Questions tagged [cryptography]

Questions on the mathematics behind cryptography, cryptanalysis, encryption and decryption, and the making and breaking of codes and ciphers.

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Fermat's last Theorem and elliptic curve cryptography

AFAIR, elliptic curve cryptography became popular soon after Fermat's last Theorem had been proven. Is it just a coincidence, or some important cryptographic properties of elliptic curves follow from ...
38 views

Fully homomorphic encryption textbook suggestion

I am looking for mathematics textbooks which include a rigorous introduction to fully homomorphic encryption and especially CKKS / TFHE algorithms at the level of Boneh and Shoup's A Graduate Course ...
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1 vote
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Conway Polynomial for p=2, n=3?

Im doing an exercise on Conway polynomials. As far as im concerned, for p=2, n=3 both $f(x)=x^3 + x^2 + 1$ and $g(x)=x^3 + x + 1$ satisfy every condition. According to every source i found, the latter ...
31 views

How can I be certain of the existence of elliptic curves of certain order when the parameter a is fixed?

My question came up while researching an attack on Elliptic Curve Cryptography (described in Computer Security - ESORICS 2015. I'm given an elliptic curve $E$ defined by $y^2=x^3+ax+b$ over the finite ...
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Determine Whether a Pseudorandom Generator Is Secure

Let $G: \{0, 1\}^s \to \{ 0, 1\}^n$ be a secure pseudorandom number generator (with $s$ seed bits and $n$ output bits). I have attached a problem below that I am confused about; Which generator $G'$ ...
32 views

Problems about Probability Analysis of the Success Rate

I am currently reading a paper on linear cryptanalysis and I am a bit confused by the probability analysis of its success rate. I wonder if I can seek advice here? Let $N$ be the number of given ...
1 vote
101 views

distribution of square roots of unity $mod n$ | Factoring with inverse pair

I am writing a proof related to the RSA cryptosystem, specifically showing that given an inverse pair $d, c$ under multiplication mod $\phi(N)$, where $$dc \equiv 1 \pmod{\phi(N)},$$ there exists a ...
1 vote
29 views

Proof of Golomb's three randomness postulates for binary sequences [closed]

I want to prove that the binary sequence generated by a max-length linear feedback shift register (LFSR) satisfies Golomb's balance, run and autocorrelation postulates: The numbers of zeros and ones ...
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Can we do any better bijective mapping of a permutation series which is only bijective for a probabilistic subset of its input domain?

So we want to bijectively map one path to another. But depending on start and target node we can only choose from a subset of all transitions. It would look like this: We also do not know where one ...
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1 vote
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What does $\mathbb{Z}_2^3$ mean? [closed]

What does $\mathbb{Z}_2^3$ mean? Is the subscript $2$ a modulo and the superscript $3$ dimensions of each element? I am studying lattice cryptography and set theory and I would like to know the how ...
58 views

Finding the period of the BBS sequence

Let $n=pq$, where $p,q$ are primes and $p \equiv q \equiv 3 \mod 4$. Choose an integer, $x_0$, such that $x_0$ and $n$ are co-primes. We define the sequence: \begin{align} x_i = x_0^{2^i} \mod n \end{...
120 views

Unexpected Result from Finite Field Calculations in GF(2^8)

I'm performing calculations within the finite field $GF(2^8)$ and I can't seem to get the expected results. This is my first time working with finite fields, so my understanding is quite basic. I ...
1 vote
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Distinct derivations of polynomial over finite field

I am a student studying algebra and cryptography. I wonder below question is possible. Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is ...
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Prove that if $e.d \equiv 1 \bmod (p-1)(q-1)$ then it’s impossible to have $e.d \equiv 1 \bmod pq$

I am studying R.S.A. cryptosystem and here is the question that came to my mind. Let’s pick $p, q$ to be two primes and $n = p * q$. From that we calculate Euler’s totient function:  \phi(n) = (p - ...
51 views

Time to Find an Elementary Antiderivative of an Elementary Differential Form?

So, in encryption theory, a basic principle is that one has an operation that can be computed in a "forward direction" relatively quickly but for which computing in the "reverse ...
31 views

Testing a Pseudo-Random Number Generator Algorithm

I created a pseudo-random number generator that creates random bits from given numbers. For better visualization, suppose that we have inputs "a", "b", "ab", "abc&...
92 views

About calculating isogeny between two elliptic curves

I'm trying to understand Vélu formulas for calculating isogenies. I took an elliptic curve $E: y^2 = x^3 + 3x + 5$ over $GF(7)$. So I've got the following points on this curve: \{\...
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224 views

Binary multiplication in Galois Field GF(2^8)

I am working on a project (high school), and I need to explain the process of AES MixColumns for one of the parts. I am trying to show an example of the matrix multiplication in MixColumns that uses ...
44 views

Are high-dimensional versions of NTRU cryptosystem more secure?

The basis for this question is a 1-dimensional NTRU cryptosystem. After some literature inspection I have found out it can be also generalised into higher algebras: quaternions (QTRU) and octonions (...
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47 views