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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determining the next random number pseudorandom number generator?

I have given 3 numbers let's say basic example x_0=5, x_1=6 and x_2=2 and modulus p is 7, ...
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92 views

Rank of Quadratic Form

Let $n,m, s \in \mathbb{Z}$ be integers satisying $n=s^2$ and $m=2n$. Let $\newcommand{\bigmatrix}[1]{ \begin{pmatrix} #1_1 & #1_2 & \cdots & #1_s \\ #1_{s+1} & #1_{s+2} & \cdots ...
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Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...
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107 views

Given odd number $n$ count the bases to which $n$ is Euler pseudoprime

As the title says we are given an odd number $n$ and wish to find the number of bases $b$ such that $n$ is an Euler pseudoprime; That is, $\gcd(b,n)=1$ and $b^{(n-1)/2} \equiv \left( \frac{b}{n} ...
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$c$ primitive root, $a \in \{1,\ldots,p-1\}, w/ j \in \mathbb Z^+, a \equiv c^j \pmod p), a^{\frac{p-1}{2}} \equiv 1 \pmod p\implies j\text{ even}$.

Suppose c is a primitive root modulo $p$. Suppose you have a particular integer $a \in \{1,2,\ldots,p-1\}$ and you have found $j \in \mathbb Z^+$ such that $a \equiv c^j\pmod p$. Show that if ...
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136 views

Mathematical foundation crisis and the RSA

I am currently in my last year of high school and I am writing a report on cryptography from a idea historical and mathematical perspective. I am including a few of the subjects: Cantor's diagonal ...
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112 views

Understanding Quadratic Sieve Algorithm

I am studying Cryptography and came upon the quadratic sieve algorithm. However, I am having hard time understanding how the algorithm works. I kind of understood how the steps are followed through ...
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54 views

If P = NP can asymmetric key exchanges still exist?

One functions are easy to compute (ie polynomial time checking) but hard to reverse. if P = NP does that mean that asymmetric key exchanges will be reduced from polynomial computation time and ...
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98 views

Decrypting a message without the Private Key

I am given 5 different encryption modulus, N, each ranging from 78 to 88 numbers long. Then for the encryption exponent, each has the same which is 5. Then I am given 5 different encrypted messages, ...
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69 views

Asymmetric block ciphers?

Any block cipher transforms a block of $N$ bits into another block of $N$ bits based on a $\mathcal{K}$ bit key. This can be considered to be a substitution cipher on an alphabet consisting of $2^N$ ...
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102 views

Decryption of an Encrypted Message

Suppose we are given sending a message to two people: A and C. A and C have the same RSA encryption modulas: R=(some arbitrary number, say) 454564515456465465465156. But A and C have two different ...
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Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
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42 views

Discrete Logarithm Problem in $\mathbb{F}_{p}$ and using Elliptic curves

I want to learn about the hardness of the DLP in $\mathbb{F}_{p}$ and using Elliptic curves, and the best attacks against each. I want to be able to compare the hardness of the problem in the two ...
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59 views

Questions regarding the use of Index Calculus for finite fields and elliptic curves

Ok I have a few questions that hopefully some people can answer: For the Index Calculus applied to the Discrete Log Problem in $\mathbb{Z}_p^*$. I first thought that if we could find the ...
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70 views

Is discrete ultralogarithm harder than discrete logarithm?

Is computing $g^{xy} \bmod{s}$ from $g^{x} \bmod{s}$ and $g^{y} \bmod{s}$ easier harder or the same level of difficulty as computing $g\uparrow\uparrow(xy) \bmod s$ from from $g\uparrow\uparrow x$ ...
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Adding and multiplication in jacobian coordinates

Please tell me how i can to derive formulas for adding and multiplication of 2 points in jacobian coordinates $((x,y)=(\frac{X}{Z^2},\frac{Y}{Z^3}))$ over elliptic curve? Thanks a lot beforehand. I'm ...
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Efficient decoding of irreducible binary Goppa codes and the role of matrix P in McEliece cryptosystem

If we assume that the support for an irreducible binary Goppa code $\gamma_1, ..., \gamma_n$ is publicly known, when is it possible to efficiently decode the code? I know it's possible if one knows ...
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94 views

Mathematical Basis of OAuth Encryption

There are numerous explanations of the common public-private key system available online, explaining how large primes are used to encrypt messages. Is there any similar guide to the mathematics of ...
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383 views

Diffie-Hellman key exchange public key calculation

I encountered a question that I can't seem to get around it. Lets say user A and B uses the DHKE defined over $GF(2^8)$ induced by the irreducible polynomial $x^8 + x^4 + x^3 + x^2 + 1$ and the ...
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137 views

How to proof this equation without calculating the values it self

I have the following equation. $$(X + a)^n\equiv(X^n + a)(X^r - 1)\bmod n.$$ This is part of the AKS algorithm. The problem is, that I'll have to solve this equation for every $1\leq a<10$ and ...
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Algorithms for Performing Large Integer Matrix Operations w/ Numerical Stability

I'm looking for a library that performs matrix operations on large sparse matrices w/o sacrificing numerical stability. Matrices will be 1000+ by 1000+ and values of the matrix will be between 0 and ...
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104 views

A variant of the “closest vector problem” (CVP) in lattice-based cryptography

Consider a public-key scheme on lattices, such as GGH. The private key is a basis $\mathbf{B} \in \mathbb{Z}^{m \times n}$ of a lattice $\mathcal{L}$ with good properties (such as short nearly ...
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Finding the random $r$ in a Paillier encrypted message with knowledge of the private key.

In the Paillier cryptosystem, suppose that I know a Ciphertext encrypted with some unknown random $r$ i.e. $$C = (g^m r^n) \bmod n^2 $$ I know $g, n$, the prime factorization of $n$, i.e., $pq$. I ...
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15 views

Converting Walsh coefficients to values of a function

I assume I know the Walsh coefficients of a function f: $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2}$. Is there any efficient possibility to get the values of the function f ?
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Type(s) of Hashing function that keeps the ordering information

I am asking this question from a perspective that we need to store a set of hashed data that can be queried later for in an ordered fashion. My situation is that some data has to be encrypted, I am ...
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26 views

How to find the period of a recurrence relation

Given the recurrence relation $s_{i+5}=s_{i+1} + s_i$ over $\mathbb{F}_2$ with initial states $s_0 = 1, s_1 = 1, s_2 = 1, s_3 = 0, s_4 = 1$ What is the best/quickest way to find the period of the ...
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22 views

Binary solutions of multivariate polynomial system in special (factored) form.

In my personal research I've run into a system of multivariate polynomials (with coefficients in a field). I am aware that there is no polynomial time algorithm (in the number of indeterminates) for ...
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17 views

$S=\{1\le a \le n:(a,n)=1,a^{n-1}\not\equiv 1\pmod n\}$, $T=\{1\le b \le n:(b,n)=1,b^{n-1}\equiv 1\pmod n\}$ iwth composite and prime numbers

I have two sets with $n>2$ natural number: $S=\{1\le a \le n:(a,n)=1,a^{n-1}\not\equiv 1\pmod n\}$ $T=\{1\le b \le n:(b,n)=1,b^{n-1}\equiv 1\pmod n\}$ Can anyone explain me if there are prime ...
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Questions about RSA cryptosystem primitives: key construction and signature

i) Construct a pair of private/public key RSA, where the prime numbers that you will use are $p=11$ and $q=13$ ii) Describe how the owner of the above keys calculates a signature RSA in the message ...
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21 views

ElGamal signature for finding the private key

Alice uses an ElGamal signature with base the group $Z^*_{107}$ and parameter $g=3$ of order $q=53$.The private key of Alice is some $x \in \{0,1,.....,52\}$ and the public key of her is $y=10$.To ...
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29 views

Exponential cryptosystem

The cryptosystem works as follows: The plaintext message is first replaced by ciphers (a=00, b=01, etc.) and then encrypted in blocks of four digits. So if the message is "hi", the plaintext number ...
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Framing a lattice problem from information available on multiple runs of GLV decomposition

I have posted a similar question here. The GLV method [ref] is used to speed up ECDSA signature generation. In this method, an input scalar $k$ is decomposed into two scalars, $k_1$ and $k_2$. Then ...
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33 views

Rabin's cryptography - when the message $M$ isn't coprime to $n = pq$

Say the message $M$ is a product of one of the primes $p$ or $q$, won't the $gcd$ of $M$ and $n$ (the public encryption key) give me $p$ or $q$? say $p = 11$ $q=19$ $n=11*19=209$ and $M=33$. ...
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12 views

How to define a one-parameter family of probability distributions

I am trying to evaluate a noise-source as a means of providing entropy to a random number generator. I am running into trouble when it comes to determining the probability distribution that has the ...
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22 views

How do I construct the S boxes of the following boolean function?

$f(z) = \dfrac{az+b}{cz+d}$ Where $ab-cd$ is non zero. I have already constructed the sixteen element Galois field, but how do I use the function to construct the $S$ boxes?
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67 views

How can we find the private key?

Alice uses the ElGamal signature scheme with the variables $p=47$, $q=23$ and $g=2$. For two different messages $m_1, m_2$ with $h(m_1)=4, h(m_2)=3$ she produces the signatures $(r_1, s_1)=(14, 8)$ ...
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42 views

Pohlig-Hellman algorithm

I am studying Cryptology right now and I am facing some difficulties to understand the Pohlig-Hellman algorithm. Could you explain to me how the algorithm works?? $$$$ EDIT: I read an ...
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27 views

How can we find the decrypted message?

Let's suppose that $A$ uses the encryption system of ElGamal with with public key $(p, g, y)=(53, 2, 27)$. $B$ sends to $A$ the encrypted message $(15, 34)$. Find the original message. We have that ...
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24 views

Does RSA use two one-way functions?

I'm trying to understand the concepts behind RSA right now. From what I've learned so far, it's pretty much all about a one-way function with a trapdoor: Raising the message to the e'th power modulo ...
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15 views

Torelli Shanks Algorithm - Repeated Squarring Method

This algorithm is using when you want to find a square root of a number in a given moduli. I can't see the idea behind this algorithm, so can someone explain it in a simple way?
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Problem with DES Encryption

If the input string to a round of DES is 11001100 · · · 1100 = ‘1100 × 16′ and if the round key is 1111 . . . 111 (‘1 × 48′), Then how can I calculate the 20th and 33rd output bits ? This was an ...
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23 views

Decryption of RSA

I am given the following information about an RSA-encryption: $e=31671865305320609$ (public key) and $n=10e+3$. Then I am given the ciphertext $c$ which I omit here due to his length. The task is to ...
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How to decrypt a ciphertext by using the mutual index of coincidence?

I am trying to decrypt a Vigenére cipher text. I have found the key length by computing Index of Coincidence of substrings. The key length is 12. I know the letter frequencies the string and the ...
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19 views

What is the proper way to generate a key in Merkle-Hellman Knapshack Cryptosystem?

This article says that, if a message is 8-bit, then there should be 8 elements in the Super Increasing Sequence. ...
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35 views

how to calculate number of points on an Elliptic curve over prime field? suggest any best method

$y^2=x^3+a*x+b\pmod p$. For this elliptic curve over prime field, how to calculate number of points lies on the curve? suggest any best method.
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68 views

Solving RSA cipher without calculator

I have a question: Encrypt the message UPLOAD using RSA with $n=3\cdot 31$ and $e =17$. My question is, how can I solve this with a calculator and in an efficient manner due to being in an exam ...
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23 views

Why AES uses polynomials instead of numbers

In AES, the numbers actually represent polynomials and all operations like addition, multiplication have rules according to modular polynomial arithmetic. I don't understand the need to have ...
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Primitive vs Irreducible

Are all irreducible polynomials primitive? If not can anyone give an example of such a polynomial that is irreducible but not primitive?
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How does the factor command on the TI-89 works?

So to put my question in context, I am working on the following problem. Let $N=1291233941$. Eve's magic box tells her the following three encryption/decryption pairs for $N$: $$(1103927639, ...
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43 views

What's the expected number of hashes before you find a match?

For any request to a server, the server responds by sending the requestee a random number $r$ and another number $n$. The requestee must produce a solution $s$ such that $\operatorname{HMAC_r(s)}$ ...