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

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Montgomery and Galois fields

I'm a little bit confused about the design of a RSA module in VHDL. My question isn't directly related to hardware design. I've read a lot of publications and I bought also a book. In one publications ...
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109 views

Why are some $x$ coordinates unsuitable for an ECDSA generator point?

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ...
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2answers
72 views

Formal security of recycled random blinding in a Paillier scheme

This question is a follow-up/variant on a previous question. Supposing that we are trying to generate a large number of (indistinguishable) ciphertexts of a given plaintext and want to avoid the ...
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45 views

How bad would it be to reuse the random blinding factor in a scheme like Paillier?

A secure and somewhat fast way to "re-encrypt" (refresh? anonymise?) a Paillier ciphertext, $c$, is to multiply it by an exponentiated random value: $c \gets c \cdot r^n \mod n^2$ (with $r \in ...
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63 views

Programming language for modular arithmetic over large numbers [closed]

I'am trying to implement algorithms on integer factorization.This involves dealing with integers of 200-500 digits and doing modular arithmetic over them.Which programming language has inbuilt support ...
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1answer
120 views

Montgomery Algorithm

I'm trying to undestand how it works and how to implement the algo described in this paper. The paper shows a methods to compute a modular multiplication where it is used multiplier with a resolution ...
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1answer
50 views

subgroup of quadratic residue

if a number is said to be a subgroup of a quadratic residue of Z∗p, can I affirm that it is a generator of a cyclic group ? ie, say i am looking for a number which is an element of order q in Z*p, ...
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205 views

Random Galois fields

Note: I have now answered this by my own research and can generate random fields up to G(2^9) in reasonable time. I would need to find more speedups for larger fields. At this time G(2^8) takes a few ...
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59 views

RSA - Why should we sign hash rather than raw content? [duplicate]

Apart from obvious performance considerations, is there any mathematical or crypto reasons that imply that we sign a hash rather than a raw content ? I read that it was because the hash must fit in ...
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64 views

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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2answers
67 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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1answer
50 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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1answer
68 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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1answer
84 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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1answer
67 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
69 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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49 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
88 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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31 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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67 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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2answers
95 views

Point decompression on an elliptic curve

I'm programming an elliptic curve cryptosystem and I'm having difficulty with decompressing points. The following information is from my project specification as to my understanding: Given a point ...
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1answer
86 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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196 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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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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481 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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625 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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219 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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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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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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100 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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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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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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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 ...
2
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407 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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96 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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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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666 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
71 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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374 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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209 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
107 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
193 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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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
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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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2answers
210 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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178 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 $r$ ...