a cipher text space. And we have a secure ΠMAC =(GenMAC ,TagGen ,TagVer ) and it has its own message space, key space and the tag space and the goal here is to come up with a genetic way of composing these 2 primitives such that the way we are generically composing these 2 primitives always leads to an authenticated encryption scheme. So, what I mean by generic way of composing here is that I would not focus on the underlying instantiation of ΠCPA and the underlying ΠMAC. We are just going to consider them as a black box, we assume that we are given some instantiation and arbitrary instantiation of a CPA secure scheme and an arbitrary instantiation of secure MAC scheme and our goal is to generically compose them such that we are always guaranteed to obtain an authenticated encryption scheme by that generic composition. It turns out that there are several ways of composing an arbitrary ΠCPA secure symmetric encryption and an arbitrary secure ΠMAC but all those ways of composition need not give you necessarily an authenticated encryption cipher. So, what we are going to do in this lecture and the subsequent lecture is that we are going to discuss 3 of the approaches and in this lecture; we are going to discuss only the first approach. And we are going to show that this first approach always leads you to an authenticated encryption cipher whereas, a remaining 2 approaches need not always lead you to an authenticated encryption cipher. (Refer Slide Time: 03:04) So, let us discuss the first approach namely, the encrypt then authenticate approach. So just to recall what we have given here is; we are given an arbitrary ΠCPA secure symmetric encryption and an arbitrary secure ΠMAC and we are going to compose these 2 primitives to obtain an authenticated encryption cipher which has its key generation algorithm, encryption algorithm and encryption algorithm. The message space of this composed scheme will be the message space of the underlying CPA secure scheme, the key space of the composed scheme is going to be a pair of keys, were the first key will be picked as; where the first part of the key will belong to the key space of the CPA secure encryption process and the second part of the key will belong to the key space of the underlying message authentication code. And a cipher text space of the composed scheme will again be a pair, where the first component of the pair will belong to the cipher text space of the CPA secure scheme and the second component of the cipher text will belong to the tag space of the underlying message authentication code. So that is a message space, key space and a cipher tag space of the composed scheme. Now, let us see how exactly the composing happens in this encrypt then authenticate approach, so the key generation algorithm of the composed scheme is as follows: it independently runs the key generation algorithm of the CPA secure scheme and output is key as per that key generation algorithm which I denote by ke and independently it runs instance of the key generation algorithm of the message authentication code and obtains a key as per the key generation algorithm of the MAC which I denote by km. And the overall key or the overall output of the composed key generation of the key generation algorithm for the composed scheme is considered to be the pair (ke, km). I stress here that the key (ke, km), they are independent of each other because they are obtained by running independently the key generation algorithm of the CPA secure scheme and the key generation algorithm of your MAC. We are assuming that these 2 primitives namely the CPA secure scheme and the MAC, they are independent of each other. Now, let us see the encryption algorithm of the composed scheme, so imagine we are given a plain text m which we want to now encrypt as per this composed scheme and as per our notion, the key here basically consist of 2 parts; the first part of the key is for the CPA secure scheme and the second part of the key is for computing a MAC as per the message authentication code. So you can imagine it as a box here, which takes as input as the message and the key, and as the name suggest encrypt then authenticate, so what we are going to do first is we are going to encrypt the message as per the CPA secure scheme using the ke part of the overall key for the composed scheme and it will produce a cipher text c. Now the cipher text c is treated as a message for authentication and what we now going to do is; we are going to compute a tag for this c; creating the c as a message and the tag is now computed using the second part of the key namely km and whatever comes out of this tag generation algorithm, we denote it as t and now, the overall cipher text for this composed scheme will be the c part that we have obtained by encrypting the message and the t part that we have obtained by computing the tag on the c part, so that is how overall cipher text that we have obtained by this encrypt then authenticate approach. Now, let us see how the decryption happens, so again the decryption algorithm will take a key which will further consist of 2 parts and it takes the cipher text which again consist of 2 parts; a c part and the t part. And it takes the key, now to decrypt the cipher text in the composed scheme what we do is; we first perform the tag verification. So we are just performing the reverse operation what we have done for the encryption in the encryption process, so in the encryption process, we have first encrypted and then we have computed the tag, while decryption we will just perform the operation in the reverse order. So, we will first check if the TagVer(,) = 1. So if the output is 1, that means, the t part of the cipher text is indeed a valid tag on the c part of the cipher text, then what we do is we decrypt the c part of the cipher text using the ke part of the overall key and recover the plain text. Whereas, if the output of the tag verification is 0, then we do not perform the decryption as per the CPA secure scheme be simply output box, so depending upon whether the tag verification internally outputs 1 or 0, we either end up outputting the plain text m or we output part indicating that the c concatenated with t, the cipher text that we are receiving in the decryption algorithm is an invalid cipher text. So that is a way we are going to compose the CPA secure encryption and a secure MAC as per the encrypt then authenticate approach. (Refer Slide Time: 08:49)Now, we want to formally prove that if the CPA secure scheme; if the symmetric encryption that we have taken in this composition is CPA secure and if the MAC that we have taken in this composition is secure say, strong CMA secure, then irrespective of what exactly is the underlying instantiation of the ΠCPA secure scheme and what exactly is the underlying instantiation of the ΠMAC component, the overall composed encryption process that we have obtained is always an authenticated encryption scheme for a message space which is same as the message space of the underlying CPA secure scheme, so that is a theorem which we want to prove. So, before going into the formal details of the proof, let us first try to understand, what exactly is the intuition, why exactly this theorem statement is true. So, trivially it is easy to see that the composed scheme is CPA secure because the composed scheme basically consists of a composed encryption process, it internally consists of the underlying CPA secure scheme. The only difference in the authenticated encryption scheme is that as per the definition of authenticated encryption, we require it to be having both cipher text integrity property as well as CCA security property. So, I claim here that any cipher text (c*, t*) which is different from several legitimate cipher text which an adversary has seen in the past is always going to decrypt back to an invalid output and this is because if at all there is a malicious adversary who wants to forge a new cipher text, say (c*, t*) which is different from previous cipher text that it has seen in the CCA experiment as a response from the encryption Oracle service. Then that means that internally, adversary has to create or it has to compute a tag t* on the message c* under the unknown key km, only if that is the case that is possible for an adversary, then only the adversary could come up or it can forge a cipher text (c*, t*) which is different from all the legitimate cipher text that it has seen as a response from the encryption Oracle service. But that will be a violation of the strong CMA security of the underlying MAC, so remember we are assuming that the underlying MAC is strong CMA secure and if it is strong CMA secure, then it is very unlikely that a poly time adversary based on several cipher text that it is seen in the past could come up with the new cipher text of the form (c*, t*), where t* is a tag on c*, where c* is treated as the message. So that is the first observation here that means, if it is difficult for an adversary to come up with the new legitimate cipher text which decrypts to a non-null output, then it means that any CCA attacker from the viewpoint of that CCA attacker, the decryption Oracle service is completely useless because if at all it tries to create a new cipher text, it cannot do that because creating a new legitimate cipher text is as good as forging the underlying message authentication code which it can do only with very less probability. And that means that any new cipher texts or any new bit strings, c concatenated t which it tries to create and submit to the decryption Oracle service will always return back bot to him and that means decryption Oracle service is completely useless for such a CCA attacker and hence such a CCA attacker basically reduces to an instance of a CPA attacker and that automatically means that if our composed scheme Π is CPA secure. That means, basically what we are now doing is; we are reducing the CCA attacker to a CPA attacker and since our composed scheme Π uses and underlying CPA secure scheme for computing the c part of the cipher text, it will be ensured that even c concatenated t will also be CPA secure, it will give you CPA secure encryption. So that is how overall idea, the crux of the proof here is to basically show that the way we are composing the MAC and the CPA secure scheme, decryption Oracle service is going to be useless for any CCA attacker and any CCA attacker basically reduces to a CPA attacker and since we are using a CPA secure scheme internally to perform the encryption in the composed scheme, the composed scheme also will be CPA secure. However, it turns out that even though this intuition is very simple to understand formally establishing this through reductions is the challenging task. (Refer Slide Time: 13:44) So, let us go into the details of the reduction proof here, so I am retaining the encryption process and the decryption process of the composed scheme and imagine that you have an arbitrary CCA attacker who is participating in an instance of the CCA experiment against this composed scheme Π, so as per the rules of the CCA experiment is adversary ACCA will have access to both the encryption Oracle as well as the decryption Oracle service. And what I have done here is I am showing the interaction of this adversary with the encryption Oracle service with the first box and with the second box, I am basically denoting the interaction of this adversary with the decryption Oracle service. When I am saying encryption Oracle service; why it is an encryption Oracle service because the adversary does not know the value of the random ke part and the km part of the key for the composed scheme which the encryption oracle is going to use to encrypt the plaintext which the adversary is going to submit. In the same way, when the adversary is asking for the decryption of cipher text, adversary would not be knowing the ke part and the km part of the key of the composed scheme which the decryption Oracle of the composed scheme is going to use to respond to the decryption Oracle queries of this adversary. So, now before going in to the formal reduction, let me introduce some definitions here. So, in any instance of the CCA game, I will call a cipher text (c | t) to be a new cipher text, if the following conditions hold. The first condition is that ACCA has not obtained the cipher text from its encryption Oracle and or the cipher text (c | t) is different from the challenge cipher text received by this adversary in the CCA game. So, if any of these 2 conditions hold, then I will call that cipher text (c | t) to be a new cipher text. Basically, the definition of a new cipher text is that it is new in the sense that adversary has produced this string on its own that means, it has not seen this string as the encryption of any message of its choice which it has submitted to the encryption Oracle service or it is different from the encryption of the challenge pair of plaintext which the CCA attacker is going to submit in the CCA game. If any of these 2 conditions hold, then I call this bit string (c | t) to be a new cipher text, now let me define in event which I called as valid query and this event valid query denotes the event that this CCA attacker ACCA submits a new cipher text to its decryption Oracle such that the decryption of this new cipher text under the unknown key ke followed by km is a legitimate output that means, its output is bot, so that is the event valid query. So, I claim that if the underlying MAC which I am using in the composed scheme is strong CMA secure, then the probability with which the event valid query occurs is negligible, so we are going to prove this later. For the moment assume this claim is true, I also pause here for a moment and I would like to stress here that what exactly is the event valid query.If you see closely the way I have defined the event valid query that is almost the same as the even that the CCA attacker could break the cipher text integrity property in the presence of the encryption Oracle service as well as in the presence of the decryption Oracle service because valid query means that this attacker ACCA is able to come up with a new cipher text, which is different from any cipher text which it has seen as a response from the encryption Oracle service as well as the cipher text which is different; as well as it is different from the challenge cipher text such that its decryption is a non bot output. That is precisely what we require when we say that an adversary wants to break the cipher text integrity property but in the cipher text integrity property, the goal of the adversary is to come up with the new cipher text, a new valid cipher text, only based on the encryptions of messages of its choice that means, only in the presence of an encryption Oracle service. But now, this event valid query is the event that the CCA attacker could come up with a forged cipher text whose decryption is a legitimate output, even the adversary has got access to the encryption Oracle service as well as decryption Oracle service. That means, if at all this claim which I am claiming here or making here is true that means, the probability that the event valid query occurs with the negligible probability, if this claim is true. That automatically, implies that my encryption process that I have done; I have designed here by composing the CPA secure scheme and the MAC has also the cipher text integrity property that means, the probability with which the cipher text integrity property will be satisfied is always upper bounded by the probability with which the event valid query can occur in the presence of any CCA attacker. And if I indeed prove that the probability that the event valid query occurs with the negligible probability, it automatically implies that the cipher text integrity property is also satisfied by my scheme. So we will again touch upon this fact later on when we will prove the cipher text integrity property of the composed scheme. (Refer Slide Time: 19:28)But for the moment, assume that this claim is true. So now I want to prove that the composed scheme that we have designed here is indeed CCA secure, so let us try to calculate that what is the probability that any CCA attacker attacking this composed scheme is able to win the instance of the CCA game that means, what is the probability that the output of the CCA experiment is 1? The probability that this adversary can win the CCA game can be further divided into 2 cases namely first case where the adversary can win the CCA game under the presence of the event that event valid query occurs and second case where the adversary can win the CCA game even when the event valid query does not occur. That is an overall probability of the CCA attacker winning the CCA game. And then what I can do is that this first quantity here I can always upper bounded by the probability that event valid query occurs, so that is how I obtain an upper bound on the overall probability with which the adversary can win the CCA game. So that is an overall probability and what I am going to show next is that if my underlying symmetric encryption process in the composed scheme is CPA secure, then what I am going to prove is that the probability that the CCA attacker can win the game even in the absence of the event valid query is upper bounded by 1/2 + negligible.Intuitively this is true because if the event valid query does not occur that means, all the new cipher text which the CCA attacker is going to submit to the decryption Oracle is going to give him back an output part, then basically this CCA attacker reduces to an instance of CPA attacker and is my underlying symmetric encryption process is CPA secure, then it automatically means that overall probability of the CCA attacker winning the game is upper bounded by 1/2 + negligible. So, that is a claim I am going to prove next, so now assuming for the moment that indeed this claim is true, then I am already claiming that the probability with which the event valid query can occur is negligible and if I indeed prove this new claim which I am making here that the probability that this CCA attacker can win the game in the absence of the event valid query is upper bounded by ½+ negligible. Then, I obtain that overall the probability that the CCA attacker can win the game is upper bounded by ½+ negl(n)+ some other negligible function and a sum of 2 negligible function is also a negligible function, so this will prove that the probability that CCA attacker is able to win the CCA game is ½+ negl(n) and hence the scheme is CCA secure. So, now let us see the proof of this new claim, which I am making here. (Refer Slide Time: 22:33)So, I want to prove that if the underlying symmetric encryption which I am using in the composed scheme is CPA secure, then this probability is upper bounded by ½+ negl(n) and this I am going to establish formerly through a reduction. So, imagine an arbitrary CCA attacker who is playing an instance of the CCA game against the composed scheme Π, using such an attacker I am going to design another attacker which I denote it as ACPA, whose goal is to win an instance of the CPA game against the underlying symmetric encryption process which is used in the composed scheme. So, basically the goal of this adversary ACPA is to win an instance of the CPA game against the underlying symmetric encryption process which I am using in the composed scheme. So in the left hand side part basically, of the overall reduction you have an instance of the CPA game but against the underlying symmetric encryption scheme which is used in the composed scheme and on the right hand side part of the experiment you have an instance of the CCA game. This adversary ACPA is going to play a dual role; on the left hand side part of the reduction he is acting as an adversary in an instance of the CPA game, whereas in the right hand side part of the experiment he is acting as a verifier for an instance of the CCA game against the composed scheme. So, now let us see how exactly the interactions happen in this reduction, so when this adversary ACPA invokes the adversary ACCA. The adversary ACCA can demand for encryption Oracle queries, where it submits several plain text and asked for the encryptions of those messages as per the composed scheme, to respond to those queries, what adversary ACPA does is it simply forwards those queries as if it wants to know the encryption of those messages as per the underlying symmetric encryption scheme. Now, what the verifier for the CPA game is going to do is it is going to respond to this queries and to respond to those queries, it picks the key as per the key generation algorithm of the underlying symmetric encryption scheme which it I denote by ke and it is going to encrypt those messages and the cipher text here which are returned back to the adversary ACPA are basically the encryption of these messages mi’s under the key ke.Now, this adversary ACPA, he has to act as a verifier and he has to respond to the encryption Oracle queries that he has seen in the CCA part of the experiment so, what it is going to do is to respond to the encryption Oracle queries, this adversary ACPA itself picks a uniformly random key from the key space of the underlying MAC as per the key generation of the algorithm of the MAC component and what it is going to do is: it is going to compute tag ti’s on the ci cipher text which it had seen from the verifier of the CPA part of the experiment of the reduction. Each of those ci’s concatenated with the ti’s are sent back as the response from the encryption Oracle service to the adversary ACCA. So, if you see closely here what is happening here basically, adversary ACCA it expects the encryption of these messages m1, m2,.. ms as per the composed scheme Π. And how exactly the encryption for these messages would have been computed, it would have been computed by first encrypting these messages as per the underlying symmetric encryption scheme, so that is what is the ci part here and on top of that a layer of message authentication tag will be computed and that is what this adversary ACPA itself is doing. So, if I just consider the view of this adversary ACCA, it view is exactly the same as it would have expected when it participates in the instance of the CCA game and it sees the response of this encryption Oracle queries from the corresponding experiment. So, distribution wise the probability distribution of the cipher text at adversary ACCA finally sees in response to his encryption Oracle queries is exactly same as per an instance of the CCA game that it expects. The difference here is that now the adversary's interaction basically is with the CPA attacker who is computing the layer of MAC tag on the cipher text which it is seeing from the verifier or the experiment or the CPA experiment. So, that is how the encryption Oracle queries are handled by this adversary ACPA. Now suppose this CCA attacker ask for the decryption Oracle queries namely suppose, it submits a cipher text (c | t) and asked for the decryption of such cipher text. Now, what this adversary ACPA has to do: he has to act as a verifier for the CCA experiment and he has to decrypt those cipher text. But it cannot decrypt those cipher text completely because for decrypting those cipher text, you need to have both the km part of the key as well as the ke part of the key. The km part of the key is available with this CPA attacker but the ke part of the key is available with the experiment; with the CPA experiment which it cannot fetch because if it can fetch the ke part of the CPA game, then trivially this adversary ACPA can win the CPA game. The challenge for this adversary ACPA is basically to win the CPA experiment even without knowing the key ke, so the way this adversary ACPA is going to respond to this decryption Oracle queries is as follows: it sees whether this string (c | t) which has been submitted by the CCA attacker for the decryption Oracle service is already present in the list of the cipher text which this adversary ACPA has sent back to the adversary in response to an encryption Oracle queries. Basically, it checks whether the cipher text is a result of encryption of any of the previous messages for which the adversary ACCA has asked the encryption Oracle service, if it so happened that indeed this is the case then what this adversary ACPA does is; it just response back with the message m as the output of the decryption Oracle service for the cipher text (c | t). On the other hand, if this adversary ACPA sees that the cipher text (c | t) is a new string which is not present in the list of the cipher text which it has already sent as a response to the encryption Oracle queries to the adversary ACCA, then basically the response for this decryption Oracle service for such new cipher text is bot, that is what is the strategy of this adversary ACPA to respond to the decryption Oracle service. Now, as per the CCA game, this adversary ACCA will submit a pair of challenge plaintext (m0, m1) and what this adversary ACPA does is it simply forward those plaintexts as the challenge plaintext in the CPA part of this reduction. What this CPA experiment does is it randomly decides one of the message for encryption and encrypt text message as per the key ke and what this adversary ACPA does is it adds its own layer of message authentication tag on c* treating c* as the tag as per the key km and that sent as the challenge cipher text to this adversary ACCA. Again if you see the view of the adversary that means, the probability distribution of the challenge cipher text and adversary ACCA is seen is exactly the same as this adversary ACCAwould have seen by participating in a real instance of the CCA game. The cipher text (c*, t*) would have been computed by first encrypting the message mb under a key ke, which is uniformly random. And then follow it by adding a layer of tag under a key km, which is also uniformly random and that is how exactly the distribution of the challenge cipher text (c*, t*) that this adversary ACCA is seeing in this experiment.In this lecture we have seen how to compose CPA secure symmetric encryption and a secure MAC and using the encrypt then authenticate approach such that the composed scheme is always provably
Log in to save your progress and obtain a certificate in Alison’s free Cryptography: Encryption Schemes and Cyclic Groups online course
Sign up to save your progress and obtain a certificate in Alison’s free Cryptography: Encryption Schemes and Cyclic Groups online course
This is the name that will appear on your Certification
Please enter you email address and we will mail you a link to reset your password.