# Tagged Questions

Questions on cryptography and cryptanalysis, encryption and decryption, and the making and breaking of codes and ciphers. Consider posting your question at Cryptography.SE.

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### Factoring of composite numbers of two primes

Let n=pq, with primes $p=x^a +1$ and $q=x^b+1$, for $x$, $a$, $b$ integers with $a$ not equal to $b$. Is $n$ hard to factor? If not what would be an algorithm and its complexity?
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### Why is the discrete log problem intractable?

I have read the other questions on SE on this subject and they were not helpful to me, partially because I am not familiar with advanced mathematical notation. Let me explain the way I'm thinking of ...
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### Perfect secrecy of affine cipher

How can I show that the affine cipher has perfect secrecy if the key $(k,a)$, where $k\in\{1,3,5,7,9,11,15,17,19,21,23,25\}$? I know to show perfect secrecy I need to show that ...
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### Inverting Modular Exponentiation

How can I go about solving the equation $4 = y^4 \bmod{7}$? Do I have to try all of the possible $y$'s in between $1$ and $7-2$ or is there a smarter way that can be generalized for larger numbers?
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### Markov Chain and cryptanalysis

Where I will be able to found papers to read the state-art of the use that Markov chain in cryptanalysis. I found this Canteaut, A. and Chabaud, F. (1998). A new algorithm for finding ...
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### maximum number of homogeneous linearly independent equation over $\mathbb{F}_{2^m}$

In Field $\mathbb{F}_{256}$ we are given homogeneous equations in 255 variables from the field. It is said that maximum number of linearly independent such equations we can get are 247. Why ? To be ...
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### How do I find the multiplicative inverse of a finite field polynomial using Euclidean Algorithm?

How would I use the Extended Euclidean Algorithm to find a multiplicative inverse of a polynomial in a Galois/Finite field...say, for example, $GF(2^3)$ ? Is it the same process as how it's ...
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### Is cracking MD5 hash a form of P VS NP problem? [closed]

I have a question,Is cracking MD5 hash a form of P VS NP problem?
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### Explanation of this modular arithmetic example in “Understanding Cryptography”

Nothing is more frustrating than a book example that doesn't seem to make sense. I have been tasked to make an elliptical encryption accelerator, and it seemed prudent to read a book on cryptography, ...
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### Solving $x^e =c$ in $\mathbb{F}_{p}$

Find all solutions to the equation $x^3=7$ in $\mathbb{F}_{13},\mathbb{F}_{19}$ and $\mathbb{F}_{35}$. In An Introduction to Mathematical Cryptography (Hoffstein et al), we have that proposition ...
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### Digital Signature Algorithm (DSA) intuition and its relation to RSA

I already understands how to use the RSA algorithm to sign messages, as you can see in this post of mine. Searching about elliptic curves, I found a general algorithm called Digital Signing Algorithm. ...
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### How Do Computers Factor Semi-Primes [closed]

How do computers factor large semi-primes? I know it's difficult but what process do they use? Is it simply a matter of dividing by all odd numbers under the square root of the semi-prime till they ...
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### Special-purpose hash functions

I am trying to create a special purpose hash function that will have as few collisions as possible. $99\%$ of the input will be sequential numbers, from $1$ to $N$. The size of the hash table will be ...
18k views

### Why are very large prime numbers important in cryptography?

Firstly, you guys are awesome, and I learn quite a bit just from reading the questions of others. Secondly, a friend asked me recently why large primes are important for data security, and I was ...
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### Proofs of congruence relations

Exercise 2.3 from "Introduction to Mathematical Cryptography" Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ of order $r$. (a) Suppose that $x = a$ and $x = b$ are both integer ...
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### What are the values of b such that the matrix [(1,1)(b,1)] is invertible mod 26.

What are the values of b such that the matrix [(1,1)(b,1)] is invertible mod 26. I figured that the matrix is only invertible if its determinant and the n value 26 's gcd is 1, meaning they are ...
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### Primitive Root Theorem

Let $p$ be a prime and let $q$ be a prime that divides $p − 1.$ (a) Let $a \in F_p$ and let $b = a^{\frac{p−1}{q}}$. Prove that either $b = 1$ or else $b$ has order $q.$ (Recall that the order of $b$ ...
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### Find GCD of polynomials over GF(101)

Hello all I'm teaching myself cryptography, and I'm struggling with polynomial arithmetic over finite fields. I've some what been able to teach myself how to do the arithmetic over $GF(2)$, but when ...
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### Why are supersingular elliptic curves useful for cryptography?

I don't know very much about cryptography and would like to learn more. I know the basics of RSA alogrithm and how elliptic curves over finite fields can be used to do something similar. But I would ...
32 views

### Puesudorandom generation

Hi i have created a stream cipher that creates a'random' stream of numbers(1-99) as subkeys for the message.The generation algorithum is as follows:(key mod (iv+counter)) mod 99.The key is the main ...
111 views

### What numbers are relatively-prime to $256?$

Given the numbers are in the range $1$ to $256$, which ones AREN'T co-prime, would be an easier question$?$ This question may be very specific and hopefully trivial for somebody on the maths board, ...
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### How many times do I loop Solovay--Strassen primality test

First, I am aware of this former thread: math.stackexchange Yet it doesn't answer my question. If I want to check if an integer $n$ is prime using the Solovay--Strassen test, how many times do I ...
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### Where do hash functions come from?

I have some basic understanding of how hash functions work, however, I have no idea of how mathematicians created them. Were them a byproduct of a non cryptografics related research or were them a ...
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### Diffie-Hellman protocol

So I get the basics of diffie-hellman, discrete logarithms, modular arithmetic etc but I feel like I am missing something substantial or I would not be able to reverse it so easily, unless it is due ...
14k views

### Example for Cyclic Groups and Selecting a generator

In Cryptography, I find it commonly mentioned: Let G be cyclic group of Prime order q and with a generator g. Can you please exemplify this with a trivial example please! Thanks.
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### Euclid's algorithm to solve (e x d) mod p = 1

I need to use Euclid's algorithm to find d in the following equation. Given values for e and p $$(e\times d)\mod p = 1$$ I have used Euclid's algorithm to find the gcd of two numbers but can't see ...
10k views

### Calculator model with mod function?

I'm wondering does anyone know of a scientific calculator with a mod function? In C# this is shown as follows (just in case there are any other mods that a mathematical non-savant such as myself ...
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### Fermat's theorem as primality tester when powers are too large

As part of cryptography, if I wish to test whether a given number is probably prime I use the formula: $$a^{p-1} \equiv 1 \bmod p$$ where $p$ is (potentially) a prime number. However, when it ...
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### Fermat's Theorem as a primality tester doesn't work for all primes?

I'm studying cryptography. According to Fermat's theorem... $$a^{p-1} \pmod p = 1$$ .. when $p$ is a prime number. The above should prove whether a number is prime or not yet it doesn't work for ...
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### Reversing Rotation + XOR

I have this cypher which is as follows : Take 2 numbers : A=1011 and B=1010 if the ith bit of X is 1 then shift Y* i times to the left. So in the end you will get ...
### Factor the RSA modulus $n = 3844384501$ knowing that $3117761185^2 \equiv 1 \pmod{n}$
As per the title, the task is to Factor the RSA modulus $n = 3844384501$ knowing that $$3117761185^2 \equiv 1 \pmod{n}\text{.}$$ $n$ being an "RSA modulus" means that it is a product of two ...