52 Things: Number 9: What are Shannon's definitions of entropy and information?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know To Do Cryptography': a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. This blog post tries to introduce some fundamental concepts in Information Theory: What are Shannon's definitions of entropy and information?

Information Theory was founded by Claude E. Shannon in 1948. It was originally developed to study signal processing but its application has been broadened to various disciplines through decades. This blog is intended to briefly introduce two fundamental concepts, entropy and information. If you are interested in more details, I personally suggest you to find more in [1].

Entropy

Entropy is a measurement to evaluate the uncertainty[3] of one or more variables.

Assume we are investigating the first web page people visit when they open a web browser. We use samples from two groups of people: 4 cryptographer from Bristol Cryptogroup and 4 passenger grabbed at Bristol coach station. Let's make a more ambitious assumption that the cryptographers always visit http://bristolcrypto.blogspot.co.uk/ first when they open a web browser, while the others each visits a different portal.

Now let's evaluate the uncertainty of their answers: apparently, the answers from cryptographers are quite certain (low uncertainty) whilst it can hardly be guessed if an answer is from a passenger (high uncertainty). In other words, we  say the answers in the cryptographers group has a low entropy but  that in the passengers group has a high entropy.

So one of the most remarkable inventions of Shannon's is his definition of entropy(Shannon's Entropy):

\begin{equation}

H = -{\sum}_{i}{p_i \log_b{p_i}}

\end{equation}

where $p_i$ is the probability of a message (an answer in previous example) appears. In computer science, we usually use $b = 2$ (bits).

If we compute the entropy in our example, we will have:

\begin{eqnarray}

H_{cryptographer} = -{\sum}_{1}^{4}{1 \log_2{1}} = 0\\

H_{passenger} = -{\sum}_{1}^{4}{{1 \over 4} \log_2{1 \over 4}} = 2

\end{eqnarray}

So the passengers' answer do have a higher entropy than the cryptographers'!

Information

Formally, the definition of Shannon's information is given in [2] as:

    " Information is a measure of one's freedom of choice when one selects a message."

To explain this, let's make a minor modification to our previous example. Let's grab another 4 passengers from Bristol train station and assume their answers are also random portals just like the passengers' in coach station.

Here is the question: Given an answer $y$, can you tell which group the answer is from?

We can instantly tell that the answer is from our cryptographer group if $y$ is http://bristolcrypto.blogspot.co.uk/. However, we will be struggling if $y$ is a portal; therefore, we could say the message of http://bristolcrypto.blogspot.co.uk/ contains more information (less freedom) than a portal (more freedom).

So how does it relate to entropy?

Extending the definition of entropy, we define the Conditional Entropy as:

\begin{equation} H(Y|X) = \sum_{x \in X}{p(x)H(Y|X = x)} \end{equation}

which describes the entropy of $Y$ when given condition $X=x$. To make it more explicitly, since entropy is the uncertainty of a variable; hence the previous definition of conditional entropy is in fact the uncertainty of $Y$ when given the "clue"(condition) $X$.

Observation: consider two variables $X$ and $Y$. If $X$ contains only a minimal information of $Y$, then given an exact value of $X$ should not help us much on deducing the value of $Y$, that is, it does not obviously reduce the uncertainty of $Y$; on the other hand, if $X$ contains essential information of $Y$, then the entropy of $Y$ is expected to be low when $X$ is given. Therefore, the conditional entropy can be viewed as a rational measurement to the information $X$ has of $Y$!

Another important measurement is called Mutual Information. It is a measure of mutual dependency among two variables. One way to define it is the loss of entropy(uncertainty) when given a condition:

\begin{equation}  I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) \end{equation}

Cryptographic Example

The concepts of information theory is widely used in cryptography. A classic example is to view a cryptographic process as a channel with plaintext and ciphertext being input and output respectively. Research of side channel analysis also benefits from the usage of information theory.

References

[1] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory
    2nd Edition. Wiley-Interscience, 2 edition, July 2006.


[2] S. Vajda, Claude E. Shannon, and Warren Weaver. The mathematical
    theory of communication. The Mathematical Gazette, 34(310):312+,
    December 1950.

[3] http://en.wikipedia.org/wiki/Entropy_%28information_theory%29

52 Things: Number 8: How does interaction help in computation, and what is the class IP?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know To Do Cryptography': a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. This blog post concentrates on the power of interaction in computation and the Complexity Class IP.

To answer the questions, we first give give a brief introduction of the interactive proof systems. As we know, zero-knowledge proofs currently play an important role in the study of cryptographic protocols. This concept was introduced by Goldwasser, Micali and Rackoff in [1]. Such proofs have the fascinating property of revealing nothing other than the validity of assertions being proven. In order to achieve this, Goldwasser, Micali and Rackoff developed the interactive proof systems by adding two ingredients to the classical proof systems. The first ingredient is randomisation, the verification of proofs can be probabilistic and the verifier is allowed to err with small probability. The second one is interaction, the static proof is replaced by dynamic prover who will interact with the verifier to convince it the assertion is true. Combining the classical proof systems with the two ingredients has a huge effect: the set of languages of interactive proof systems is in a large complexity class IP. 

What is a proof?

Loosely speaking, a proof is a way for a party to convince another party that a certain assertion is true. The two parties involved in a proof system are called a prover and a verifier.

Classical proofs

A classical mathematical proof is a fixed sequence of statements, which can be all written down by the prover then the verifier can easily check the validity of the assertion. There is no interaction between the prover and the verifier. 

Any proof system should have the following properties: 

1. Efficiency: the verification procedure can be carried out efficiently. 
2. Soundness: for any false assertion, invalid proofs are hard to find. 
3. Completeness: for any true assertion, there exists a proof. 

Recall the complexity class NP can be viewed as a class of languages whose members all have certificates that can be easily checked (the non-members do not have certificates). Hence we have NP is exactly the class of languages of classical proofs. 

Interactive Proofs

In an interactive proof system, the prover and verifier are allowed to interact with each other by exchange of messages. Before introducing the concept of interactive proofs, we first give an example (which can be found in [2]) to show how does interaction help in computation. 

Example: Graph Isomorphism and Graph Non-isomorphism.

Two graphs G and H are called isomorphic if the nodes of G can be reordered so that it is identical to H. We define the language:

ISO = {<G,H>|G and H are isomorphic graphs}

It is clear that ISO is in NP. Even though the number of nodes can be very large, the membership of ISO can be easily verified by given the instructions of reordering.

Then we consider the complement problem of Graph Isomorphism, namely Graph Non-isomorphism. We define the language:

NOISO = {<G,H>|G and H are not isomorphic graphs}

And the question is, using a classic proof, how can a prover convince a verifier that two graphs are not isomorphic? We don't know how to provide short proofs that the graphs are not isomorphic and the verifier can't check every possibilities in polynomial time. Thus, we don't know how to prove that NOISO is in NP. Nevertheless, consider the following interactive protocol, the prover can convince the verifier the fact that the given two graphs are not isomorphic. 

Both the prover and the verifier have a pair of graphs $(G_1, G_2)$ as common input. The verifier randomly choose a random bit $b \in \{0,1\}$ and a permutation $\pi$. Then it applies $\pi$ on $G_b$ to get a graph $H$. The verifier sends $H$ to the prover. Next, upon received $H$, the prover sends $b' \in \{0,1\}$ to the verifier. Finally, the verifier accepts if and only if $b'=b$. 

The idea behind the protocol is, if the given graphs $(G_1,G_2)$ are not isomorphic, then the prover should be able to identify $H$ is from either $G_1$ or $G_2$. However, if the input graphs are isomorphic, even with unlimited computational power, the best choice of the prover is to make $b'$ a random guess. In this case, the prover accepts at most $\frac{1}{2}$.

From the above example, we conclude that NOISO cannot be proved to the verifier via a classic proof, whereas it could be proved via an interactive proof(protocol). We can see there is some power in interaction. 

Now we give the formal definition of interactive proofs and the complexity class IP. 

Interactive proof system: A pair of interactive machines $(P,V)$ is called an interactive proof system for a language L if machine V is polynomial-time and the following conditions hold:

• Completeness: For ever $x \in L$,

$Pr[<P,V>(x)=1] \geq \frac{2}{3}$

• Soundness: For every $x \notin L$

$Pr[<P,V>(x)=1] \leq \frac{1}{3}$

The class IP: The class IP consists of all languages having interactive proof systems. 

By the definition, it is obvious that any language in BPP is in IP. And if we restrict the exchange of messages between the machines to be $1$, then we have NP is in IP. Actually, IP is a surprisingly large class. In 1992, Shamir revealed that PSPACE=IP [3]. 

In addition, notice that in the protocol, the prover has the ability to toss a private-coin. If the prover is allowed to access to the verifier's random string leads to the model of interactive proofs with public-coins, which is related to a similar complexity class AM [4]. 

[1] http://dl.acm.org/citation.cfm?id=63434

[2] http://www.amazon.co.uk/Introduction-Theory-Computation-Michael-Sipser/dp/0619217642

[3] http://dl.acm.org/citation.cfm?doid=146585.146609

[4] http://en.wikipedia.org/wiki/Arthur%E2%80%93Merlin_protocol

Post-Quantum Signatures

This week's Study Group was led by Peter Scholl who spoke about hash-based signatures and in particular a recent paper by Bernstein, Hopwood, Hulsing, Lange, Niederhagen, Papchristodoulou, Schwabe and Wilcox O'Hearn, called SPHINCS: practical stateless hash-based signatures [1].

Most real-world signature schemes like RSA and DSA can be broken by a quantum computer, due to Shor's algorithm [2]. One could use lattice-based signatures but their security is not well understood: the authors of [1] note that "their quantitative security levels are highly unclear" even in the pre-quantum setting.

On the other hand, hash-based signature schemes using 'quantum-resistant' one-way hash functions are secure in the quantum setting [3], although the symmetric security parameter needs to be doubled for quantum resistance due to Grover's algorithm [4]. The trouble is that such schemes tend to be inefficient or stateful, where the latter means one cannot have a signing key shared across multiple devices and the scheme may be vulnerable to 'restart attacks' where the scheme is forced to re-use a secret key, compromising security. The paper of Peter's talk seeks to address this by constructing a (relatively) efficient, stateless, hash-based signature scheme called SPHINCS. The construction is proved secure “based on weak standard-model assumptions, avoiding collision resistance and the random-oracle model”.

One Time Signature Schemes

First, we revisited the Lamport One Time Signature (OTS) scheme. (This isn't the OTS used in the paper but it will serve as a reasonable approximation.) Here your secret key is a sequence of pairs of bitstrings $(x_0, y_0), ..., (x_n, y_n)$ and the public key is the sequence of pairs of hashes $(H(x_0), H(y_0)), ..., (H(x_n), H(y_n))$. To sign a message $m$ one takes each bit $m_i$ of $m$ and selects $x_i$ or $y_i$ according to whether $m_i$ is 0 or 1. To verify a signature one hashes each $x_i$ or $y_i$ in the signature and checks the output against the public key. This is called “One Time” since the secret key can only sign one message before security is seriously compromised: your signature reveals half of the elements of the secret key.

From 'One Time' to 'Many Times'

A natural way to build “many time” signature schemes is to iterate OTS schemes. The trouble is that doing this in a naive way means the verification algorithm needs a huge number of public key components, enough for all the bits of all the messages you ever want to sign! Instead, we compress each OTS public key using a “Merkle tree” where the (original) OTS public keys are the leaf nodes and parents are constructed by hashing the concatenation of the children (together with a bitmask for added security). We want the (new) public key to be the root of this tree and so our signature must supply enough information (in as brief a way as possible) for the verifier to recover the root. This is how we do it:

• Given the path from a leaf to the root, let the sequence of siblings of each node on the path be called the authentication path of the leaf. 
• In a signature, supply the index of the leaf node used and its authentication path.
• With the leaf and its sibling, the verifier can construct the parent.
• Then with the sibling of the parent, the verifier can construct the grandparent.
• The verifier continues like this, using the siblings given in the authentication path to recover the root node i.e. the public key.  

So we now have a scheme with a practical-sized public key. We can also shrink the secret key by just storing a seed to a Pseudo-Random Generator which will then output the many OTS secret keys which, when hashed, give the leaves of the Merkle tree. However, we haven't “eliminated the state”, to use the authors' phrase: one must store a counter recording which OTS secret keys have been used so far. 

"Eliminating the State"

Goldreich's [5] answer to this problem was to deterministically choose which OTS key to use next, rather than just use them in order, i.e. some hash of the message determines the index of the OTS secret key to use in signing. Of course, now one needs a much bigger tree in order to avoid accidentally using a “one time” key more than once. In fact, for security, there needs to be exponentially many OTS key pairs, so we can't just have one leaf of the tree for each OTS secret key or the tree would be far too big for efficient signing. Instead, we associate every node with an OTS key pair and at each step from a leaf to the root, sign the hash of the public keys of the child nodes with the secret key of the parent node. The new (longer) signature contains the index of the leaf node used and the OTS signatures of all the nodes on the path to the root. The trouble with this scheme is that the length of the signature is cubic in the security parameter. This is where the authors of [1] come in.

From 'One Time' to 'A Few Times'

The main idea of the paper is to use a 'few times signature' scheme (instead of an OTS) at the bottom of the tree to reduce the number of leaves needed for security and hence the overall height of the tree, thus shortening the signature. Their choice of scheme for the bottom of the tree is called HORS: Hash to Obtain Random Subset. In HORS, the secret key is the tuple $(s_1, ..., s_t)$, the public key is the hashes $(H(s_1), ..., H(s_t))$ and a message $m$ determines a (“random”) subset $S$ of $\lbrace1, 2, ..., t\rbrace$ with fixed size $k$ (much smaller than $t$). Then the signature for $m$ is the set of secret key components corresponding to $S$, i.e. $\lbrace s_i | i \in S\rbrace$. Now we can use the secret key 'a few times' before security is compromised as only a small number of the components $s_i$ of the secret key are revealed in each signature. But again we have the problem that the public key needs to be very large and so a Merkle tree is once again employed: the new public key becomes the root node, recoverable from the index of a leaf node and the corresponding authentication path. Notice that this means the bottom of the tree in SPHINCS, the construction proposed in [1], is itself a tree. So SPHINCS consists of a hyper-tree (a tree of trees). 

The SPHINCS Tree of Trees

The SPHINCS hyper-tree has total height $h$ consisting of $d$ layers, each of height $h/d$. The index from the hash of the message determines a HORS tree on the bottom layer of the hyper-tree and a leaf of this HORS tree from which we compute the HORS signature of the message. Then this signature is signed according to the OTS scheme on the next layer (where the initial index in some way determines the tree and the leaf to use). We repeat this OTS signing on each layer and finally output the SPHINCS signature consisting of the index, the HORS signature (which contains all the information needed to recover the HORS public key) and each OTS signature and each authentication path needed to recover the root at each layer. 

Real World Considerations (in the Quantum Computing World!) 

After proving its security, the authors demonstrate the practicality of SPHINCS with certain choices of parameters: the hyper-tree has total height 60 consisting of 12 layers (each of height 5), the number of HORS secret key elements is $2^{16}$ and 32 of these are revealed in each HORS signature (i.e. $t=2^{16}$, $k = 32$). There is also a parameter that affects the OTS scheme but we haven't detailed it here. With these choices, the signatures have size 41KB, the keys have size around 1KB and one can sign hundreds of messages per second on a modern quad-core computer.

By most accounts, quantum computers are something of a pipe-dream at the moment. Nevertheless, it's reassuring to know that security is still achievable - and indeed practical - against adversaries who can exploit the enormous power of quantum computers, whenever that day comes.

[1] - http://eprint.iacr.org/2014/795.pdf

[2] - http://en.wikipedia.org/wiki/Shor%27s_algorithm

[3] - http://eprint.iacr.org/2014/709.pdf
[4] - http://en.wikipedia.org/wiki/Grover%27s_algorithm

[5] - Foundations of Cryptography: Volume 2, Basic Applications (2004)

52 Things: Number 7: How does randomness help in computation, and what is the class BPP?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know To Do Cryptography': a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. This blog post concentrates on the Complexity Class BPP.

So far, during this blog series Ryan has introduced us to complexity classes, and in particular to P:

• P  is the class of languages decidable in polynomial time by a deterministic Turing machine.

Then, Guy introduced us to complexity class NP: 

• NP is the class of languages decidable in polynomial time by a nondeterministic Turing machine.

This week I am going to introduce the complexity class BPP:

• BPP is the class of languages that are recognised by a probabilistic polynomial time Turing machine with an error probability of $\frac{1}{3}$.

Probabilistic Turing Machine

A probabilistic Turing Machine[1] is a type of nondeterministic Turing Machine which randomly chooses between the available transitions at each point according to a probability distribution. What this means is that a probabilistic Turing machine can have stochastic results. On the same input, it could have different run times, accept it on one occasion and reject it on another. It could also never stop. This Turing Machine gives rise to other complexity classes such as RP, ZPP and, the one we're discussing in this post, BPP.

A little about the complexity class BPP

So as we have seen from the definition, the class BPP (Bounded-Error probabilistic polynomial time) contains the decision problems that are solvable in polynomial time by a probabilistic Turing machine with error probability $\frac{1}{3}$. Note that this error probability can be chosen to be any value strictly between $0$ and $\frac{1}{2}$ due to a result named the amplification lemma which I will not discuss further here. The class BPP contains P, the class of problems solvable in polynomial time by a deterministic Turing Machine, this is due to the fact that a deterministic Turing Machine is a special case of the probabilistic Turing Machine (taking the only possible path with probability 1). As talked about in Guy's post, there is an open (Millennium) problem conjecturing as to whether P = NP. There is a similar question with BPP, being P = BPP?  The number of problems known to be in BPP but not known to be in P is decreasing.

An example of a BPP Problem

One of the most famous problems known to be in BPP  but not known to be in P was determining whether a number was prime. However, recently (2002) a deterministic polynomial time algorithm was found[2] for this problem meaning that it is indeed in P. Another problem known to be in BPP and currently still not known to be in P is polynomial identity testing[3], the problem of determining whether a polynomial is identically equal to the zero polynomial.

There are still many very important unanswered questions on the topic of Complexity Classes. Some of which, if answered, could have a large impact on shaping the future of Cryptography and Computer Science.

[1] - http://en.wikipedia.org/wiki/Probabilistic_Turing_machine

[2] - http://en.wikipedia.org/wiki/AKS_primality_test

[3] - http://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_lemma

Study group: Network Security Risk Assessment Using Bayesian Belief Networks.

This weeks study group was given by Shan on assessing network security risk using Bayesian Belief Networks, following the paper of Kondakci titled "Network Security Risk Assessment Using Bayesian Belief Networks".  The model developed can be applied to a variety of security evaluation tasks, risk assessment and other decision making systems.  The most important thing to understand on this topic is exactly how Bayesian Belief Networks (BBN's)[1] work and how they can be used to calculate security risks caused by various threat sources. To achieve this understanding, we need to talk a little about conditional probability[2]. We know that given an event $A$, the probability of an event $B$ occurring is
$$\mathrm{P}(B|A) = \frac{\mathrm{P}(B \cap A)}{\mathrm{P}(A)} $$
We can use this to consider an attack on two systems $A$ and $B$. Let us assume that the probability of an attack  $\mathrm{P}(N)$ on either system is 0.1 (meaning the probability of no attack  $\mathrm{P}(N')$ is 0.9). Let us also assume that the probability of system $A$ failing if there is an attack is 0.8 and if there isn't an attack 0.1. If there is an attack on system $B$ it fails with probability 0.6 and if no attack 0.5.
From these values we can very easily calculate the chances of the various systems failing. The probability of A failing  $\mathrm{P}(A)$ is calculated as follows
$$\mathrm{P}(A) = \mathrm{P}(A|N)\mathrm{P}(N) + \mathrm{P}(A|N')\mathrm{P}(N') = 0.1$$
Similarly we can show the probability of B failing as 0.53. Using Bayes Theorem[3] the probability that an attack has occurred given that system A has failed can be calculated as
$$\mathrm{P}(N|A) = \frac{ \mathrm{P}(A|N)\mathrm{P}(N) }{ \mathrm{P}(A)} = \frac{(0.8 * 0.1)}{0.17} = 0.47 $$
As you would expect, the observation that System A has failed has increased the probability that an attack has occurred. Let $M = N|A$ be the event of there being an attack given that $A$ has failed and $M' = N'|A$ the event of there not being an attack given that $A$ has failed. Using the law of total probability[4], the probability that system B has failed can be calculated as
$$\mathrm{P}(B) = \mathrm{P}(B|M)\mathrm{P}(M) + \mathrm{P}(B|M')\mathrm{P}(M') = 0.55 $$
Again, observing that System $A$ has failed increases the probability that System $B$ has also failed.

This simple example shows how Bayesian Belief Networks can be used to get a better understanding of the state of systems. As a real world example, consider mobile phone applications. If we know that an application has failed on a certain phone then we can calculate the failure risk on other phones. Obviously these models can easily increase in size and complexity, the hardest thing about this model is building the conditional probability table associated with the various systems involved, in our case, finding the values of $ \mathrm{P}(A|N)$ etc. These are built using historical data. Systems can be threatened by various types of threats, the first step is to create a directed acyclic graph (DAG)[5] showing all the relations involved in the system. This can then be converted into a BBN. This way we can combine certain threats into a joint impact parameter, for example we could have external threats as well as internal threats and within that we can have different types of internal threats for the system.  The model presented is a clever, generic way of computing the risk of the various classifications of threats of IT systems and computing environments using conditional probability through Bayesian Belief Networks.

[1] - http://en.wikipedia.org/wiki/Bayesian_network
[2] - http://en.wikipedia.org/wiki/Conditional_probability
[3] - http://en.wikipedia.org/wiki/Bayes'_theorem
[4] - http://en.wikipedia.org/wiki/Law_of_total_probability
[5] - http://en.wikipedia.org/wiki/Directed_acyclic_graph

52 Things: Number 6: How can we interpret NP as the set of theorems whose proofs can be checked in polynomial time?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know' to do Cryptography: a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. We continue the Complexity Theory section with an alternative definition of NP...

This question is very much a follow up question to the previous one. Last week Guy answered the question of ``What is meant by the complexity class $\mathbf{NP}$?'', while this week I will be answering the related question of ``How can we interpret $\mathbf{NP}$ as the set of theorems whose proofs can be checked in polynomial time?''.

Now to me this is a more intuitive definition of what it means for a problem to be in $\mathbf{NP}$. Not only is it a more intuitive definition but it should (hopefully) also be clearer as to why this is a complexity class that is useful both for cryptography and the wider world. Now before we go into what we can use the class of problems for, the definition is as follows:

$\mathbf{NP}$ is the class of languages that have polynomial time verifiers.

OK but what does this actually mean? Basically if we have an element $x$ and we want to know if $x\in L$ (where $L$ is some $\mathbf{NP}$ language) we have a prover $P$ which given $x$ outputs a witness $w$, this may take exponential time to find $w$ given $x$. Then if we give $x$ and $w$ to our verifier $V$, $V$ can tell if $x\in L$ in polynomial time.

This definition seems very different from the one given last week but they are in fact equivalent. Informally they are equivalent because the witness $w$ can just be the sequence of decisions the NDT made at each branching point to show that $x\in L$. For a (slightly) more formal proof of their equalence [1] is a reasonable online source (with references to textbooks).

So why might this be useful in cryptography? Well essentially we have a class of languages which can take exponential time to check if you do not know a witness but with a witness it can be done in polynomial time. This has the ``feel'' of a lot of cryptographic algorithms - take Encryption (formalisation to follow in future weeks' blogs) for example; you want it to be exponentially hard to get the message from ciphertext without the key (similar to a witness) but with the key you want it to be efficient (polynomial time) to extract the message.

A warning: While it sounds like a good move to use an $\mathbf{NP}$ problem for cryptography it may not be as simple as it first appears. This is because languages are in $\mathbf{NP}$ based on the worse case complexity where as for encryption we need it to be hard on average. For example we may have an $\mathbf{NP}$ language which has one element that takes exponential time to solve but all others are really efficient to solve - this will not make a good basis for an encryption scheme because we want encryption to be secure for all messages not just one.

Now I am aware that integer factorization isn't known to be $\mathbf{NP}$-complete and isn't known to be in $\mathbf{P}$ (See Ryan's blog here for a description) but it makes for a nice example of the point I am trying to make about choosing your $\mathbf{NP}$ instances carefully. In general finding a factor of a number is easy (half of them are divisible by two!) but if we choose something sensible we can make it hard to factor. Let us focus on numbers of the form $N=p\cdot q$ for $p,q$ prime (a.k.a numbers with only two proper factors). Now if either of these numbers is small then it is again easy, so we want the numbers to be of equal size (this is what we do for RSA). From this it is possible to build an encryption scheme over the top.

[1] - http://en.wikipedia.org/wiki/NP_(complexity)#Equivalence_of_definitions

52 Things: Number 5: What is meant by the complexity class NP?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know' to do Cryptography: a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. We continue the Complexity Theory section with NP...

Last week, Ryan introduced us to complexity classes, and in particular to P:

• P  is the class of languages decidable in polynomial time by a deterministic Turing machine.

This week, we introduce another complexity class: 

• NP is the class of languages decidable in polynomial time by a nondeterministic Turing machine.

What is a Nondeterministic Turing Machine (NDTM)?

Well, an NDTM is a Turing Machine for which the transition function can have multiple values for the same tape value/state pair (meaning it is not technically a function any more, so the correct thing would be to call it a transition relation). Thus evaluation of an NDTM on any particular input can be thought of as a tree, branching at each point the transition function provides more than one possible value. Now, an NDTM evaluates all of these possible branches at once, and accepts if at least one of these computations halts in the accept state. This definition generalizes from language membership to decision to computational problems in the same way as P did in last weeks blog.

Some Examples of NP problems

We begin with an easy example: path finding. Given a directed graph (on n vertices) is there a path from vertex A to vertex B. How do we know this is in NP? Well, there is an NDTM that solves it, which basically works by simply trying every route by branching each time there is a junction. If one of these branches (ie one of the trial routes) reaches B, then that branch terminates in the accept state. Any branch that does not reach B within n steps (ie after traversing n edges) terminates in the reject state. Since any path will involve at most n-1 edges, any valid route will be detected, and so this machine correctly decides whether the path exists.

One of the key examples of an NP problem is the satisfiability problem:

• SAT: Given an expression in n boolean variables, is there an assignment of the variables such that the expression evaluates to True?

For example, the expression $(A \land B) \lor (A \land \lnot B)$ is satisfiable, with one valid assignment being A=B=True. Note that in it's standard form, SAT is a decision problem: it suffices to decide whether such an assignment exists, we don't have to find it.

Ok, so there are problems in NP. What is interesting about it?

Firstly, we see that P⊆NP since a DTM is an NDTM that just happens not to ever branch (in fact, our first example can actually be solved by a DTM and so is within P). So, the real question is what sort of things can we do in NP that we couldn't do in P? Well, this is the root of the P$\overset{?}{=}$NP millenium problem, which is a major open problem. There are certainly problems that we have found that are known to be in NP that we do not know to be in P, but perhaps future research will show that these are also in P.

Lots of interesting cryptographic systems (particularly in the public key setting) are secure based on the assumption that a computational problem is "hard", which generally means "at least as hard as any problem in NP". That is, many schemes are based on problems which we think are difficult to solve, and that if you can create an algorithm that solves them you could also use this algorithm to solve lots of other problems that we currently believe to be difficult.

The Cook-Levin theorem provides an interesting result in this direction: No problem in NP is any harder than SAT. What his means is that if I had an oracle (basically an algorithm that I can see the input/output of but none of the workings) that solved SAT, by asking it to solve cleverly constructed queries I could use it to solve any other NP problem. This made SAT the first example of an NP-complete problem. So, to show that a problem X is at least as hard as solving an NP problem (it is NP-hard), it suffices to show that if I could solve X, I could use it to solve SAT.