# 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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### Help with next step in the Quadratic Sieve

So I am at the same step as someone from math.stackexchange but he never recieved an answer so I will copy-paste his question here: Say, for N = 90283, I compute bound 𝐵=𝑒(12+𝑜(1))(ln(𝑛)ln(ln𝑛√))...
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### Why is this attack complexity equal that exact number of bit operations?

in the this paper,section 3,autors attack hamsi-256. Im trying to make a parametrized version, so i need to understand how do they estimate the complexity of attack in bit operations,that reads as ...
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### Show that $\text{FACTORING} \le_P \text{SQROOT}$

I tried to prove that $\text{FACTORING} \le_P \text{SQROOT}$ in a general setting, so $n = p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdot \ldots \cdot p_k^{\alpha_k}$. THEOREM:Let $n$ be a composite ...
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### What are standard cryptographic assumptions?

I am struggling to understand what is meant by "standard cryptographic assumption". The Wikipedia artice on the Goldwasser–Micali system (GM) reads "GM has the distinction of being the first ...
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### Complexity of a brute-force search

I am currently thinking about the complexity of a brute-force attack on a cipher. Let the key length of the cipher be 64 bit. Then there are $2^{64}$ different keys in the keyspace. If you do a brute-...
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### Partial key recovery from linear equations

I have searched for my question but I didn't find any relevant answer to my situation. I guess maybe it is too easy but I am a newbie in crypto and I can't figure out the answer. Here is the exercise: ...
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### 'random' function $r_{n+1} = f(r_n)$, $|\{r_i,\forall i\}|=N$, $r_N = r_1$ where a function $g(r_k)=k$ is harder than in ECC?

Is there a deterministic pseudo random function $f$ with $r_{n+1} = f(r_n)$, $|\{r_i,\forall i\}|=N$, $r_N = r_0$, (for given random initial value $r_0$) where a function which derives the index of a ...
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### Question on the Quadratic Residuosity Assumption

I am reading the Handbook of Applied Cryptography and on page 99 the authors write , after showing that $QRP \le_P FACTORING$: It is believed that the $QRP$ is as difﬁcult as the problem of ...
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### What if an AES Whitebox 1024-bit (or larger key) is created? Does it increase complexity consistently?

Following the Chow et al paper and Muir's tutorial, I was able to implement the AES algorithm using tables embedding keys of 128, 192 and 256-bit sizes, later extended to 1024, 2048 and 4096-bit sizes....
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### Definition of support in the context of an encryption scheme and usage of P = NP assumption in lemma 9.2 (Arora, Barak)

EDIT: I've migrated the question by deleting the same question I asked on mathematics stackexchange. 2 questions: (1) I am confused about the definition of support used in the proof given in the ...
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### (Whitebox Crypto) Using ChaCha20, is it safe to reduce the nonce length in a single block cipher?

I am willing to write a Whitebox Crypto unit using ChaCha20 algorithm (Bernstein, D. 2008) for an input consisting of a single block. The fact it is going to be a single block cipher is of special ...
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### Are there post-quantum cryptosystems with a gap between classical and quantum security?

Is there a gap between classical attacks and quantum attacks against some post-quantum security assumptions? (I'm particularly interested in asymmetric cryptography.) I understand that there is no ...
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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?
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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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### 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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### 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 ...
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 ...