0
votes
0answers
39 views

RSA Decryption - finding Private Key

Let p and q be primes and $e \in \mathbb{Z}^+$, with $\gcd (e, (p-1)(q-1)) = 1$. Let d be the inverse of $e \mod (p-1)(q-1)$. The decrption process where M is plaintext and C is ciphetext ...
0
votes
1answer
33 views

Modular arithmetic to find the mod of a large number

If $x \equiv 23 \bmod 317$ and $x \equiv 25 \bmod 331$, what is $x \bmod 104927$? What techniques are typically used to solve problems of this nature? It doesn't seem clear to me that it can be solved ...
1
vote
1answer
96 views

How do I row reduce a matrix mod 26 when it is singular mod 26?

Cryptography assignment question: matrix $A$ is \begin{equation} A = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 3 & 1 \\ 0 & 2 & 5 ...
0
votes
1answer
64 views

ElGamal Public Key Cryptosystem and Digital Signature Scheme

I'm tryting to understand how ElGamal algorithm works, and I got the following example, and I couldn't understand one part of this: A) P=23, g=5. B) x=3, then y=10 (for 53 mod 23=10 ). C) Sign for ...
0
votes
2answers
76 views

Explanation of $d^{-1}$ in modular arithmetic [duplicate]

I wasnt quite sure what to name this question, so that's what it is. I'me working on an encryption system, and I need modulus. I already asked a question on this, here, and I cannot figure out the ...
0
votes
0answers
129 views

Why does RSA have to use Euler's Totient function?

$$\begin{aligned}m^{ed} &\equiv m\bmod n\\ ed &\equiv 1 \bmod \phi(n)\\ \end{aligned}$$ Why does the modulus of the modular multiplicative inverse have to be the totient function? Won't any ...
0
votes
3answers
57 views

Which divisors produce unique moduli? (for RSA encryption)

Sorry if this question is confusing, I'm still confused by the whole thing. I'm trying to understand how RSA encryption works, but I'm having trouble with the modulus part. For RSA to work, $c=m^e ...
0
votes
1answer
51 views

Find the RSA factorization

I want to solve this exercise: Assume you have to do with an RSA System whose public parameters are (n,e)=(55,17). Now you can compute d. -->That's easy I've got d=33. You know a computer uses CRT ...
1
vote
2answers
39 views

Modulus Function

I am watching a tutorial an i saw how to use the modulus they said if 20/7 = 2.8571422857 you must subtract the whole number then multiply it by the divisor now am trying to understand a Public key ...
2
votes
1answer
33 views

Given four numbers $(a,b,e,n)$ is it possible to find $k$ such that $k^{e}a = b \pmod n$?

Let $n$ be a large given number. Also, $n = pq$ for some unknown primes $p,q$. The Euler Totient function, $\varphi(n) = (p-1)(q-1)$ is not known or easy to calculate. $e$ is a given number co-prime ...
1
vote
1answer
18 views

Given three numbers $a, b$, and $n$, is it possible to find a number $k$ such that $ka\equiv b\pmod{n}$?

I came up across this problem working on some (purely academic) attack on cryptographic schemes. Given any three integers $a,b,n$ such that $a<n,b<n$, I am interested in an integer $k$ which ...
0
votes
1answer
358 views

RSA encryption/decryption scheme

I've been having trouble with RSA encryption and decryption schemes (and mods as well) so I would appreciate some help on this question: Find an e and d pair with e < 6 for the integer n = 91 so ...
5
votes
3answers
64 views

Public Key Cryptography

Assuming that a message has been sent via the RSA scheme with $p=37$, $q=73$, and $e=5$, what is the decoding of the received message "34?" So far, I have $x^5 \pmod{37\times73} \equiv 34$. How do I ...
0
votes
0answers
50 views

If $n$ is a Carmichael number then there exist at least one $a: a^{(n-1)/k} \equiv 1$ (mod n)

If $n$ is a Carmichael number then there exist at least one $a: a^{(n-1)/x} \equiv 1$ (mod n) such that $a^{n-1} \equiv 1$ (mod $n$) and x is prime such as $x |(n-1)$. I am solving the bigger proof ...
1
vote
1answer
46 views

computing the discrete log of $23^x \equiv 102 \pmod {431}$

I've been working on this problem for a while now. Could someone please help me see where I'm going wrong? "Alice and Bob agree to use a Diffie-Hellman key exchange with values p = 431 and primitive ...
0
votes
2answers
146 views

What is a perfect square in mod n

I have been stuck with a question on eliptic curves lately. I need to know whether perfect square mod n is different than a normal perfect square. And also is 3 a perfect square in mod 13?
2
votes
3answers
61 views

computing $2^{170}+ 3^{63}\pmod {19}, 3^{175} + 2^{73} \pmod {17}$, etc… by hand

I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for ...
0
votes
0answers
42 views

Must the “n” in mod(n) always be prime?

I'm experimenting with mod(n) and have the following questions even after reading the Wiki page and numerous articles about the subject. Must mod(n) always be prime for cryptographic purposes? Is ...
1
vote
2answers
44 views

Proving that if $ed ≡ 1 \pmod{\frac12 φ(n)} $, then $y^{ed} ≡ y \pmod{ n}.$

This is actually the third step of the problem. It's preceded by these questions that I'm sure are supposed to lead me to solution. $n = pq$, p and q distinct odd primes First I'm supposed to show ...
1
vote
3answers
220 views

RSA encryption without a calculator

I'm doing an RSA encryption and to get part of the solution I need to solve $$C=18^{17} \pmod{55}$$ How would I solve this problem without a calculator Thanks in advance
3
votes
4answers
159 views

how do I calculate inverse modulo of a number when the modulus is not prime?

I came through Fermat's Little theorem, and it provides a way to calculate inverse modulo of a number when modulus is a prime. but how do I calculate something like this 37inverse mod 900?
2
votes
2answers
98 views

Why modular arithmetic in secret sharing?

I learned about how secret sharing works in my math class today. From what I understand about the way I was taught it's possible to implement it, I can choose a secret number $N$ and generate a ...
2
votes
1answer
80 views

Modular exponentiation WITHOUT modular exponentiation

Given that 719 is prime, find the least positive residue of $11^{721} (mod\ 719)$, without using modular exponentiation. So, I know how to use modular exponentiation and have done it to get the ...
12
votes
6answers
641 views

How to find the inverse of 70 (mod 27)

The question pertains to decrypting a Hill Cipher, but I am stuck on the part where I find the inverse of $70 \pmod{ 27}$. Does the problem lie in $70$ being larger than $27$? I've tried Gauss's ...
0
votes
1answer
206 views

RSA and calculating huge exponents

I am writing an Extended Essay on RSA encryption and in the essay, I am going through a worked example of all of the stages involved (key generation, encrypting and decrypting). I am using very small ...
1
vote
0answers
34 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 $A \in \mathbb Z^{n \times m}_q$ ...
3
votes
1answer
97 views

Is the number of quadratic nonresidues modulo $p^2$, greater than the number of quadratic residues modulo $p^2$?

Let $p$ be a prime. The number of quadratic nonresidues modulo $p^2$, is greater than the number of quadratic residues modulo $p^2$. Is that statement true or false? Why? Thank you.
0
votes
2answers
48 views

Determine amount of congruent numbers

I found the claim in a paper that there are at max 8 integers mod $2^{130}-5$ congruent to one integer mod $2^{128}$. $$u \pmod {2^{130}-5} \equiv g \pmod {2^{128}} \quad\text{ with }u \in U \quad ...
1
vote
1answer
59 views

RSA encryption theory - modulo theory

I'm a bit mathematically challenged and have been working on the RSA cipher (good start). I can find the public and private keys and know how to work do modulo operations on a calculator. The problem ...
1
vote
1answer
50 views

Maximum order for $x$ in $g^x \equiv 1 \mod {n}$, when n=pq

I am currently trying to learn about the ElGamal Digital Signature scheme. It is based on the discrete logarithm problem, where it is computationally infeasible to find $x$ in $y=g^x \mod{p} $), if ...
1
vote
2answers
105 views

Why is a prime number needed for the Diffie-Hellman key exchange? (modular arithmetic)

I'm writing a cryptography essay, and am wondering why you need a prime number for the deffie-hellman key exchange? Any help would be appreciated :) this is a link to a previous post which quickly ...
1
vote
1answer
109 views

How to solve exponential format modular equation have the same base

I'm reading the paper of Taher Elgamal whichs talks about his digital signature scheme. For example a user needs to sign a document $m \in [0, p-1]$ where $p$ is a large prime number. His private key ...
4
votes
0answers
50 views

Prove that there are $736$ $2 \times 2$ matrices ($A$) where $A=A^{-1}$ [duplicate]

I'm doing some assignments to teach myself cryptology. I am still at the introductory cryptology level, where a lot of it is discrete mathematics, so I believe - and hope - that it is a somewhat ...
11
votes
5answers
301 views

Why are there$ 736$ $2\times 2$ matrices $(M)$ over $\mathbb{Z}_{26}$ for which it holds that $M=M^{-1}$?

I'm currently trying to introduce myself to cryptography. I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for ...
1
vote
1answer
86 views

Why just 4 square roots given ($x^2 \bmod N$)

Oblivious transfer algorithm's page on Wikipedia claims: The receiver picks a random $x$ modulo $N$ and sends $x^2 \bmod N$ to the sender Note that $\gcd(x,N)=1$ with overwhelming probability, ...
0
votes
2answers
72 views

Why is it safe to assume M < all Ns in Håstad's Broadcast Attack

I am reading the Wikipedia article on Broadcast attack. In the prove, the editor made an assumption that M is less than all N. Why is this assumption safe?
0
votes
2answers
59 views

System of Linear Equations for Congruency

So this is my question: Find all x such that $4x=3 \pmod{21}$, $3x=2 \pmod{20},$ and $7x=3 \pmod{19}$ So I know I have to use chinese remainder theorem and I know how to do it if $x$ didn't have a ...
2
votes
1answer
205 views

System of Linear Equations using Mod

I just want to check that I did a certain problem correctly. This is it: $$a+b=3 \pmod{26}\\2a+b=7 \pmod{26}$$ Solve for $a$ and $b$ Now I setup the augmented matrix: $$\left[ \begin{array}{ccc} 1 ...
1
vote
0answers
132 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 ...
1
vote
2answers
343 views

Solving $4$ simultaneous linear equations modulo $26$

I'm trying to solve these equations to solve a ciphertext which is encrypted by the Hill Cipher. I tried to solve these equations algebraically. (First choose two of them and try to eliminate one ...
0
votes
1answer
190 views

Simple modulus algebra - rabin karp weird implementation

I'm studying the Rabin Karp algorithm and something isn't clear about the modulus algebra: Let's suppose I have all base-10 numbers for simplicity's sake $14159 = (31415 - 3 \cdot 10^4) \cdot 10 + ...
0
votes
2answers
56 views

Security of a particular cryptosystem

I recently came across this problem, and while I'm fairly certain the solution is not too 'conceptually-challenging', I've been stumped at finding the right trick/manipulation to make any solution ...
0
votes
2answers
152 views

Need help with finding matrix inverse in $\mathbb{Z}/26\mathbb{Z}$

I am trying to learn the Hill Cipher and I am facing difficulties understanding how to find the inverse of a matrix in Modulo 26. What I've learnt so far is that I need to apply elementary row ...
1
vote
2answers
218 views

Power computation in modulo

I have a confusion regarding power computation in modular arithmetic Lets say I want to compute $(7^5)^4 \pmod {17}$ there are many ways to compute this and I get different answers with each ...
16
votes
1answer
17k views

Finding a primitive root of a prime number

How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
1
vote
1answer
399 views

RSA Decryption of a huge cipher text and exponent

I am trying to work an assignment question I have been stuck on for some time. The question is to decrypt the message with given decryption key and mod $n$. $$374484638351^{320986308343} \bmod ...
0
votes
3answers
1k views

How to calclulate multiplicative inverse of e mod $\phi(n)$?

In this wikipedia article about RSA, At step 5, How are they calclulating value of $d$? Can anybody give me a step-by-step explanation? Compute $d$, the modular multiplicative inverse of $e ...
0
votes
2answers
90 views

modular cipher proof

the above is a textbook question I found and believe it is very similar to what I have except n=1 mod p-1 and that remainder 1 is something I dont have in my question... I am terrible at proofs but ...
2
votes
1answer
357 views

Question about finding the nth root in a modulus.

Some notation: $\mathbb Z_n$ denotes the set of integers up to n i.e. $\mathbb Z_n=\{i│i ∈ \mathbb Z,0≤i < n \}$. $\mathbb Z_n^*$ denotes invertible elements of $\mathbb Z_n$ i.e. $\mathbb ...
1
vote
0answers
65 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 ...