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0answers
23 views

(Counting problem) very interesting Modular N algebraic eqs - for combinatorics experts [migrated]

We have some attempt to numerically solve this math problem, which means that we like to count the number of independent solutions of this set of six of modular N algebraic equations: $$ (1) x_1 ...
4
votes
1answer
105 views

Polynomial GCD over a ring (with composite characteristic)

I'd like to implement the "Franklin-Reiter Related Message Attack" (see section 4.3 of Boneh's paper). As part of the implementation, I require to compute the GCD of two polynomials over ...
1
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1answer
237 views

Faster GCD Implementation

Is there any chance to write a faster GCD than the built-in one in Mathematica? @Mr.Wizard has written one in this question (although it's not for this purpose) which is 6 times slower on a 100k ...
5
votes
1answer
722 views

Solving a system of linear equations modulo n

I have a system of linear equations $$ a+b+c \equiv 31 \pmod{54} $$ $$ 4a+2b+c \equiv 3 \pmod{54} $$ $$ 9a+3b+c \equiv 11 \pmod{54} $$ What should I input (I'm using ...
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0answers
83 views

Solving for $d$ in $x= E^d mod(n)$

With regards to Public Key Cryptography, I have been tasked with the problem of attacking some information given in an assignment and discovered a private key used to digitally sign a message. Known: ...
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0answers
95 views

Is there a way to speed up Simplify and/or PolynomialReduce Modulus-> 2?

I'm trying to simplify a series of equations with at most 64 input terms. As the number of terms involved in the equations increase, the runtime seems to grow exponentially. Does anyone know of ways ...
1
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2answers
293 views

Finding shortest non-zero vector $x$ satisfying $Ax=0 \pmod q$

Let $n$, $m$, and $q$ be positive integers (with $m > n$), and $A$ be a matrix over $\mathbb{Z}_q^{n \times m}$. Using Mathematica, I want to find the shortest non-zero vectors $x \in ...
4
votes
2answers
387 views

Inverse of a polynomial in a polynomial ring

Let $N$ be a prime, and $q$ be a positive integer. Given a polynomial $f(x)$ in $R = \mathbb Z[x]/(x^N-1)$, I want to find another polynomial $f_q(x)$ in $R_q = \mathbb Z_q[x]/(x^N-1)$, such that ...
9
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1answer
431 views

Modular arithmetic - efficiently calculating the remainders of factorials

When working on this question regarding the divisibility of the sum of factorials, I decided to write some code to test "small values" of the problem using the following code. ...
9
votes
2answers
422 views

Solving/Reducing equations in $\mathbb{Z}/p\mathbb{Z}$

I was trying to find all the numbers $n$ for which $2^n=n\mod 10^k$ using Mathematica. My first try: Reduce[2^n == n, n, Modulus -> 100] However, I receive ...
5
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1answer
385 views

Montgomery Modular Exponentiation

I'm trying to write a Montgomery exponentiation based on this which can compete with Mathematica PowerMod. We know that PowerMod ...
2
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1answer
497 views

Linear Solve with Modular Arithmetic

I am interested in using LinearSolve[m,b] which will find a solution to the equation $m.x=b$, where I am in mod 2 arithmetic. Is there any way to perform this ...
0
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1answer
62 views

How can I seperate specific elements from the list? [duplicate]

Possible Duplicate: Question about MapThread and Dynamic I write a function name as inputFieldsList and it returns both ...
6
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1answer
119 views

Implementing Remainder Tree

I want to implement Remainder Tree based on this. With the answers on SE I've come up with: ...
3
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0answers
240 views

Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
9
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5answers
400 views

Incrementing a number where each digit has a different base

Let's say I have a list, for instance {10,5,3}, indicating the bases for each digit of my 3-digit number. Using this basis, if I wanted to increment {8,4,1} a couple of times, here's what I would get: ...
1
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1answer
469 views

Matrix Multiplication Modulo 2

I would like to perform matrix multiplication modulo 2. Hence, instead of the usual: A.B I did: ...
2
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2answers
357 views

decompose a number (less than 255) in a sum of powers of 2

Is there a built in function that would take a number and decompose it into a sum of powers of 2? The numbers will be non negative less than 256. For what it's worth I'm trying to understand a paper ...
6
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1answer
496 views

How to Simplify equations over a Ring with Mathematica?

For example, when we work over a ring, the equation x^3=0 does not imply x^2=0 or x=0, but ...
11
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2answers
469 views

Factorizing polynomials over fields other than $\mathbb{C}$

I'd like to take a polynomial in $\mathbb{Z}_5[x]$ of the form $ax^2+bx+c$ and factor it into irreducible polynomials. For example: Input... x^2+4 Output... ...