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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934 views

Modular Arithmetic - pairs of additive inverse pairs and multiplicative inverse pairs

I am taking a Cryptography class and we are working on modular arithmetic. I am still unsure on how to find pairs of additive inverse pairs and multiplicative inverse pairs. I've seen some videos and ...
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316 views

Does there exist some relations between Cryptography and Algebraic Topology?

We know that there are many application of Cryptography in our real life. Are there any relation between Cryptography and Algebraic Topology? If yes, please suggest me some link or books. Thanks ...
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67 views

provably secure hash function

I have the following question related to proving a hash function is secure if discrete log in group $\mathbb{G}$ is hard. The hash function (Gen,H) goes as follows: Gen: on input $1^n$, run to obtain ...
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43 views

Find a polynomial of degree $8$ with integer coefficients with given root

algorithm to find a polynomial $f(x)$ s.t (1) $degree(f(x))<9$ (2) Integer coefficients (3) Absolute value of coefficients $< 10^5$ (4) f(39.770525) =~ 0 about to be <<10^-10 I ...
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43 views

Trouble Understanding Pinocchio (Verifiable Computing) Sparse Polynomials

I hope I'm asking the question properly. I've never asked anything on this exchange before, but I didn't know where else to ask. The paper in question I've almost got all the pieces to understand can ...
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66 views

Drawing a 5-stage binary LFSR with feedback Sm+5= Sm + Sm+1

Any guidance on how to draw this would be greatly appreciated I know this is more of a visual thing but I also want to go on to determine all the possible (different) cycles that are generated by this ...
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105 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
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56 views

The number of Balanced Boolean functions

Suppose we have n-variable Boolean function (BF) and we know that the weight of a Balanced BF is $2^{n-1}$. The total number of BFs are $2^{2^n}$, Affine BFs are $2^{n+1}$ and Linear BFs are $2^n$. In ...
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46 views

Find a particular function given certain restrictions

This maybe more of a computer science problem but maybe the solution lies in number theory. Given integers $x,y$, define a function $f$ so that $$f(x,y) = \begin{cases} 1 & \text{if $x=y$} \\ 0 ...
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33 views

What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$?

Let $w$ be a primitive element of $\mathbb F_{5^4}$. Let $\alpha=w^{13}$. Define, $F:\mathbb F_{5^4}\times \mathbb F_{5}\times \mathbb F_{5} \Rightarrow \mathbb F_{5} $ as, $$F(x,y,z)= Tr (\alpha ...
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307 views

Solving the discrete logarithm using index calculus, finite fields and factor bases.

(a) Let $p$ be the prime 1073741827, with $\Bbb{F}_p$ the corresponding finite field. A primitive root in $\Bbb{F}_p$ is equal to $g=2$. Use a factor base of primes up to 13 to find the discrete ...
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48 views

statistical analysis of discrete (non-uniform) p-values: cryptographical random data test

i'm doing a statistical analysis of a well-known cryptographic algorithm and have hit an anomaly. i need to prove that what i have found is statistically significant. i am taking block sizes of 256 ...
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29 views

Single-Digit Errors

I've been assigned the following homework problem: Given an eight digit number $a_1a_2...a_8$ and a check digit $a_9$, $7a_1+3a_2+9a_3+7a_4+3a_5+9a_6+7a_7+3a_8+9a_9 \equiv 0 \mod{10}$ ...
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72 views

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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101 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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73 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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139 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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55 views

$c$ 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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142 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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139 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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59 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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112 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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72 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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116 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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90 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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65 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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72 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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61 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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102 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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430 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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138 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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55 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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117 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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95 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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Find normal basis of the field $GF(3^6)$ and find the normal matrix

I am working with a homework is about normal basis on fields GF and I want opinions and maybe if you can help me in some doubts. 1) Find normal basis of the field $GF(3^6)$ which is understood as a ...
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23 views

Algorithm to find password from hash value

I'm currently trying to solve this exercise (sorry for link to image, but there's a bit text): http://i.imgur.com/ETaCK0H.png But there's a few things in the exercise I don't understand. For ...
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11 views

Security of $(k, 2k)$-bit generator for small seeds

Here is the problem I am working on for context. I have $\epsilon \le 1 - 2^{-k}$ and $\epsilon$ approaches 1 as $k \to \infty$ but I'm stuck on part c). The $f$ is secure iff there does not exist ...
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36 views

Using an exponential cipher system, encipher the word HALT. where $p = 29, k = 11$, and $m = 1$.

Using an exponential cipher system, encipher the word HALT. where $p = 29, k = 11$, and $m = 1$. The progress I have made so far: H A L T $07, 00,11,19$ Since, $m =1$, we break this up into $2*m$ ...
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29 views

Calculating block size for RSA

I am trying to encrypt some text via an RSA system, however I am having trouble working out how I decide what size the message blocks should be. p = 641 and q = 751 (n = pq) n = 481391 e = 347393 ...
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9 views

Extended Golay codes are self dual

Show that extended Golay code $G_{24}$ and $G_{12}$ are self dual. To show it have to show that any two rows of $G_{12}$ and $G_{24}$ are orthogonal, that is inner product of any two rows are zero. ...
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In stinson's (2,n)-VCS How to calculate weight of rows of $S^1$ where all the binary n-vectors of weight $\lfloor{\frac {n}{2}}\rfloor$

Stinson introduced a new type of (2,n)-VCS. The $n\times m$ basis matrix $S^1$ is realized by considering all the binary n-vectors of weight $\lfloor{\frac {n}{2}}\rfloor$. Hence the pixel expansion ...
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34 views

Question about the following notation (groups and homomorphism)

So I was reading a paper on homomorphic encryption, and it in turn introduces some concepts that I didn't know much about before (primarily groups). I have a few questions but I'll first post the ...
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21 views

I am currently working on the implementation of the Powerline System which is based on the Chor-Ri

I am currently working on the implementation of the 'Powerline System' which is based on the Chor-Rivest cryptosystem for my number theory project. There is a step in the key-generation phase of the ...
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11 views

how to impersonate A (Alice) in this protocol based on replaying a compromised session key. Is my answer correct?

Construct an attack based on replaying a compromised session key that allows an intruder to impersonate A. given protocol: ...
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39 views

Computing square roots in modular arithmetic

Could someone please help me with this. Calculate the $4$ square roots of $170 \bmod 253$. I understand that Rabin encryption involves 'extracting' square roots from the ciphertext. Is the ...
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27 views

How can you distinguish modular exponentiation from random?

Let $N$ be the product of two primes and let $P$ be the smallest prime larger than $N$. Let the algorithm $R(N,s)$ return $s^{1/P} \pmod{N}$. Let the algorithm $\widehat{R}(N,s)$ pick a ...
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9 views

Complexity of general factorisation algorithms

How does the complexity of the general number field sieve, the quadratic sieve and the elliptic curve method change with the bit length of p and q when factorising composite numbers n=pq?
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28 views

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 ...