# Tag Info

17

Unfortunately, this doesn't seem to work (see below for details), and it seems hard to find a way to make this kind of idea yield a provably secure scheme. The problem with your general idea You're not the first to think of the idea of trying to base cryptography on NP-complete problems. The problem is that NP-hardness only ensures worst-case hardness, ...

16

The key phrases you are probably looking for are "information-theoretic cryptography" and "quantum cryptography". Searching the literature on these topics will turn up lots of work of the sort you are looking for. Some example highlights below: For confidentiality: the one-time pad, the Wyner wiretap channel, secret sharing, quantum key exchange, etc. For ...

15

1) What is meant by necessary is that one way to generate a $k$-wise independent distribution is to break the input in blocks of $k+1$ bits, and let the $(k+1)$th bit of each block be the parity of the other $k$ bits in the block. Obviously this distribution can be broken just by computing parity on $k$ bits. The result you claim follows from the fact that ...

11

Two answers that I learnt while writing a blog post about this question No: In black-box variants, quantum query/communication complexity offer the Grover quadratic speedup, but not more than that. As Ron points out, this extends to computational complexity under plausible assumptions. Maybe: Nash equilibrium is arguably the flagship problem of "Christos ...

10

This is not an actual answer; I'm just sharing some results (which do not fit in one comment). Goldreich, Micali and Wigderson (J. ACM, 1991) proved that every language in NP has a zero-knowledge proof of language membership (assuming OWFs exist). To this end, their presented a ZK proof for graph 3-colorability. Later, Bellare and Goldreich (CRYPTO '92) ...

9

No, the intuitive observation "There are about $\sqrt{n}$ prime factors to try" does not imply a lower bound on the complexity of factoring. There is absolutely no reason that a factoring algorithm must try every possible prime factor, or even that the algorithm's behavior should resemble "trying" different factors at all. Even though precisely the same ...

9

SHA-1 was SHattered by Stevens et al. They demonstrated that collisions in SHA-1 are practical. They give the first instance of a collision for SHA-1. It is an identical-prefix collision attack that enabled the attacker to forge two distinct PDF documents that have the same SHA-1 hash value. I.e. They extended a given prefix $p$ with two distinct near-...

9

(I am interpreting this question in the following way due to the comments.) In implementations of cryptographic systems, hash functions like the SHA-256 are defined by a deterministic algorithm mapping arbitrary-length bit-strings to some fixed-length bit-strings. This situation is troublesome for formal analyses (reductions) of these systems because, when ...

9

I assume you want an efficient algorithm, e.g., one whose running time is polynomial in $n$. (If you don't care about running time, then heuristically about $4n/\lg m$ samples should suffice to uniquely determine $a,b,m,p$, and you can use exhaustive search over all possible $2^{4n}$ parameter choices to find the correct one. Of course, the running time of ...

9

The main thing missing from your list is the beautiful 2006 paper of Klivans and Sherstov. They show there that PAC-learning even depth-2 threshold circuits is as hard as solving the approximate shortest vector problem.

9

Boaz Barak addressed this in a blog post My takeaway from his post (roughly speaking) is that we only know how to design cryptographic primitives using computational problems that have some amount of structure, which we exploit. With no structure, we don't know what to do. With too much structure, the problem becomes efficiently computable (thus useless ...

9

There are essentially only two algorithms that I'm aware of: Use repeated-squaring, along the lines you mentioned. Factor $n$ using a state-of-the-art algorithm, then use the Chinese remainder thoerem. If $p$ is prime, you can compute $a^{b^c} \bmod p$ efficiently by computing $b^c \bmod p-1$ using fast exponentiation, call the result $d$, then computing $... 9 I will attempt to elaborate a bit on why CHKPRR shows that$\mathsf{PPAD}$is plausibly hard for quantum computers. At a high level, CHKPRR builds a distribution over end-of-line instances where finding a solution requires to either: break the soundness of the proof system obtained by applying the Fiat-Shamir heuristic to the famous sumcheck protocol, or ... 8 (wow! after three years of time passing, this is now easy to answer. funny how that goes! --Daniel) This "Learning with (Signed) Errors" (LWSE) problem, as invented-and-stated above by me (three years ago), trivially reduces from the Extended Learning with Errors (eLWE) problem first introduced in the work Bi-Deniable Public-Key Encryption by O'Neill, ... 8 Polylog independence may not be the only way to fool$AC^{0}$circuits. To illustrate this example, consider the class of linear polynomials. Any zero set of a linear polynomial is$(n-1)$-wise independent but of course this doesn't fool linear polynomials. Hence,$(n-1)$-wise independent distributions do not fool this class. This of course doesn't mean that ... 8 Here is a "canned" answer that might be useful, but has no cryptographic depth (hopefully we get answers with depth as well). What makes for a good candidate OWF? The naive answer tends to boil down to "something that looks hard to invert to me", but the expert's response is usually more like "something that many smart people have tried to invert but failed"... 8 (Note: Andreas Björklund suggested a solution in the comments that I believe is better than the one described below. See http://eprint.iacr.org/2017/203, by Ball, Rosen, Sabin, and Vasudevan. In short, they give proofs of work based on problems like Orthogonal Vectors whose hardness is well understood and to which many problems (e.g., k-SAT) can be reduced ... 8 The application you mention is called "proof of useful work" in the literature, see for instance this article. You can use a fully homomorphic encryption scheme (where the plaintext is the CNF instance) to delegate the computation to an untrusted party without disclosing the input. This doesn't answer exactly your question, since such scheme doesn't map a ... 7 If there is an Arthur-Merlin protocol for knottedness similar to the [GMW85] and [GS86] Arthur-Merlin protocols for Graph Non Isomorphism, then I believe such a cryptocurrency proof-of-work could be designed, wherein each proof-of-work shows that two knots are not likely to be equivalent/isotopic. In more detail, as is well known in the Graph Non ... 7 The answer to your question is the same as with many other such assumptions in cryptography: despite a lot of effort no one has found any substantially faster quantum algorithms for lattice problems. Why do we assume that RSA is secure? We don't have any particular justification for its classical hardness other than the fact that no one has found any fast ... 7 One can encrypt an n-qubit state using a 2n-bit classical secret key. The idea is to use the key to select a random Pauli operator, and apply that operator to the secret as an encryption. (The inverse operator is applied to decrypt.) The resulting scheme is perfectly secure -- if the key is selected uniformly at random, then even an attacker who know a ... 6 Secure cryptographic protocols that can be executed by humans are considered in this paper: http://link.springer.com/chapter/10.1007%2F3-540-45682-1_4 (Hopper and Blum, "Secure Human Identification Protocols"). Reading the abstract, it sounds like they don't fully solve the problem. I suspect a satisfying solution is not known, but if you want to look into ... 6 In a CPA game, a key pair$(sk,pk)$is generated, and$pk$is given to the adversary. The adversary outputs a pair of messages$(m_0,m_1)$in the message space, such that$|m_0|=|m_1|$. A random bit$b \in \{0,1\}$is then generated, and$m_b$is encrypted and returned as the cipher$c = E_{pk}(m_b ; r)$, where$r$is the randomness used for encryption. The ... 6 All of what I am going to say is well-known (all the links are to Wikipedia), but here it goes: The approach used in RSA using pairs of primes can also be applied in a more general framework of cyclic groups, notably the Diffie-Helmann protocol that generalizes$\left(\mathbb{Z}/pq\mathbb{Z}\right)^{\times}$to an arbitrary group, notably elliptic curves ... 6 Yes, if the encryption algorithm achieves IND-CPA security (semantic security), this implies that an adversary cannot predict any linear combination of encrypted bits better than random guessing. The easiest way to see this is to note that IND-CPA (left-or-right indistinguishability) implies real-or-random indistinguishability under chosen-plaintext attack:... 6 Yes, you can use Levin universal search to construct a "universal one-way function" (e.g., these lecture notes). From this one-way function you can then construct symmetric-key encryption primitives (pseudorandom generators, block ciphers, CPA/CCA-secure encryption) using standard theoretical constructions. One-way function$\to$pseudorandom generator: ... 6 To expand somewhat on Sasho Nikolov's comment... LWE is at least as hard as finding approximate solutions to SVP, but the approximation factors for which the reduction from SVP to LWE works are larger than the approximation factors for which we know NP-hardness ...the following is a reasonably complete view of things: This was part of Oded Regev's talk ... 6 Here is the problem: if$M$has low entropy (for example, if the attacker has side information that narrows$M$down to just two possible messages), then conditioned on$M+K$, the key$K$also has low entropy (there are only two possibilities for$K$). If the eavesdropper stores the first message (which was an encryption of$K\$), then she can use it to ...

6

Feigenbaum in, Encrypting Problem Instances, proposes a definition (Def. 1) of encryption function for NP-complete problems which satisfies your requirements. She proves that the NP-complete problem Comparative Vector Inequalities admits such encryption function. She concludes with the main theorem that all NP-complete problems that are p-isomorphic to CNF-...

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