Questions tagged [modular-arithmetic]
The modular-arithmetic tag has no usage guidance.
78
questions
2
votes
2answers
296 views
How to ask Mathematica to solve a simple modular equation
What's the most straightforward way to ask Mathematica to find all solutions of an equation like
$$3x + 2y + 4z = 0 \pmod {11}$$
(for instance), where either $x$, $y$, $z$ can be considered to be ...
3
votes
1answer
48 views
Can Mathematica automatically optimize an expression with PowerMod?
PowerMod[x,p,m] is significantly faster than Mod[x^p, m] for a large p
I have an expression, ...
1
vote
1answer
56 views
A modular version of LinearRecurrence?
I'm playing around with implementing the Lucas probable prime test (mainly so I can understand it better), and would love a version of the LinearRecurrence function that used addition modulo n (with n ...
0
votes
1answer
42 views
Find relationship between values within finate range of integer domain
Update:
There are some misakes in the coefficient of left and right.
But still get similar result.
I understand the mathematica ...
2
votes
1answer
41 views
How to simplify quotient and mod in integer domain
I want to simplify (a*32 + a%32)/32, and a is integer.
So I write:
...
2
votes
1answer
57 views
Non-valid modulus when using LinearSolve
I have a system of linear equations I want to solve mod 4, and I happen to know the solution, but I get an error when trying to solve it using LinearSolve. I define the matrix ...
11
votes
3answers
1k views
How can I program the last digits of Fibonacci numbers?
Is there a way to compute the last digits of an arbitrarily large Fibonacci number?
For the $10^n$th Fibonacci number, we can just find the $2^n$-th Fibonacci number (if that isn't too large) $\bmod n$...
2
votes
1answer
56 views
How to get the modulus of two functions [closed]
I might be overlooking something, because what I want to do is quite elementary.
I want the following term Mod[2(1 + z), (1 + z)] to evaluate to ...
1
vote
0answers
55 views
How to apply PolynomialMod to a SparseArray?
I have a list bdrs of SparseArrays, whose entries are polynomials in the variable t with ...
2
votes
0answers
42 views
Why gives Modulo of complex numbers different answers for rationals and reals
When running
Table[1.*Mod[(1 + I)^(1 + k*I), 1 + k*I], {k, 49, 50, 0.5}]
Table[1.*Mod[(1 + I)^(1 + k*I), 1 + k*I], {k, 49, 50, 1/2}]
I get for the first table
<...
4
votes
3answers
107 views
how can I calculate $a_{n+3}=a_n+a_{n+1}+a_{n+2}$?
recently I need to calculate this: $a(n+1)=((a(n-2)+a(n-1)+a(n)) \bmod 10000)$
and get $a(20000000)$ (for example).
I know RecurrenceTable,but
...
0
votes
0answers
52 views
Finding a solution to a modular problem efficiently
I want to find a naïve hash function of the form Mod[a x + b, 2^p] which would produce a value less or equal to v for all ...
5
votes
1answer
632 views
modulo with n instead of 0 without using if-statement
Is there a way to to express the following without using an if statement, but with mathematical operations instead?
{ 5 if x % 5 = 0, x % 5 otherwise }
2
votes
1answer
280 views
How to Plot modulo function Mod on a function of two variables? [closed]
I know that Plot[Mod[x, 2], {x, -6, 6}] is a Mathematica command to plot the remainder of x over 2 when I range x over reals -6 to 6. How do I plot the same ...
5
votes
1answer
98 views
FindEquationalProof to prove divisor theorem
I'm trying to implement proofs of concepts for Equational Proofs on some basic number theory theorems.
One such example is:
"let a and b be positive integers and let d = gcd (a, b). If t divides ...
1
vote
1answer
191 views
Polynomial PowerMod
Is there equivalent of PowerMod for polynomials in Mathematica?
We have Mod[a^e,m]==PowerMod[a,e,m], $a$, $e$ and $m$ all ...
3
votes
1answer
168 views
List of invertible congruence classes
I am attempting to create a list of the invertible congruence classes $\bmod 120$.
The code I have is ...
1
vote
1answer
67 views
Is this decryption function idiomatic?
This is my first non-trivial Mathematica notebook, which I wrote for a cryptology homework. It encrypts/decrypts strings using an affine cipher. Is the code readable/idiomatic? How could it be ...
4
votes
3answers
350 views
How to reverse in modular arithmetic
Given an RGB color {127, 255, 212} where scale is 0-255, VBA's RGB function find a long integer x such that
...
0
votes
1answer
44 views
Looping on ranges (Modular iterations)
Is it possible to make the following code modular, so that it works up till depth n, and returns an array of all possible combinations for used{}?
...
2
votes
1answer
154 views
How to solve a system with GCD?
For example,
Solve[{x + y == 150, GCD[x, y] == 30}, {x, y}, Integers]
Solve::nsmet: This system cannot be solved with the methods available to Solve.
The same ...
6
votes
1answer
375 views
RowReduce modulo 3
When I wanted to compute a row echelon form of a matrix over the finite field ${\Bbb F}_9$ with 9 elements (which is the three element field ${\Bbb F}_3$ adjoined a square root of $-1$ which I denote ...
0
votes
0answers
108 views
Variables defined modulo $\pi$ and avoiding singularities; phase plotting for a map
I have the following $2$-dimensional implicitly defined map
...
1
vote
1answer
116 views
Solving a modular equation with a large modulus [duplicate]
I am having trouble solving this equation:
$17^n\equiv 1 \mod 640826899755722841329632946651705039010285107743541637238400825304943424424715705661705191669759$
for the smallest integer $n > 0$
I ...
0
votes
1answer
51 views
Get missing cyclic numbers via modulus?
Consider a list of numbers:
list = {1,2,3,4,5};
Taking a random set of three of these numbers:
...
7
votes
2answers
1k views
How can I solve a certain congruence equation?
I want to solve the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$.
I know that one way to solve this is to first solve the congruence:
$t^2 \equiv 5 \pmod {19}$, because $b^2 - 4ac \equiv 24 \equiv ...
4
votes
1answer
244 views
Matrix operations “Modulus->non-prime” problem
I am using Mathematica 10.4.1.0. When I specify the "Modulus" option to be a non-prime, the functions RowReduce and MatrixRank return the error
...
2
votes
1answer
166 views
How to remove symmetries from this list?
First of all, I work with modular arithmetics and the elements of the arithmetics are lists themselves.
I recognize two kinds of symmetries - shift and mirroring. We could define shift of a list like ...
9
votes
1answer
176 views
Modular equation problem
I have problem solving this modular equation $67^n \equiv 67 \pmod {317026939759222841944}$ with $n>1$. I have tried my Laptop and Wolfram Alpha engine, but I don't get any solution, I'm very ...
1
vote
0answers
69 views
Last digits via PowerMod [closed]
Do you have an idea why this produces different results?
PowerMod[2003, 2002^2001, 1000]
241
...
6
votes
2answers
199 views
Memory problem when solving a system of modular equations [duplicate]
I need to solve the following system of modular equations, but the computation can't finish because I run out of memory (I have 12 GB of RAM). Is there any workaround to this problem? I am using ...
0
votes
0answers
53 views
Function to add two matrices mod 2 [duplicate]
I have:
lightstep[m_, gg_] :=
Which[gg == {1, 1}, m = Mod[m + m, 2]]
Then I entered:
A = {{1, 1, 0}, {1, 0, 0}, {0, 0, 0}}
...
2
votes
0answers
95 views
Finite fields package doesn't simplify
I'm trying to do arithmetic using the FiniteFields package. Supposing a is a generator of a Galois field (say GF(2^5)) I want to be able to simplify things like a^32 and (a^8)^-1. I don't care about ...
0
votes
1answer
58 views
Putting some calculations together to form an algorithm
Considering these below where n is an odd number, k is also an odd number less than n:
<...
3
votes
1answer
196 views
Coppersmith method of small integer solutions to multivariate polynomials
In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous"...
3
votes
2answers
326 views
Evaluate modular fractions
If I have a list of numbers l = {256/11, 258/11, 263/11, 263/11, -22, -22, 251/11, 0, 261/11, -22, 265/11, 259/11, 0, 259/11} which I want to evaluate mod 26, how ...
-2
votes
1answer
72 views
11
votes
2answers
858 views
MatrixPower with Modulus
I have a matrix that will be multiplied with itself by an extremely large amount of times under $Z/pZ$. The matrix itself contains small numbers, so ...
4
votes
0answers
169 views
Sparse matrix multiplication mod p
Let A and B be sparse matrices with integer coefficients between 0 and ...
2
votes
1answer
352 views
Get Mathematica to solve Modular Arithmetic problem [closed]
How would I get Mathematica to solve something like this for $x$?
$4x \equiv 1 \pmod 5$
2
votes
0answers
126 views
What does Mod^(1,0) mean in mathematica output [closed]
I am trying to take he derivative of mod(remainder).
In:= D[Mod[x, y], x]
Out = Mod^(1,0)[x,y]
I cannot understand the meaning of Mod^(1,0). What does the ...
-1
votes
3answers
121 views
Create table of elements which belongs to specific sets
I would like to produce elements (preferably by Table or For-If commands) that they belong to specific sets. For example I want to write positive integers from 1 to 100 such that they are $2 \pmod 3$, ...
2
votes
1answer
94 views
A Difficult Application of Modular Arithmetic
Looking to input a $p$ for the first line:
p=6;
Build a list of primes from the 3rd to $p-1$:
...
2
votes
1answer
144 views
Twin Prime Max Gaps (Performance Tuning)
Ok, let's build a foundation here:
A common way of testing primality, is dividing by all primes smaller than the number's square root. For instance, $97$ is prime because dividing by none of the ...
3
votes
2answers
125 views
Using “Outer” with “ChineseRemainder”?
I'm looking to build a $7\times3$ matrix using ChineseRemainder. Trying to use
${0,1,2,3,4,5,6} \pmod 7$
and
${3,15,21} \pmod {30}$
so that the first element of ...
8
votes
1answer
301 views
Taking one list Mod a second list
Just on my free trial and can't find this anywhere in online docs. Trying to use the Mod function as follows:
Mod[{list},{list}} where it takes each number in list ...
8
votes
1answer
149 views
Bug in PiecewiseExpand and Mod with assumptions
Bug introduced in 9.0 or earlier and fixed in 10.4
The following code using PiecewiseExpandand Modgives the wrong answer
<...
1
vote
1answer
87 views
Simplify Mod to subtraction when possible
FullSimplify[Mod[t,10], t>101 && t<109]
It's a long shot, but can I get Mathematica to return t-100 or something ...
27
votes
1answer
2k views
Fast calculation of discrete logarithms
Does Mathematica have any built-in fast algorithms for calculating discrete logarithms over $(\mathbb{Z}_p)^\times$ (the group of integers modulo $p$)?
Essentially, for a fixed large prime ...
2
votes
0answers
81 views
Sum of zeros in finite Galois field
In the following code:
<< FiniteFields`
GF[2][{0}] + GF[2][{0}] == GF[2][{0}]
GF[2][{0}] + GF[2][{0}] + GF[2][{0}] == GF[2][{0}]
Why does the second line ...
