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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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 ...
11
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2answers
88 views

Practical differences between circuits and turing machines for cryptography

In formal cryptography, we model algorithms (mostly our adversaries) as (Probabilistic) Turing Machines or as boolean circuits. In our lecture on formal cryptography, we learned that circuits are more ...
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61 views

Majority encryption algorithm?

Assume that I want to leave an encrypted message to a group of $n$ people in a way that they can only decrypt it if they work together in the following sense: For some fixed $k < n$ every ...
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1answer
106 views

Performance of Fully homomorphic encryption VS Paillier encryption in Practice

Consider two schemes both have computation complexity linear to the input size (i.e. number of inputs). One scheme is based on Paillier encryption and the other one is based on fully homomorphic ...
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1answer
4k 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 ...
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2answers
692 views

Is AES solvable by reducing to SAT?

Consider a known plaintext attack on AES — just so we have an actual system of equalities that we can feed to a SAT solver. Is AES solvable in this way? In other words, will the algorithm eventually ...
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38 views

Best/average/worst-case complexity for cryptography

I understand that the complexity of a problem can be measured by it's best-case, average-case, or worst-case complexity. Am I correct in thinking that, for cryptographic purposes, each of these is of ...
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1answer
41 views

Running time of Shamir's secret sharing scheme

Let $p>n$ be a prime number. The key steps in the $(t,n)$ Shamir's secret sharing is as follows: Steps of dealer: Choosing $s \in \mathbb{Z}_p^*$ Selecting $b_i \in \mathbb{Z}_p^*$ for ...
4
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3answers
338 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 ...
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1answer
65 views

Finding a secret cipher given the key and known plaintext?

Let $x,y,k$ be plain text, cipher text and key respectively. Also suppose $\operatorname{Enc}$ is the algorithm of encryption for block cipher with size $n$. So we have $$\operatorname{Enc}_k(x)=y$$ ...
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1answer
61 views

Function with no fast path and with fast proof

When a function is iterated and each time a previous result is used as input to the next iteration (feedback), so that there is a limited benefit from parallel computing, is there such function that: ...
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119 views

Complexity using RSA

You encrypt a message using the RSA encryption system as $t^e \pmod n$ , where $t, e < n$ and $t$ is the numerical equivalent of the message. The message is written in a $27$-letter alphabet and is ...
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72 views

Find a constant $C$ in $O(Cn^2)$?

I am writing a paper on RSA, and I am calculating some encryption running times. I have been told that the encryption running time is $O(n^2)$ where $n$ is the modulus. I have also been told that ...
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36 views

DSSP reduction to DSSI

In “Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies" by DeFeo, Jao and Plut, a reduction from the Decisional Supersingular Product (DSSP) problem to Decisional ...
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68 views

P-Complete hashes, hashing to a larger set

Historically hashes have been from a large set (say 256 characters) to a smaller set (256 bits). Also, hash functions that are P-complete have no known parallel algorithm; they must be computed ...
4
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1answer
139 views

Usage of Zero-knowledge proofs for NP-complete languages

It is well known that if OWFs/PRGs exist, then there is a zero knowledge proof for any NP-complete language, say G3C (graph coloring in 3 colors). The zero-knowledge notion maintains that any ...
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1answer
134 views

computational complexity class of decryption of AES [closed]

I haven't really seen what computational complexity class of decryption of AES is. Can anyone provide reference papers or answers here?
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1answer
85 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 ...
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2answers
74 views

How can complexity be increased or decreased in AES?

I have been studying data compression for a while. For educational purposes, after a lot of reading, I managed to create a software that performs encryption and authentication using AES256-GCM. ...
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1answer
252 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 ...
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1answer
77 views

What is meant by - computational complexity $2^x$?

Recently Bi-clique Cryptanalysis allowed to obtain following results on full AES ES that claim to have The first key recovery attack on the full AES-128 with computational complexity $2^{126.1}$ The ...
5
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1answer
824 views

Why does applying 56-bit DES twice only give 57 bits of security? [duplicate]

Given two 56-bit keys, $k_1$ and $k_2$, why does $E_{k_1}(E_{k_2}(M))$ only give 57 bits of security? So basically I'm unsure why it only gives 57 bits of security; I understand that one key will ...
0
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1answer
284 views

uniform vs. non-uniform PPT

I'm trying to understand PPT and in particular what the differences are in uniform and non-uniform PPT's. First, this is how I see it: A Probabilistic Polynomial-Time algorithm A is an algorithm that ...
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1answer
573 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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1answer
559 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 ...
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2answers
336 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 ...
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258 views

Is the complexity of Caesar (shift) ciphers “n * n!”?

Can we say that any shift cipher to be decrypted needs an algorithm of complexity “n * n!”? (where n is number of possible ...
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2answers
313 views

How fast would a polynomial time factoring algorithm compute?

I know factoring is the chief means of breaking RSA keys. I know an algorithm that runs in polynomial time would be able to break an RSA key pair "quickly". But how quickly is "quickly"? Note, I'm not ...
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1answer
149 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 ...
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3answers
634 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 ...
3
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1answer
939 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 ...
4
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1answer
230 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 ...
3
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1answer
111 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 ...
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52 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 ...
3
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2answers
452 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} ...
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1answer
135 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 ...
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153 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 ...
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237 views

Solution with high decryption cost and low encryption cost

I am looking for any cryptographic solution that will meet those requirements : Only known method to get the encrypted string need to be brute force. Decrypting on modern computer not more than ...
3
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2answers
211 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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1answer
299 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: ...
3
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1answer
186 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 ...
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1answer
252 views

How to calculate bit strength of Integer Factorization Cryptography (IFC) such as RSA using Python

I would like to know how to calculate the bit-strength of Integer Factorization Cryptography (IFC) such as RSA by using Python. I gathered it is based off the complexity of factorizing the modulus ...
3
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1answer
330 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
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1answer
466 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 ...
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1answer
230 views

PRG existance and P versus NP

how can we prove that if there is a secure PRG then P!=NP or in reverse order?(is there any reduction?) a secure PRG is a pseudo random generator that for every eficient (running in probabilistic ...
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3answers
402 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. ...
2
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1answer
169 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 ...
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303 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 ...
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1answer
331 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: $E(brute)$ = ...
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1answer
373 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 ...