# Np completeness and cryptography software

To show the nphardness of sat is some work but it was done in 1971 by stephen cook. Worstcase hardness of npcomplete problems is not sufficient for cryptography. It asks whether every problem whose solution can be quickly verified can also be solved quickly. More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly in polynomial time, such that the output for any input is yes if the solution set is nonempty and no if it is empty. Basically it all boils down to the mismatch between averagecase hardness required for cryptography, and worstcase hardness required for np hardness. Proof that pnp would lead to the fields inevitable demise, since it would eventually make computing someones private key, given their public one, doable in a reasonable amount of time. These algorithms will becomes useless if a polynomial algorithm is found. Automata, computability, and complexity electrical.

Some cryptosystems based on nphard problems have been proposed such as the merklehellman cryptosystem based on the subsetsum problem, and the naccachestern knapsack cryptosystem based on the knapsack problem, but they have all been broken. Most of the problems that well see that are np hard are also np complete. Constructing these objects relies on the theory of \\mathbf np \ completeness. Npcompleteness computational science stack exchange. The latter problem is a polynomial programming problem, which is known to be nphard, since this program class contains quadratic programming, which is also nphard. Np completeness has had tremendous impact even in areas where, in some sense, it should not have. They are symmetric they are asymmetric they are completely unbreakable it is infeasible to break them in a finite amount of time, given todays techniques and computing resources. Incidentally, the biggest problem is in saying but i have some idea that the people who do know about factoring, similarly, have pretty good reasons to feel authoritative about their own big assumptions. So np completeness can be thought of as a way of making the big pnp question equivalent to smaller questions about the hardness of individual problems. A problem is nphard if it follows property 2 mentioned above, doesnt need to follow property 1. It is now common knowledge among computer sci entists that npcompleteness is largely irrelevant to publickey cryptography, since in that area one needs sophisticated cryptographic assumptions that go.

Trying to understand p vs np vs np complete vs np hard. A variant of the 3satisfiability problem is the oneinthree 3sat also known variously as 1in3sat and exactly1 3sat. Postquantum crypto is somewhat relevant, although pnp is more devastating than practical quantum computers. The focus of this book is the pversus np question and the theory of np completeness. Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has exactly one true literal and thus exactly two false literals. Important complexity classes will be defined, and the notion of completeness established through a thorough study of np completeness. The course will explain measures of the complexity of problems and of algorithms, based on time and space used on abstract models. Pnp is essentially the question of whether we can find solutions quickly if we can define or know there is a solution quickly in laymans terms, it means we know. At this juncture, it is interesting to compare proofs in cryptography to proofs in complexity theory. Let us recall how a proof that a language l is np complete. Most of todays encryption, such as the rsa, relies on the integer factorization, which is not believed to be a np hard problem, but it belongs to bqp, which makes it vulnerable to quantum computers.

Department of software systems consequences if p np a proof that p np could have stunning practical consequences. In computational complexity theory, a problem is npcomplete when it can be solved by a. Even if npcomplete problems are hard in the worstcase p. Most of the problems that well see that are nphard are also npcomplete. If any npcomplete problem is in p, then it would follow that p np. So when we prove this, we prove that there is basically no polynomial time algorithm for that problem. Np may be equivalently defined as the set of decision problems that can be solved in polynomial time on a nondeterministic turing machine. In fact, proving the secureness of an encryption scheme more specifically, proving the existence of one. This means that a proof for pnp and algorithms that use it will appear at the same time.

Sims 255 foundations of software design complexity and np. You cant prove a cryptographic algorithm unbreakable, but you can prove the probability of it being broken is so small to be imp. The computational complexity of simultaneous diophantine. For the case of rsa, security of the system is based on hardness of the integer factorization problem. For instance lots of encryption schemes and algorithms in cryptography are based on number factorization, the best known algorithm having exponential complexity. Its hard to say too much more without knowing the specifics of. So, cook did this using the turing machine concept. After watching a harvard lecture regarding the understanding of p, np, and np complete,they also talk about our encryption algorithms being cracked or useless once we solve the mathematics side of. Npcomplete problems exist and their existence is one of the greatest.

Why hasnt there been an encryption algorithm that is based on the. It also provides adequate preliminaries regarding computational problems and computational models. This course provides a challenging introduction to some of the central ideas of theoretical computer science. You could use the turtle block cipher with cmac, maybe. The turtle block cipher claims a cipher design that is based on an npcomplete subproblem. The pversus np question asks whether or not finding solutions is harder than checking the correctness of solutions. Let us recall how a proof that a language l is npcomplete. By contrast, np completeness does not seem to be useful for designing cryptosystems. What i intended to ask is for a public key encryption algorithm with the.

Generally speaking, p problems are ones that can be solved rather. If the shortest program that can solve subsetsum in polynomial time is b. Are there public key cryptography algorithms that are. As pointed out by many other posters, it is possible to base cryptography on np hard or np complete problems. These npcomplete problems really come up all the time. Once that one npcomplete language was known, it was relatively simple to show the npcompleteness of other languages via reduction.

But practically, we might still be able to construct useful cryptosystems. The bottom line is on the nature of the problems known to be npcomplete. A language b is np complete if it satisfies two conditions. A r is npcomplete b r is nphard c q is npcomplete d q is nphard answer. A np complete problem has to be in both np and nphard. Thus this theory that originally was designed to give a negative result show that some problems are hard ended up yielding positive applications, enabling us to achieve tasks that were not possible otherwise. The turtle block cipher claims a cipher design that is based on an npcomplete subproblem block ciphers can be used to produce authentication tags via algorithms such as cmac im not sure if this combination necessarily qualifies the authentication as being based on an npcomplete subproblem too. Therefore, npcomplete set is also a subset of nphard set. Np and what this may mean to cryptography hannos blog.

The short answer is that a proof that pnp doesnt necessarily mean that all cryptography is insecure in practice it does mean we would need to reevaluate carefully the basis of security of our cryptographic schemes. I wonder, why has there not been an encryption algorithm which is based on an known nphard problem. And when you find one polynomial algorithm, you can use it to solve all other npcomplete problems by reducing the problems to a common form. Np is the class of decision problems for which it is easy to check the correctness of a claimed answer, with the aid of a little extra information. The pversusnp question asks whether or not finding solutions. B correct because a np complete problem s is polynomial time educable to r. Based on npcomplete is a bad way of expressing what im interested in. However, the common methods for cryptography are going to be based on difficult mathematics difficult to crack, that is. Lecture 16 of scott aaronsons great ideas in theoretical computer science says something about this, which i think you should take. For any problem y in np, there is a reduction from y to x. Cryptography public key, rsa, dh, digital signatures algorithms, secret sharing, coin flipping, passwords, practical and useful ones complexity of computation p, np, npcompleteness, randomized algorithms and complexity classes groups and finite fields example, important facts, with some proofs, polynomia rings. Most of todays encryption, such as the rsa, relies on the integer factorization, which is not believed to be a nphard problem, but it belongs to bqp, which makes it vulnerable to quantum computers. Obtaining a reduction as the other answer seems to do correctly is useful, but unnecessary, since.

A proof would involve finding a polynomial time algorithm for an npcomplete problem. The term np refers to a class of problems which could be solved by a nondeterministic computer in an amount of time which varies as some polynomial in the size of the input hence np, for nondeterministic polynomial time. Many modern systems rely on public key cryptography popular implementation is rsa used in all web browsers for secure connections. Are there public key cryptography algorithms that are provably np. Im not sure if this combination necessarily qualifies the authentication as being based on an npcomplete subproblem too. The turtle block cipher claims a cipher design that is based on an np complete subproblem. That would kill pretty much all of modern cryptography, including symmetrickey cryptography and publickey cryptography. It does depend on your definition of possible since security is inherently a matter of probability. If x is npcomplete and a deterministic, polynomialtime algorithm exists that can solve all instances of. To add to the other answers, some of the hard problems in crypto are randomly selfreducible within a given modulus, curve, or other security parameter. Complexity and cryptography an introduction provides a neat and easily readable introduction to cryptography from a complexity theoretical perspective. Hence, we arent asking for a way to find a solution, but only to verify that an alleged solution really is correct.

Beginning in antiquity, the course will progress through finite automata, circuits and decision trees, turing machines and computability, efficient algorithms and reducibility, the p versus np problem, npcompleteness, the power of randomness, cryptography and oneway functions. This article is intended to be an easytounderstand introduction to cryptography for students and people with a semitechnical. Achieved the haya freedman prize for best dissertation. Completeness theorems for noncryptographic faulttolerant. P and np are problem classes that say how hard it is to solve a problem.

P, np, and the search for the impossible on this topic, from a laymans view, then see below for comparative differences. The p versus np problem is a major unsolved problem in computer science. Np complete problems exist and their existence is one of the greatest. Relation of encryption to p, np, and npcomplete mathematics. In particular, for an undergraduate audience, the most familiar reductions proofs involve proving the npcompleteness of problems. Npcompleteness reductions are confusing at first, as they seem somewhat backwards. If a language satisfies the second property, but not necessarily the first one, the language b is known. A np complete problem has to be in both np and np hard. The latter problem is a polynomial programming problem, which is known to be np hard, since this program class contains quadratic programming, which is also np hard. The focus of this book is the pversusnp question and the theory of npcompleteness. A r is np complete b r is np hard c q is np complete d q is np hard answer. There are some complexitytheoretic reasons to believe that cryptography cant be based on np completeness. I wonder, why has there not been an encryption algorithm which is based on an known np hard problem. The asymptotic running time of an attack against a cryptosystem is.

In particular, for an undergraduate audience, the most familiar reductions proofs involve proving the np completeness of problems. Coupled with some good old fashioned ignorance of cryptography and software bugs. Np completeness reductions are confusing at first, as they seem somewhat backwards. A problem p in np is npcomplete if every other problem in np can be transformed or reduced into p in polynomial time. In cryptography, a boolean function is said to be complete if the value of each output bit depends on all input bits. Suppose we found a on4time algorithm for a npcomplete problem like sat, where the constant hidden by the bigo notation isnt too large.

What happens to public key cryptography if pnp is proven. This is a desirable property to have in an encryption cipher, so that if one bit of the input is changed, every bit of the output has an average of 50% probability of changing. The class np consists of those problems that are verifiable in polynomial time. So if we believe that p and np are unequal, and we prove that some problem is np complete, we should believe that it doesnt have a fast algorithm. Np problem gets solved, does modern cryptographic software become obsolete. Npcomplete authentication systems cryptography stack. Probabilistic algorithms with particular applications in cryptography np completeness.

Block ciphers can be used to produce authentication tags via algorithms such as cmac. You need funny board shapes for the problem to be npcomplete. What does np completeness imply for cryptographic systems. Examples include most problems related to discrete log. Probabilistic algorithms with particular applications in cryptography npcompleteness.

Impagliazzos five worlds paper discusses a bunch of things like pnp, the existence of oneway functions, etc. Suppose we found a on4time algorithm for a np complete problem like sat, where the constant hidden by the bigo notation isnt too large. If x is known to be np complete, then you can reduce another np problem y to x, and then y is np complete. Completeness theorems for noncryptographic faulttolerant distributed computation extended abstract michael benor shafi goldwassert hebrew university mit avi wigdemon hebrew university abstract every function of n inputs can be efficiently computed by a complete network of n processors in such a way that. The phenomenon of np completeness is important for both theoretical and practical reasons. If a string is not in l, then m may not halt think of an infinite loop on a program. A standard assumption in cryptography is the known plaintext attack.

Specializing in graph theory, cryptography, np completeness, randomized and approximation algorithms, combinatorial optimization. Chess playing programs work by traversing this tree finding what the possible. Oh, one more thing, it is believed that if anyone could ever solve an npcomplete problem in p time, then all npcomplete problems could also be solved that way by using the same method, and the whole class of npcomplete would cease to. Mathematical aspects of modern algebraic cryptography. Npcomplete problems are the hardest problems in np set. While rsa and other widelyused cryptographic algorithms are based on the difficulty of integer factorization which is not known to be npcomplete, there are some public key cryptography algorithms based on npcomplete problems too. If x is known to be npcomplete, then you can reduce another np problem y to x, and then y is npcomplete. It is now common knowledge among computer sci entists that np completeness is largely irrelevant to publickey cryptography, since in that area one needs sophisticated cryptographic assumptions that go.

In computational complexity theory, a problem is npcomplete when it can be solved by a restricted class of brute force search algorithms and it can be used to simulate any other problem with a similar algorithm. Npcompleteness is about the theoretical limits of computing. If language a is known to be nphard, then showing that a. If p np, it would make our current theoretical definitions of security obsolete. Im not sure if this combination necessarily qualifies the authentication as being based on an np complete subproblem too.

Npcomplete problems are in np, the set of all decision problems whose solutions can be verified in polynomial time. Computer science stack exchange is a question and answer site for students, researchers and practitioners of computer science. Npcompleteness refers to a very specific level of difficulty, and most of the problems used in encryption are not quite that difficult. Oh, one more thing, it is believed that if anyone could ever solve an np complete problem in p time, then all np complete problems could also be solved that way by using the same method, and the whole class of np complete would cease to exist. Because if we prove npcompletenessi mean, really we care about nphardness, but we might as well do npcompleteness. Decision vs optimization problems npcompleteness applies to the realm of decision problems. Martin antonov frontend game dev team lead pariplay. What does npcompleteness imply for cryptographic systems. Proof that pnp would lead to the fields inevitable demise, since it would eventually make computing someones private key, given their public one, doable in.

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