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. ...

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1answer
540 views

What does “running in polynomial time” really mean?

I'm currently learning private-key cryptography. I've been able to see that perfect secrecy is achievable if no assumption is made about the computational power of the attacker. However, perfect ...
6
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1answer
285 views

Formal definition of “explicit” algorithm?

A long time ago, I read that the definition of "cryptographic hash function" is "collision-resistant one-way function". (A similar definition shows up in the FIPS standards for SHA-1 etc.) But this ...
6
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1answer
238 views

Finding where I am in a linear recurrence relation

Suppose I have a linear recurrence relation $$a(n) = c_1 a(n-1) + \dots + c_k a(n-k) + d,$$ where the constants $c_1,\dots,c_k,d$ are given and the initial values $a(0),\dots,a(k-1)$ are given as ...
5
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3answers
986 views

Why is the complexity of RSA-1024 80 bit and not 86 bit?

Why is the complexity of RSA-1024 80 bit and not 86.76611925028119 bit? Here is the complexity for the GNFS (pulled from the linked Wikipedia article): $$\exp\left( ...
5
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4answers
1k views

What is the largest performed/possible bruteforce attack to date?

I've read that cracking 128-bit key is currently out of reach of all humanity. However, I can't seem to find any information on what scope of brute force attacks have been performed or are possible at ...
5
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2answers
269 views

Pen-and-paper one-way function for externally-anonymous survey

When conducting surveys, an Administrator might send an Enumerator to survey a Respondent. For "sensitive" questions (e.g. about embarrassing behavior), the Respondent may be fine with the truth being ...
4
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2answers
239 views

Can we say that if $P=NP$ there is no CPA secure public key encryption?

I've learned that public key encryption is based on the problem of Discrete Log (as regard to group theory) which believed to be hard. But, can we say that it doesn't matter on which problem our ...
4
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3answers
455 views

P = NP and current cryptographic systems

I've recently heard some people claiming that if the fact that P = NP is proven, most (all?) the current cryptographic algorithm considered secure like RSA will be unusable in secure systems. My ...
4
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3answers
354 views

Discrete log problem, when we have many examples

Suppose I have many instances of the discrete log problem, all using the same unknown exponent. Is this problem easier than the standard discrete log problem? Oh, heck, I should be more precise. ...
4
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1answer
178 views

Are post-quantum cryptographic ciphers *also* secure if the P=NP conjecture holds true?

First of all, I only understand the P versus NP debate on a rather shallow level as I am not a computer scientist. So perhaps the answer to the question is straightforward but if not, I would be ...
4
votes
1answer
283 views

What is meant by $\tilde\Omega(\lambda^4)$?

I'm currently reading the paper (Leveled) fully homomorphic encryption without bootstrapping , and the following paragraph was near the start: What is meant by the symbol used? Is it merely to ...
4
votes
1answer
195 views

Solve a Modular Exponentiation

It might be common, but if we had to solve an equation like this $m=s^{e}$ mod $n$ where $m,e,n$ are known. How can we find $s$. What optimisations could be applied? And what would the complexity of ...
4
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2answers
300 views

Is there any research about cryptography on nondeterministic Turing machines?

I know it's a highly theoretical topic, but I was wondering if there was any research out there about what cryptography would be like assuming that we had access to nondeterministic Turing machines. ...
4
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1answer
299 views

A discrete-log-like problem, with matrices: given $A^k x$, find $k$

Let $p$ be a large prime; we will work in $GF(p)$. Let $A$ be a $n\times n$ matrix. Also, let $x$ be a $n$-vector and $k$ a positive integer. Suppose we are given $p$, $A$, $x$, and $y$. The goal ...
4
votes
1answer
283 views

Complexity class of an idealised version of Bitcoin's proof-of-work (hashcash)? [closed]

To formulate this question precisely, I will define an idealized hypothetical "perfect" hash function $H(n)$ which has nice scalability properties, and will formulate a problem PERFECT HASHCASH in ...
3
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3answers
549 views

Complexity of arithmetic in a finite field?

I am wondering what the complexities are of adding/subtracting and muliplying/dividing numbers in a finite field $\mathbb{F}_q$. I need it to understand an article I am reading. Thank you
3
votes
2answers
142 views

iterated discrete log problem

Consider the following problem: given $g_1 \ldots g_i,h_1 \ldots h_i \in G$, $\forall i$ find $x_i$ such that $g_i^{x_i}=h_i$ For $i=1$ this is the discrete log problem and is assumed to to have ...
3
votes
1answer
69 views

Is xgcd faster than Fermat for calculating $d$ in RSA?

I want to calculate $d$ from $e$ when generating RSA keys. What is faster? Calculating $\operatorname{xgcd}(e,p)$ and $\operatorname{xgcd}(e,q)$ and CRT. Or calculating $e^{p-2}\bmod p$ and ...
3
votes
1answer
242 views

Fully Homomorphic Encryption over the Integers - Runtime Question

I have a question regarding the paper "Fully Homomorphic Encryption over the Integers" (http://eprint.iacr.org/2009/616.pdf): On page 6 after they set their parameters, it says "This setting results ...
3
votes
2answers
383 views

Proving that a function is not invert-able (one way function)

I am having problems with proving if the one-way function (http://crypto.stackexchange.com/tags/one-way-function/info) is hard to invert or not. $2^\sqrt {m} $ one-way function $ f: \{0, 1\}^{2m} ...
3
votes
1answer
136 views

Is it a good idea to use Lagrange/Newtonian interpolation for encryption?

I see that Lagrange interpolation is commonly used for secret sharing, but could it be used for encryption? The goal is to reduce database I/O and compute new values on the fly. Suppose the use case ...
3
votes
1answer
584 views

How can I calculate the time complexity of modular arithmetic?

I'm doing modular arithmetic in a Java program, and I want to calculate the time complexity of the individual operations. $$ c= a · b \bmod n $$ $$ m = a^{-1} · b \bmod n $$ How do I get an ...
3
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2answers
219 views

Parallel-resistant proof-of-work scheme?

I am looking for a proof-of-work scheme which cannot be effectively parallelized. For example, in hashcash (and by extension bitcoin) you have some collision-resistant hash function $f()$, a target ...
3
votes
3answers
230 views

Are asymptotic lower bounds relevant to cryptography?

An asymptotic lower bound such as exponential-hardness is generally thought to imply that a problem is "inherently difficult". Encryption that is "inherently difficult" to break is thought to be ...
2
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1answer
2k views

Big-O Notation: Encryption Algorithms

I am currently completing a dissertation concerning the encryption of data through a variety of cryptographic algorithms. I have spent much time reading journals and papers but as yet have been ...
2
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1answer
171 views

How to calculate complexity of ElGamal cryptosystem?

How to calculate time and space complexity of ElGamal encryption and decryption as there are two exponentiation operation during encryption and one during decryption? Here is my code for encryption: ...
2
votes
1answer
134 views

Explanations for the complexity values for second preimage attack on GOST?

I've been reading the article "A (second) preimage attack on the GOST hash function" by F. Mendel et al (link) and I'm having some difficulty to grasp some of the values of complexities/probabilities ...
2
votes
1answer
209 views

Brute force attack expected running time

I am a bit confused about the expected running times of brute force attacks on different cryptosystems. So let's assume a key size of $2^n$ bits. Symmetric key cryptography: $E(brute)$ = ...
2
votes
1answer
781 views

Meet-in-the-middle with checking complexity

In regards to meet in the middle type attacks, I have been considering the amount of operations in order to successfully find a key given two sets of plaintext / ciphertext pairs. All of the sources I ...
2
votes
1answer
38 views

Differences between Work Factor and Time Complexity

I am interested to know if work factor means the same thing as time complexity. Quoting Work Factor : Uncovering keys in cryptosystems The Work Factor of a cryptosystem is related to its ...
2
votes
1answer
80 views

What does it mean for an adversary to run in PPT?

I've been reading this question where a detailed description of mine is given, I've understood that a polynomial-time adversary is an adversary for which the only feasible strategy are those that take ...
2
votes
1answer
223 views

ECC Complexity order of point addition, scalar point multiplication and selecting random point

I am facing this problem in calculating the order of a process which involves ECC point addition: $P+Q$ , scalar multiplication: $aP$, and selecting random points in the group. The group is of prime ...
2
votes
1answer
257 views

Why does Merkle's Puzzle requires Eve quadratic complexity of effort to break the system?

The way Applied cryptography 2ED explains the puzzle is as follows (I paraphrase it): Bob generates 2^20 messages of the form x,y where ...
2
votes
2answers
109 views

Computational Complexity - When is it really exponential time?

I'm currently working on the discrete logarithm problem and the relevant attacks. I'm fine on the mathematical side of things, but when it comes to estimation of running times I run into problems. ...
2
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0answers
108 views

Will non-ECC algorithms like RSA eventually become too inefficient?

The strength of symmetric and asymmetric encryption schemes scales with the key length, but there is a difference between symmetric algorithms like AES and asymmetric algorithms like RSA. For ...
1
vote
2answers
373 views

Complexity of ECB and OFB

What is the complexity of ECB in terms of Time and Memory? and also in OFB? I can't find it in the internet, so I decided to ask it in here.
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2answers
645 views

What is the time complexity of the RC4 encryption & decryption algorithms?

I'm trying to figure out what the time complexity of RC4 encryption & decryption algorithms is, in big-O-notation.
1
vote
1answer
80 views

DES-X , computation load and storage

The passage said that the computational load to attack DES-X can be reduced to approximately $2^{(56+64)}=2^{120}$ steps,and the storage of data sets should be $2^{64}$. But I can't figure why ...
1
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2answers
136 views

One-way function and $EXP$

All examples of one-way functions I have see till now are closely related to the assumption that $NP\neq P $ (or even weaker ones, such as $UP\neq P$), but why not considering the theorem $P\neq EXP$? ...
1
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1answer
3k views

How do I derive the time complexity of encryption and decryption based on modular arithmetic?

I want to calculate the time complexity of two encryption and decryption algorithms. The first one (RSA-like) has the encryption $$ C := M^e \bmod N $$ and decryption $$ M_P := C^d \bmod N. $$ ...
1
vote
1answer
210 views

McEliece Public Key Encryption

The definition of Public Key Encryption(PKE) say that: A PKE scheme is a triple of probabilistic polynomial time algorithm (PPT) (Gen,Enc,Dec). The definition of PPT say: In complexity ...
1
vote
2answers
115 views

Polynomials and efficient computability

In public key crypto, the popular definitions of security (CPA, CCA1,2) depend on PPT adversaries. I'm trying to understand why adversaries should be PPT. It's clear that adversaries should be at ...
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vote
0answers
50 views

Asymmetric encryptions' computational complexity [duplicate]

I need to know the computational complexity of the public key encryption (e.g. Paillier), please. (i.e.Paillier in his paper mentions that the computational complexity of most of public key ...
1
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0answers
113 views

Complexity of verifying OTP secret

What is the minimum number of unique pairs of digests and inputs to a one-time pass needed to verify that a secret is equal to a ...
1
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0answers
118 views

How to develop a public key cryptosystem based on a hard problem?

Recently, I found a function that is performed on a sequence to return another sequence. All known algorithms for finding the input, given the output are of exponential complexity. I want to propose ...
0
votes
1answer
226 views

Why is AES unbreakable?

Why is it said that AES is unbreakable? Brute force attacks would take years to crack it, so is it possible to crack it if the computational speed of machines increase in the following decade?
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2answers
182 views

Is solving a modular linear equation a hard problem when the coefficient is not an invertible element?

Assume that we have a linear equation like this: $$ax=b \pmod n$$ when $x$ is the unknown, and $a$ is not an invertible element in $n$. is finding $x$ a hard problem? (by solving I mean finding an ...
0
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2answers
353 views

Cryptographic Symmetric Stream Cipher

Let me know a cryptographic symmetric stream cipher system with only two functions say S() and P() and it should satisfy the ...
0
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1answer
116 views

How much (home PC) CPU time is required to generate a prime number of a given size?

How much CPU time is required on a typical home computer to generate a prime number of size 100 bit, 200 bit , 512 bit and 1024 bit using given random bits of the respective sizes? Please note that ...
0
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1answer
33 views

One-one correspondance complete function

I am reading the paper by John B. Kam and Georges I. Davida (1979) titled Structured Design of Substitution-Permutation Encryption Networks. On page 749 it reads ...