Questions tagged [complexity]

Complexity describes - in simple words - how hard (complex) it is to reach a specific goal; and under which conditions. In cryptography, this mostly ends up in using the complexity theory to analyze things. One of the main goals of complexity theory is to prove lower bounds on the resources (e.g. time and/or space) needed to solve a certain computational problem. Cryptography can therefore be seen as the complexity theory's main field of use.

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Computational Complexity: ECC multiplication vs Modular multiplication

How does performing scalar multiplication on an elliptic curve compare to exponentiation in a multiplicative group modulo a prime? I.e. on a given elliptic curve of size $|t|$, what's the complexity ...
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Group Rings on Cryptography

Let $R[G]$ or $RG$ be the group ring where $R=F_q$ and $G$ is any group. Let $Dim(V)=\vert G \vert$. It's clear that $V$ has $\vert R \vert^{\vert G \vert}$ distinct $\vert G \vert$-tuples. This ...
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Induction is problematic in computational cryptography - Why?

In Dr. Lindell's lecture The Yao Construction and its Proof Of Security, in briefly explaining the hybrid argument, he makes the statement that mathematical induction is a problem in computational ...
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Multivariate Cryptography: Security of the affine transform T

In this question, I'd like to discuss the security of the last transformation $T$ employed in the construction of a MV-scheme. MVCrypto is based on solving a system of polynomial equations, but ...
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Bit-strength of discrete logarithm for a group of integers modulo a safe prime

Preliminaries Let $p$ be a safe prime number. Let $\mathbb{Z}_p^*$ be the multiplicative group of integers modulo $p$. We have $\mathbb{Z}_p = \{\,a \in \mathbb{Z} \mid 1 \le a \lt p\,\}$ . Let $g \...
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What makes the quadratic residuosity problem hard?

The quadratic residuosity problem is the problem of determining whether, for given $r$, $m$, $\exists a.a^2\equiv r\mod m$. This problem's believed to be hard to solve in general (e.g. an efficient ...
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Time complexity of Euler's totient function

I believe there are different time complexities for Euler's totient function depending on how you execute the algorithm. The two I know of are: Iterate through 1 to k and calculate each $\gcd$: $O(n \...
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Why does the following SIS-based decision language not make sense?

I'm currently reading about important lattices problems and noticed that while CVP, SVP, and LWE have decisional versions, SIS does not. I read in the question Relation between decisional SIS and ...
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Speed of the General number field sieve

So according to the wikipedia page https://en.wikipedia.org/wiki/General_number_field_sieve the algorithm has complexity $$\exp \left( \left(\sqrt[\leftroot{1}\uproot{0}3]{\frac{64}{9}} + o(1) \right) ...
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Attack on a Feistel Cipher given the key and half of the ciphertext

Consider a classical Feistel Cipher, with the round functions given and the keys used in the ciphering process. Is it possible to reconstruct the original data if half of the ciphered text is given? ...
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If DDH is hard then CCA-secure PKES exist?

My cryptography slides describe several relations between cryptographic problems. I don't still have a good justification on the following: If decisional Diffie-Hellman problem is hard then there ...
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Big-O Encryption Algorithm

I am currently doing a research paper on the Blowfish encryption algorithm and one of the components that I need to include is time and space complexity. I have tried reading academic articles and ...
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What is the time complexity of computing a cryptographic hash function/random oracle?

I'm wondering what's the computational complexity of computing a hash function/random oracle when doing complexity analysis. For example, what's the computational complexity of computing $H(b\|r)$? ...
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What is meaning of the term “language”?

I don't have much formal background, and I could not find a suitable explanation for this after searching on Google/Wikipedia. What is the meaning of the term "language" as used in cryptographic ...
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AES-128 (CBC) brute force given 90+ rightmost bits of key, known IV and Ciphertext?

Given: Known ciphertext (in hex) (ciphertext is the exact length of the message (i.e. non-padded). It is known that the cipher was developed using CBC. There is one and only one ciphertext message ...
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Complexity of arithmetic in (the integer ring of) a number field?

What is the running time complexity (average or worst case) of common arithmetic operations in number fields? In fact, I'm only interested in the integer ring of the quadratic extension $\mathbb{Q}[\...
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If finding a private key from a public key is [co-]NP-complete, does NP=co-NP?

In a given public key cryptosystem, if the problem of determining the private key from the public key is NP-complete or co-NP-complete, does that imply that NP = co-NP? Complexity theory is ...
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Why do look up tables speed things up compared to brute force?

I'm currently reading up on lookup tables and efficiency. In my uni script it says the following: For Brute Force: Preparation time: $O(1)$ Disk space requirement: $O(1)$ Time required to crack the ...
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Determining whether functions are one-way or not

I read a book about one-way function. In this book, I saw 2 exercises that I can't understand and don't know how to solve. Can anyone help me to solve these? Is $f(x,y)=x+y$ a one way function? Is $f(...
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What does universal look like? What is the complexity(size/depth) of universal circuit

I was reading a paper about attribute based encryption. The authors showed that ciphertext-policy ABE can be constructed from key-policy ABE using universal circuit. But the universal circuit should ...
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Can you explain what an NP statement is when they refer to it in Zero knowledge proofs?

When I read about zero knowledge proof, I keep encountering the term NP-statement. I am aware of complexity classes but I am a little unclear on how it ties up to NP-statement. I came across the ...
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Algorithm complexity: $\mathcal O(n\cdot m)$ vs. $\mathcal O(max(n,m)^2)$

Suppose $A(n,m,k)$ computes for 1 < i < n do { for 1 < j < m do { /* some efficient cryptographic operation */ } } where $k$ is a ...
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What happens for factoring algorithms if P=NP?

If someone ever demonstrates that P=NP, will it give us a polynomial factoring algorithm, or will it only tell us that such an algorithm exists, but we still have to find it?
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What is the computational complexity of Coppersmith's bivariate algorithm?

Coppersmith's original paper Finding a Small Root of a Bivariate Integer Equation; Factoring with High Bits Known says the algorithm to find bivariate roots under certain conditions runs in ...
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Reduction from integer factoring to computational Diffie Hellman

The computational Diffie Hellman (CDH) problem for ${\mathbb{Z}}^*_p$ is given a prime $p$, a generator $g$ of ${\mathbb{Z}}^*_p$, and a pair $(g^i, g^j)$ to compute $g^{ij}$. The value $g$ is called ...
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What is the time complexity of the basic components of a symmetric cipher?

I have a very basic knowledge on time complexity and even less on programming, so please bear with me. I am interested to know the time complexity in big-O notation of some of the basic operations in ...
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How reassuring is 64-bit (in)security?

In Feb 2017, CWI and Google announced SHAttered hash collision attack on SHA1, which took $2^{63.1}$ work estimated 6500 CPU years, to achieve. Therefore, 64-bit should be considered now an insecurity....
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Newbie question: one-way functions in cryptography

I'm reading this article on the basics of cryptography and it says that the main principle is about taking such an algorithm that knowing the end result and the algorithm, an eavesdropper wouldn't be ...
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Double encryption : what method is the best?

Given the block-ciphers defined as follows : $1.$ $C=$ $E_{k1}($$E_{k2}$$(M))$ $2.$ $C=$ $E_{k1{\oplus}k2}$$(M)$ $3.$ $C=$ $(E_{k1}(r),E_{k2}(r$ ${\oplus}$ $M))$ where $C$ is the cipher ...
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Negative time complexity?

Just finishing an investigation into Shor's Algorithm, and the following equation, $$ O\big(\big(\log N\big)^2 \big(\log \log N\big)\big(\log \log \log N\big)\big) $$ Is given for its time complexity. ...
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Secure offline treasure hunt

Say I would like to create an app that would allow users to organize treasure hunts for their parties. The host of the party would create a list of GPS coordinates for each hiding spot and distribute ...
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If trapdoor OWF exists then f is a trapdoor OWF, is there such a construction?

Is there a known construction of f, such that given that a trapdoor OWF exists then f is a trapdoor OWF, so we can construct inefficient cryptomania, ala Levin's construction for minicrypt in "The ...
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Why do we focus on polynomial time, rather than other kinds of time?

Polynomial time seems to be mentioned quite frequently on this site. It often forms a threshold between two possible outcomes like being secure or an attack's validity. I know what $\mathcal{O}(n^c)$ ...
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Computation complexity of modified Euclidean algorithm

The computational complexity of the extended Euclidean algorithm is $O(log(b)^2)$ ($b$ being the second integer) as referenced by Wikipedia. How to compute the complexity of the modified extended ...
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Are there lower bounds to how efficient one can make obfuscated code?

I am wondering if there are any theoretical reasons why obfuscated programs cannot be nearly as efficient as the plaintext programs and whether there is any necessary computational overhead from ...
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Time complexity of Weil pairing

I ran and timed an implementation of the Weil pairing on three set of parameters. One with an order of 512 bits, one with 256 bits and the last with 161 bits. I took the Miller's algorithm to compute ...
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Why do memory-hard functions rely on a time-space trade-off?

I was reading about memory-hard functions recently. In those papers I read, they almost always introduce a time-space trade-off like this: $$ S(n) \times T(n) \in \Omega(\mathrm{Poly}(n)) $$ I ...
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Space complexity and cryptography

It appears that in cryptography a lot of definitions are based on the time complexity of various algorithms. For example, a "good" encryption scheme should be resilient against a polynomial adversary. ...
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Encryption scheme with high complexity encryption over decryption

I need an encryption operation be 1000-1000000 times more complex than decryption operation. Is it possible to achieve with EC ...
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Finding k collisions on hash function

Let $n$ be the size of the image-space of a hash function $H$. It is known that you can find a collision on $H$ in $O(\sqrt{n})$ time (by birthday paradox). How can I show that, in order to find $k$ ...
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Computational complexity of RSA

What's the computational complexity or RSA? One can assume it's O(prime length^2) it you consider multiplication by column, but speed tests slightly differ on slow operations with private key and ...
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How was this Mersenne Twister seed for a 20-character string known a priori found?

Someone generated a seed for the Mersenne Twister, with the intent of that seed producing this string: "9!dlroW ,olleH"ck,@ Which is 20 characters long. Why he ...
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Analysing communication complexity of a protocol

My question is more about the practical side of designing crypto. protocols and it is related to complexity. So, if you think it's irrelevant to this forum please kindly let me know and I will remove ...
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How many reversible gates would DES require?

How many reversible gates (said counting Toffoli and Controlled NOT, with free NOT) would be required to reversibly implement $(K,P)\mapsto(G,C)$ for the block cipher DES? $P$ is the plaintext, $C$ ...
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Making attacks on password hashes less economical

Perhaps an abstract question on complexity given the trade offs between memory vs runtime, I was wondering if it's possible to constrain only either extremes approaches to be optimally efficient, thus ...
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Hash function composition - security level

When using two hash functions, g(x)=SHA-512 and f(x)=MD5 g(x) has 512 bit output (using salt) f(x) has 128 bit output. Let's say that z(x)=f(g(x)) meaning the output is 128 bit long. The Question: ...
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Complexity lower bound in Uber-Assumption family

I try to wrap my head around calculating of complexity lower bound using Corollary 1 from The Uber-Assumption Family by X.Boyen (http://www.academia.edu/download/30698012/...
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Complexity leveraging in case of exponentially many hybrids

Complexity leveraging is a proof technique in cryptography where the reduction algorithm runs in super-poly time. (see this). Many papers use complexity leveraging when there are exponentially many ...
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Random self-reducibility and NP

I was reading the Wikipedia page https://en.m.wikipedia.org/wiki/Random_self-reducibility and it states: "If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy ...
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What is the difference between Argon2d and Argon2i?

I know that Argon2d accesses the memory array in a password dependent order and Argon2i accesses the memory array in a password independent order. What is the difference in computational complexity?