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

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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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59 views

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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106 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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134 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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108 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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52 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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90 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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100 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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66 views

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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58 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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45 views

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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56 views

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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93 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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379 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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136 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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54 views

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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102 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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77 views

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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21 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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9 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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19 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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49 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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39 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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23 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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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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27 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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63 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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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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25 views

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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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 $\operatorname{HMAC_r(s)}$ ...
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76 views

Affine cipher does not satisfy the diffusion property.

Generally, we know that substitution ciphers do not have the property of diffusion. And affine ciphers is the special case of substitution ciphers. But how can we prove that affine cipher does not ...
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39 views

Of what use is my code for finding prime numbers of a certain size?

I've developed a bit of mathematica code that can find primes within a range of numbers. For example, if I wanted all the primes between one million and two million, it could do that. Of what use is ...
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Formulate a general version of this cryptosystem

The exercise describes a public key cryptosystem that requires Bob and Alice to exchange several messages. We illustrate the system with an example. Bob and Alice fix a publicly known prime ...
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Time estimation and big-O notation

Let a and m be elements of $\mathbb{Z}$ where m is positive. How can I show that the inverse of $a$ can be computed in $O(log^3m)$ bit operations? I thought that I should find the greatest common ...
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Diffie-Hellman decision problem

I am looking at the Diffie-Hellman protocol. The Diffie-Hellman decision problem (DDH) is the following: We are given $g, g^a, g^b, g^c$ and we want to check if $g^{ab}=g^c$. where $g$ is an ...
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23 views

Verify no elements of a list have been removed

I have an sequence of $k$ elements, $\{a_k\}$. Say at any given moment I add an element $a_{k+1}$ to the sequence. Is there any way to verify the sequence has not been altered, without checking each ...
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78 views

Find a point $P$ on an elliptic curve, given $2P$

Let $E$ be the Elliptic curve given by $Y^2=x^3+5x-6$ and suppose $P$ is a point on $E$ over $\mathbb F_{65537}$ with $2P=(7283,24272)$. Find $P$. I approached this question as follows. ...
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52 views

Elliptic Curve finding point of a curve backward?

Given $E: y^2=X^3+5X-6$ over $F=(65537)$ with $2P=(7283, 24272)$ how to find $P$ Can anyone provide an example in steps?
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34 views

Rabin cipher question

I am getting wrong answer to this question. Any one else too has solved it ? Decrypt the Ciphertext message 1819 0459 0803 that was encrypted using the Rabin Cryptosystem with b= 3 and n= 47 *59
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1-2 Oblivious Transfer and its simulation by ordinary OT

The version of 1-2 oblivious transfer mentioned in Even, Goldreich and Lempel [1985] has the provision that the receiver, Bob, can discover with probability > 1/2 if the sender, Alice, sends the same ...