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

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What are some good resources to study Cryptography?

What are some good resources to study Cryptography? I have knowledge of abstract algebra .Is it enough to take up Cryptography as a special paper or I will have to undergo courses in some other ...
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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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1answer
19 views

Math behind perfect hash

I am reading material on cryptographic hash functions and it says "Collision resistant property : for a hash of length L, a perfect hash would take $2^{L/2}$ attempts." Can someone explain why? ...
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asymmetric cipher with multiple public keys [migrated]

asymmetric cryptography is well known. You generate private key. Using private key you generate public key. Then you can sign a message with private key and check signature using public key. I need ...
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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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3answers
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Why does (1/3) mod 3016 = 2011?

So I am taking a class where we are working on a cryptography section. Basically, the course says that: $$\frac 1 3 \mod(3016) = 2011$$ or when run through Python - modified with SciPi: $$\frac 1 3 ...
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1answer
24 views

How is de = 1 (mod ϕ(n)) calculated

I am reading RSA algorithm. So, I was writing a question but I saw this question and still couldn't understand it. If $$e\cdot d \equiv 1 \pmod{\varphi(n)},$$ then $$ed=k\cdot \varphi(n)+1, \qquad ...
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2answers
21 views

Proof DES is injective - is this a valid argument

Without going too much into detail into the crpytography of the matter since not every mathematician is interested or knowledgable in the field, there is an encryption process called DES (data ...
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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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1answer
23 views

Is it possible to estimate the number of primes between 0 and a 128 bit number?

I'm attempting to visualize an RSA public/private key pair, or a SHA2 hash. In order to reduce that massive number that is meaningful to humans I'm looking at this SHA2 visualization function to ...
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6answers
6k views

Mathematically, why was the Enigma machine so hard to crack?

Mathematically, why was the Enigma machine so hard to crack? In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the ...
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2answers
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Calculating all Possible Keys vs All possible numbers confusion

With a key of length n bits, there are 2n possible keys. eg: 128-bit key length will have 2128 possible keys But when calculating every possible n digit number, ...
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1answer
126 views

How fast was the Turing's machine for breaking the enigma code?

We know that, recently, personal computers make around $10^9$ calculations per second, and I'm just curious about how many calculations was able to compute the machine invented by Turing for breaking ...
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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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0answers
42 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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1answer
27 views

Why wasn't the length of key mentioned in this algebraic notation of Vigenere Cipher?

Let, $M=m_ 1 m_ 2 m_ 3 ... m_ n$ and, $K=k_ 1 k_ 2 k_ 3 ... k_ m$ Then how algebraic notations of Vigenere Cipher should be? In the following pages key-length and message-length are shown same. ...
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0answers
18 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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1answer
120 views

What do these notations mean, if we read those in English?

If m: message, M: message space, k: key, K: keyspace, c: cipher, C: cipher space and $E_k$: encryption function, such that $E_k(m) = c,\ m,m^* \in M,\ k\in K,\ c\in C.$ Then, what do the following ...
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1answer
34 views

How can I calculate Index of Coincidence of Vigenère cipher?

I have computed the letter frequency of the cipher text. However, I don't know how to apply Friedman Test to Vigenère cipher. I couldn't calculate the Index of Coincidence. Does anyone can help to me ...
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2answers
34 views

RSA and extended euclidian algorithm

I'm learning about RSA, public private key stuff, and I just found a very nice article explaining the basics. ...
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0answers
37 views

Generator of group, Computation of discrete logarithm

The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the ...
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0answers
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Given plaintext and ciphertext of the same length, how could one generate potential symmetric keys if encryption algorithm is unknown?

This question is about both encryption and about how and if one could transform data from one given form to another given form and back. I am given plaintext and ciphertext, both of which are the ...
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1answer
33 views

Exponentiation for hash function & associativity

Some cryptographic papers use $H^n(x)$ to mean $H(H^{n-1}(x))$ where $H^0(x) = x$ and $H$ is a cryptographic hash. So $H^3(x)$ would be $H(H(H(x)))$. Is this definition formally correct? It seems to ...
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1answer
55 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
52 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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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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22 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
40 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 ? ...
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0answers
30 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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0answers
39 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
22 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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1answer
65 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 ...
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1answer
86 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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2answers
29 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
31 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
50 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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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
71 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
52 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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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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1answer
41 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 ...
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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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1answer
22 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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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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1answer
32 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
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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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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? ...
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1answer
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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 ...