Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value… the modulus.

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How to verify the result of repeated squaring method

I am wondering that is there any way to check that the result of repeated squaring is correct or not Suppose we have a calculator (like Casio FX 570ES, ...), and we want to calculate x^d ≡ n If any ...
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63 views

Security concern about reducing hash value using modulo operation

As stated in the title, what I am looking for is information about a "technique" that I would like to use in some of my algorithms. Sometimes I need to map a hash function's result into a range of ...
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48 views

Modular Reduction of polynomials in GF(2^m)

Hello I am having trouble understanding the algorithm implementation in hardware of the reduction process over galois fields of F(2^163) In the following process it ...
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66 views

For which RSA moduli, precisely, is $e=d$ for all $e$?

This question shows that there are at least two valid RSA moduli $n$, namely $35$ and $91$, such that for any $e$ coprime to $\lambda(n)$, $$e^2\equiv1\mod\lambda(n)\text.$$ Reading the linked ...
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83 views

Polynomial division hardware implementation

I am beginning the implementation of the polynomial binary division algorithm now as I understood i will be checking the MSB bit if 1 to XOR and shift the sum if 0 i will only shift. What I am not ...
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65 views

Polynomial Modulus

So I am studying for finals and I came across this problem and I am completely stuck. I would really appreciate someone to clarify on the steps to take to solve these problems. Let $f(x) = x^4 + x^3 ...
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1answer
66 views

Fermats Little Theorem, primitive root [closed]

So I am studying for finals and I am not able to solve the problem: Let $ p = 3 * 2^{11484018}- 1 $ be a prime with 3457035 digits. Find a positive integer $x$ so that $2^x\equiv 3\pmod p$ Any ...
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44 views

Polynomial Inversion over Galois Field

Hello guys I am looking to calculate the Inverse of a given polynomial in Galois field I have found the little Fermat's algorithm and the Itoh-Tsujii I am getting a bit confused with both algorithm ...
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1answer
85 views

Roots in modulo field

I have a point $(X,Y)$ on an elliptical curve $E(a,b)$ where $a=-3$ and $B$ is a large number that is in hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ ...
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1answer
25 views

Modulo Square Roots [duplicate]

Here's my issue and someone can help me understand it so I can program it correctly. I have a point(X,Y) on an Elliptical Curve E(a,b) where a=-3 and B is a large number that is in hexidecimal from ...
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63 views

Arithmetic modulo 1

In the context of lattice-based cryptography, in particular the Learning With Errors (LWE) problem, I see some definitions given in terms of equations modulo 1 (see for example, Appendix A of this ...
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1answer
82 views

Addition / Multiplication modulo 13

I am currently studying Diffie-Helman protocol. In my lecture notes I have a generator g=2 and I'm in group $Z_{13}^*$ . There I calculate a remainder cycle from 1 to 12 with $r =g^i \mod 13 \forall ...
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1answer
193 views

Square roots, prime factorization

I´m stuck in my homework and since its homework, I would rather get some hints than full solution. The problem goes: factorize n = 88416763 in case that you know that the square roots of 51733469 ...
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1answer
80 views

Showing that $2^{N-1}\equiv1\pmod N$ when $N=2^p-1$ for prime $p$

I got this question on a previous exam and I got it wrong. I've gone back through it several times since then, but I can't seem to get it. I would really like to know how to do it, so if someone could ...
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2answers
99 views

RSA prove $a^{\varphi(n)/g} \equiv 1 \pmod{n}$

I should prove that $a^{\varphi(n)/g} \equiv 1 \pmod{n}$, so I thought I would prove it with primitive roots but I got stuck. Here is the assignment: Let $p, q$ be primes such that $p \ne q$. ...
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2answers
438 views

Reversing DJB2 Hashes

I have some spare time, and a few hundred DJB2-hashed values sitting around. I thought I'd try to do something "useful" and invert DJB2, such that I could calculate the plaintext of the hashes (which ...
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3answers
623 views

RSA with modulus n=p²q

I've been wondering what would happen if the RSA modulus would be $n=p^2q$ instead of $n=pq$. I feel like there is an obvious security flaw (as there is with most modifications). Would this choise of ...
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2answers
211 views

In RSA, why does $p$ have to be bigger than $q$ where $n=p \times q$?

In openSSL – during RSA key generation – if $q$ is bigger than $p$, they exchange them. Why is that?
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396 views

Exactly two of the four roots must be greater than N/2

Theorem: Let $y$ be a quadratic residue in $\mathbb{Z}_N$* where $N=pq$. There are exactly four integers $x_1, x_2, x_3, x_4$ where $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ ...
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89 views

Modulo settings for successful encryption?

I saw this awesome video which shows how encryption works using "discrete logarithm". The example says: $3^x\mod17$. I understood that $3$ is called “generator”, because it has no "straight" root and ...
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1answer
88 views

How can I find the multiplicative inverse in the first transformation of the SubBytes() transformation in AES?

In FIPS-197 §5.1.1 , it says the first transformation in the SubBytes() transformation is: Take the multiplicative inverse in the finite field $\text{GF}(2^8)$ described in Sec. 4.2; the ...
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71 views

RSA: Common modulus attack problem [duplicate]

I understand in theory how the common modulus attack works (as described here: how to use common modulus attack?) Though, I did not understand completely how it worked with a negative $s_i$. Since ...
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1answer
145 views

ECDSA: How to retrieve a non-random k

I have a question regarding the random $k$ number of ECDSA encryption. As far as I know, it is possible to retrieve $k$ (and thus the private key) from two signed messages if both used the same $k$. ...
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120 views

Three different numbers with x³=x mod p

p is a prime greater than 2 and $a \in \mathbb{Z}_p$. Why are there exactly three solutions for a³ = a mod p? Obviously 0 and 1 are both in $\mathbb{Z}$ and valid solutions, but that still means, ...
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1answer
106 views

Modulo properties of two prime numbers

I am supposed to prove that x = y mod (p*q) <=> x = y mod p and x = y mod q with p and q are prime numbers. It somewhat sounds reasonable to me, but unfortunately I don't have any clue how to prove ...
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105 views

Proxy re-encryption mod operations

I managed to implement the proxy re-encryption scheme from http://eprint.iacr.org/2009/189.pdf in python 2.7, however I am having performance issues. As it is, I can run the algorithm for key sizes up ...
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1answer
376 views

RSA: Fermat's Little Theorem and the multiplicative inverse relationship between mod n and mod phi(n)

I'm learning about the proof of the RSA encryption algorithm, and I'm clearly fudging or missing something, because for me it doesn't add up. When generating keys for RSA encryption, we make sure ...
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1answer
92 views

ElGamal digital signature. why mod p-1 in delta?

Im trying to figure out why $\delta$ in the signature part of ElGamal has mod $p-1$ when $\gamma$ has mod $p$? $\gamma = \alpha^k \mod p$ $\delta = (x-a\gamma)k^{-1} \mod p-1$
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1answer
233 views

modulo operations in crypto algorithms

Am not a mathematician. Every crypto specification I see uses the modulo operation. For example RSA - If $e$ is the public key and $m$ is the plaintext with a modulus $n$ - the cipher text is $c = ...
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4answers
621 views

Pseudocode for constant time modular exponentiation

I'm looking to implement modular exponentiation (for RSA) in constant time, but most of the examples I've found are more mathematical descriptions of the operations. Are there any references with ...
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1answer
70 views

Can I publish the accumulator trapdoor and still keep it secure?

I'm implementing the Camenish-Lysyanskaya dynamic accumulator. It seems to me that the accumulator is provably secure because the trapdoor is unknown to the attacker. The risks should be the fact ...
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321 views

How to calculate Modulo using a standard calculator for a one time pad encryption

I wrote a simple python script to preform one time pad encryption I would like to be able to decrypt the ciphertext by hand. In my script I use mod 26. This is how I was shown to calculate a mod ...
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203 views

Why does this square root algorithm work?

I've been doing some elliptic curve cryptography, and a library I'm using has this slightly bizarre algorithm for computing modular square roots: Let $x$ be some quadratic residue modulo $p$ for some ...
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1answer
106 views

Modular exponentiation with Chinese Remainder Theorem

I'm learning modular exponentiation with Chinese remainder theorem. I found a great answer from below How can I use eulers totient and the chinese remainder theorem for modular exponentiation? But I ...
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1answer
182 views

Fast modular reduction

I am looking at ways to speed up modular reduction for the polynomial $$2^{256}-2^{32}-2^9-2^8-2^7-2^6-2^4-1$$ I have read the paper "Generalized Mersenne numbers" by J.A. Solinas, but it does not ...
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103 views

Gallant-Lambert-Vanstone method

I am experimenting with the GLV method but cannot manage to get the right results according to the literature. I managed to find lambda, beta, split $K$ into $k_1$ and $k_2$ etc. for the curve I'm ...
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1answer
248 views

Decode message $m = p * q$, where $p ^ 5 \bmod N$, $q ^ 5 \bmod N$ and $N$ are known

I been going through some cryptography exercises and stumbled across this problem. Discover message $m$. You know that $m = p * q$. Also $p ^ 5 \bmod N$, $q ^ 5 \bmod N$ and $N$ are known ...
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75 views

$f : \mathbb{Z}_n \rightarrow \mathbb{Z}^\times_n$?

Is there any function $f : \mathbb{Z}_n \rightarrow \mathbb{Z}^\times_n$ that is invertible? By invertible, I mean it given $y \in \mathbb{Z}^\times_n$, it should be easy to find $x \in \mathbb{Z}_n$ ...
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209 views

PRP, PRF and modular arithmetic

Are there any arithmetic or mathematical functions that can be used as PRPs or PRFs ? Since, Conventional block ciphers like AES are that are proven to be good PRPs are not based on mathematics but ...
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172 views

Partially-known-plaintext attack of a stream cipher based on modular arithmetic

I have a function called $F$, using modular arithmetic as does RSA, defined as $$x\mapsto F(x) = g^x\bmod p$$ where $p$ is a 1024-bit prime and $g$ is a generator of $\mathbb Z_p^*$. A secret key $r$ ...
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Proof for exponentiation in modular arithemtic

If $e$ is a natural number, then this is true: $$m^e \bmod\ n = (m\bmod\ n)^e\bmod\ n$$ This is often used when encrypting, especially with RSA, since one can avoid directly calculating $m^e$, ...
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3answers
523 views

How do institutions like banks do RSA with big primes?

When encrypting with RSA it is often infeasible to decrypt by just doing c^d mod n, because for example when using the primes $(p,q)=(12553,1233)$, which are small ...
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2answers
238 views

Avoiding overflow when encrypting with RSA

When encrypting with RSA one calculates $ m^e \pmod n $ by doing the following: m^e % n Where $m$ is what we encrypt. Often $e$ is a very big number to make it ...
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149 views

Shamir Secret Sharing Modular Reduction

Say players use Shamir's secret sharing to share a value $k$ such that each player now holds $k_i$, a share of $k$. How can they securely compute $k \bmod m$ for some $m$. Of course they can ...
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72 views

Implementation of modular arithmetic?

In FIPS 186-3 appendix D.2 "implementation of Modular Arithmetic", they show shortcuts for solving $$B = A \mod m$$ for select Curves. How would you go about determining a valid short cuts for the ...
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1answer
558 views

ECC - Point Addition/Point Multiplication

So I have a very beginner-esque knowledge of ECDSA and I'm trying to write something in python to take a private key and output the public key (Basically from what I understand just trying to do the ...
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1answer
165 views

Why encrypting with private and public keys produce the same result?

Let say $e = 5$, $n = 119$ and $d = 77$. If I encrypt, for example, $m=15$ I get: $$m_1 = m^e=15 ^ 5 \mod 119 = 36\qquad\text{and}\qquad m_2 =m^d= 15 ^ {77} \mod 119 = 36$$. Why? Is it always like ...
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419 views

Where does the $\varphi(n)$ part of RSA come from?

$e d \equiv 1 \pmod{\varphi(n)}$ Where does the $\varphi(n)$ part come from? How did the inventors of RSA arrive at $\varphi(n)$?
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490 views

How does Clifford Cocks 'Non-Secret Encryption' work?

I have read Clifford Cocks "A Note on 'Non-secret Encryption'" and thought I would try to implement this, but I don't seem to be able to get it to work. I'm obviously missing something. From the ...
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2answers
162 views

How can I solve congruence modulo N?

I am trying to solve congruences of the form $$J_A \cdot a^e\equiv 1 \pmod n$$ where $n=pq$ for $p,q$ prime and $\gcd(e,\varphi(n))=\gcd(J_A,n)=1$ Solve for $a\in \mathbb{Z}$, in terms of $n,J_A$ ...