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

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1answer
57 views

Non-constant Linear Boolean Function

How can we prove that any non-constant linear Boolean function is balanced ? I know that any non-constant affine function is balanced. But i cannot expend this for Boolean function.
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1answer
63 views

Why in RSA, the public exponent $e$ must be coprime with $\phi (n)$

I'm trying to understand the RSA cryptosystem, and that's what I know so far: If we think about some number $m$ as the message, then we are searching a $e$ and $d$ such that $$m^{ed} \equiv m \ \ ...
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0answers
28 views

Latin Squares and Olderogge Code

So I have two Latin Squares, $A$ and $B$ that form a pair of MOLS of order $m$. I then have an Olderogge code formed from $A$ and $B$, where each binary vector of length $m^2$ is encoded as a codeword ...
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0answers
24 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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1answer
45 views

Basic Modular Exponentiation question

I just know this rule :ab mod n = [(a mod n)(b mod n)] mod n. How can it be proved that the following rule is true ? ...
0
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1answer
47 views

Discrete Log Problem

I've been given this key for an elliptic curve crypto -system: A:=4569782456273849 B:=74578265973825694738 p:=164516845864567592349187678956932587156973824569837657473 So the EC group is ...
3
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0answers
33 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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1answer
15 views

Determine a positive integer $e$ that satisfies $M^{17e}\equiv_{77}M$, when $(M,77)=1$.

We're doing public key cryptography this week and I just can't seem to get a grasp on it. I really don't know how to solve this problem. Can anyone point me in the right direction? I'd really ...
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1answer
24 views

Solving ANF equations

Can anyone suggest a method of solving a system of boolean equations in ANF form? Boolean equations in ANF form (Algebraic Normal Form ) are equations of the form of xor of products of boolean ...
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0answers
45 views

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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1answer
68 views

Breaking RSA code

I will be grateful for some tips on how to bite a task like so: I need to break a RSA code. I know that public key is $n=462257, e=13$. I also have cryptogram $c=139552$. The goal is to find a number ...
4
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1answer
92 views

How does the Enigma machine ensure that no letter is substituted for itself?

In Alan Turing: The Enigma Andrew Hodges describes how the letter encodings performed by a German Enigma machine "would always be swappings" (original emphasis). And goes on to say that There was ...
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1answer
33 views

Confusion about negligible and non-negligible functions in crypthography

I am learning basic cryptography from Coursera's cryptography I course and am a bit confused about the negligible and non-negligible function epsilon and how it relates to the predictability of pseudo ...
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2answers
32 views

Exponentiation in Modular Arithmetic

I feel like this is a fairly straightforward question, but I've been having a great deal of difficult computing one modular arithmetic expression. It's this: $9 ≡ 3^a \pmod{17}$ How does one go ...
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3answers
38 views

affine cipher $ax+b \mod m$

I have an affine chipher $ax+b \mod m$ For what values $a,b$ is this an injective encryption function? From what i understand thats the case when $a$ and $m$ are coprime, so $gcd(a,m)=1$ and the ...
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1answer
52 views

Artificial Integer?

Consider a function $$ f: \Bbb{Z} \rightarrow \Bbb{Z} $$ Over the integers. Furthermore consider a number E such that there doesn't exist an integer R such that $f(R) = E$ or formally stated $$ E ...
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0answers
41 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 $\operatorname{HMAC_r(s)}$ ...
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0answers
74 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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2answers
53 views

Euler's Totient Function and Cryptography Question

I'm working on a problem set for a class on intro computing and cryptography. I'm being asked to find the $n = pq$, where $p,q$ are integers (not necessarily prime), such that $\phi(n)=$ ...
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0answers
35 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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0answers
30 views

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 ...
14
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2answers
299 views

Are there practical applications to the new prime pair proof?

I've recently heard that its been proven that the set of prime pairs that are separated by no more than 70,000,000 is infinite. Does this have any impact on cryptography or another practical ...
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0answers
21 views

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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2answers
86 views

Points on elliptic curve over finite field

Find the points on the elliptic curve $y^2 = x^3 + 2x + 2$ in $\mathbb F_{17}$. Do I have to guess a first point and then use an algorithm to spit out all other points?
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1answer
33 views

Square Roots in Number Theory

Let N=pq, with p and q primes, with p congruent to 7 mod 8 and q congruent to 3 mod 8. We have seen in class that if h is relatively prime to N, then h=efs, where e= 1 or -1, f= 1 or 2, and s=r^2 for ...
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0answers
26 views

Tidy way to represent XOR over the ring of $2^{32} - 1$

I was reading about a cipher called Speck, which defines a system of equations using Addition Mod $2^{32}$ ($\boxplus$), Bit Rotation, and XOR. If we pretend that the additions were taken over ...
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0answers
15 views

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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0answers
65 views

What minimum subset of fields of mathematics is needed to understand homomorphic encryption?

Without the luxury of full undergraduate training in mathematics, if one worked part time could the community list the smallest set of mathematical fields needed to understand homomorphic encryption? ...
0
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1answer
12 views

What exactly does it mean that the key $e \in \mathcal K$ uniquely determines $E_e$?

What exactly does it mean that the key $e \in \mathcal K$ uniquely determines $E_e$ ? Does it mean that for each $e \in \mathcal K$ there exist only one function $E_e$ corresponding to $e$ ? Does ...
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0answers
37 views

Elliptic curve cryptography order

How do I compute an order a a point P on an elliptic curve? My question is specifically in reference to the attached photo. I understand how to do part a but I am totally lost in part b. I don't know ...
0
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1answer
62 views

Number theory used in cryptography [duplicate]

I am entering the realm of cryptography and encountering Number Theory related stuff a lot (As expected). I have a good knowledge and background on mathematics but I have been away for a while. So if ...
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2answers
41 views

In (m mod n = x) how to find m when you know n and x?

So I'm doing some cryptography assignment and I'm dealing with a modular arithmetic in hexadecimal. Basically I have the values for $n$ and the remainder $x$, but I need to find the original number ...
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1answer
67 views

how to solve $x^2 \equiv a \pmod{ n}$, where $n = p_1 p_2 \dots p_r$

Let $p_1,p_2, \dots , p_r$ be different odd prime numbers, and $n$ be the multiplication of them $n = p_1 p_2 \dots p_r$. Let $$a \in \mathbb{Z} / n \mathbb{Z}$$ and assume that $\gcd(a,n)$ is ...
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2answers
24 views

Help with linear recurrence.

I am trying to understand the following problem: Consider the following linear recurrence over $Z_2$ of degree four: $z_{i+4} = (z_{i+3} + z_{i+2} + z_{i+1} + z_{i}) \bmod 2$ i >= 0. ...
0
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1answer
34 views

Stream ciphers - Block ciphers

What is the difference between the stream ciphers and the block ciphers?? Is the difference the time complexity?? At the block ciphers the message is cut into parts of $n$ characters. If we have ...
0
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1answer
47 views

Who to solve this linear modular equation system?

I have this equation system: a + b + c (mod 11) = 8 9a + 3b + c (mod 11) = 2 16a + 4b + c (mod 11) = 9 Unfortunately I totally don't know how to solve it. It is in general part of Lagrange's ...
3
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2answers
79 views

In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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0answers
77 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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2answers
62 views

Function for encryption/decryption - What is $n \phi(n)$?

In my notes there are the following functions of encryption/decryption: $$E_k(x)=x+k$$ $$D_k(y)=y-k$$ ($E_k : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$) ($D_k : \mathbb{Z}_n \rightarrow ...
0
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1answer
33 views

How many multiplications are needed when one applies the algorithm to computing $132^{1023} \mod 2047$?

Suppose the integer $a$ has binary representation $b_kb_{k-1}\cdots b_1b_0$ where $b_i$ are either $0$ or $1$. The power $x^a \mod n$ can be computed using the fast exponential algorithm. The ...
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2answers
59 views

Where does 2525 and 252525 come from in RSA cryptosystem example?

This is an example from Discrete Mathematics and its Applications I understand how to encrypt, the first step is to turn the letters into their numerical equivalents(same thing we had to do for ...
0
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1answer
27 views

How to recover the original text/find decryption function?

This is from Discrete Mathematics and its Applications Here's my book section on shift ciphers. I understand the idea behind this. If you were trying to encrypt say a single letter 'b' with a ...
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4answers
84 views

How to prove this modular multiplication property to be true?

I am watching a youtube video on modular exponentiation https://www.youtube.com/watch?v=sL-YtCqDS90 Here is author's work In this problem, the author was trying to calculate $5^{40}$ He worked ...
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3answers
130 views

Is this proof for $(\mathbb{Z},+) \ncong (\mathbb{Q},+)$ valid?

In an introductory cryptography course, our teacher demonstrated a proof for $(\mathbb{Z},+) \ncong (\mathbb{Q},+)$. I'm not convinced, even though the statement may be correct (I don't know). ...
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1answer
46 views

Generating elements of a Galois Field using an irreducible polynomial

I am practicing some cryptography problems and I am having problems with one involving Galois Fields and irreducible polynomials. Here is the problem: Using the irreducible polynomial $f(x) = x^5 ...
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0answers
22 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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1answer
64 views

Finding the Order of a group and the Order of each element

I am working on a cryptography example problem. The problem is the following: For the group G = < Z26*, x> a) find the order of the group b) find the order of each element in the group c) Is ...
1
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1answer
56 views

Suppose that you had a machine that could find all four solutions for some given $a$. How could you use this machine to factor $n$?

Question: Suppose $n = pq$ with $p$ and $q$ distinct odd primes. Suppose that you had a machine that could find all four solutions for some given $a$. How could you use this machine to factor $n$? ...
2
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1answer
67 views

Prove that if the equation $x^{2} \equiv a$ (mod $n$) has any solutions, then it has four solutions.

Question: Suppose $n = pq$ with $p$ and $q$ distinct odd primes. Suppose that gcd($a,pq$) = 1. Prove that if the equation $x^{2} \equiv a$ (mod $n$) has any solutions, then it has four solutions. ...
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0answers
48 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?