Cryptography Stack Exchange
Cryptography Stack Exchange

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.

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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What's the difference between polylogarithmic and logarithmic?

I can't imagine one that is not polylogarithmic but logarithmic. $O(\log N)$ satisfies both. What about $O(\log^{3}N)$, $O(\log^{100}N)$, and $O(\log^{10000}N)$ ? Let's say $N=10^{10}$
mallea
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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. Or, inverting the image of a one-way function…
Alon Navon
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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 approximation of the time complexity here, for example for…
almodawan
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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. More specifically: if we take a generic square…
polarbear
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Cryptography based on uncomputable problems?

A lot of cryptography is based on the assumption that ${\sf P} \neq {\sf NP}$. Is it conceivable to construct a cryptography system based on a class of much harder problems than ${\sf NP}$-problems, namely undecidable problems such as the mortal…
Dominic van der Zypen
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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 $\mathcal{O}(n^c)$ means mathematically, but why do cryptographers…
Paul Uszak
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Difficulty of a congruence problem

The problem is described as follows. Let $c_1=p_1q_1+r$, $c_2=p_2q_2+r$, $\cdots$, $c_n=p_nq_n+r$, where $p_i$'s, $q_i$'s, $r$ are all large positive integers, and $p_i$'s and $q_i$'s are randomly chosen. Notice that $p_i$'s and $q_i$'s are…
user147687
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Computational Cost of bilinear maps

What is the exact computational complexity in absolute operation numbers (multiplications, exponentiations, etc) of a bilinear map evaluation both for symmetric and asymetric groups. And how this is comparable with exponentiation?
curious
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What is the difference between computational complexity and time complexity?

Computational complexity seems to be used quite a lot in cryptographic papers. The time complexity I am referring to is the one from Computational Complexity Theory. Are these two the same things?
WeCanBeFriends
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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 $\gcd$: $O(n \log(n))$ Factor n first and then use Euler’s…
dkssud10
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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) (\ln n)^{\frac{1}{3}}(\ln \ln n)^{\frac{2}{3}}…
Matt
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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 security parameter and integers $n$ and $m$ are upper-bound by a polynomial in that parameter. Algorithm…
Alpha Bravo
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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 Euclidean algorithm for basis reduction? I know that…
Mosen
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What is the "successive-configuration relation"?

This term is used in the book Foundations of Cryptography on pg 20 with regard to defining deterministic oracles, but is not previously defined and I can't seem to find a definition online easily.
z.karl
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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 interest?: Best-case: If we can find a problem…
MattBurrows
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