Questions tagged [modular-arithmetic]
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value… the modulus.
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What makes the quadratic residuosity problem hard?
The quadratic residuosity problem is the problem of determining whether, for given $r$, $m$, $\exists a.a^2\equiv r\mod m$.
This problem's believed to be hard to solve in general (e.g. an efficient ...
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Small solutions to modular arithmetic linear congruence
Let $p$ be a prime number with $N$ bits, let $a,b,c$ be constants. The problem is to find solutions to the equivalent $a x + b y \equiv c \pmod p$ with both having at most $N/2$ bits.
What ...
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Is modular exponentiation always cyclical?
In RSA clock arithmetics is used, and as Fermat's little theorem says, $a^p \bmod p = a$. The exponentiation is cyclical, $a^x = a^{x \bmod p-1} \bmod p$, the same sequence of numbers is repeated in ...
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Question about using Montgomery form for elliptic curve operations on bls12-381
Since the prime for bls12-381 is not of a form to allow easy modular reduction , is the best approach to use the Montgomery multiplication + reduction algorithm?
...
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1answer
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What makes the Discrete Logarithm Problem hard?
I am missing a crucial piece of the maths behind the DLP, and I'm hoping someone can give me a really dumbed down answer..
If $h=g^x \bmod p$ and we're working in the group $Z^*_p$, why can I not ...
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Problem with Chaum`s Untraceable Electronic Cash
For example according to protocol I need calculate this:
$b=F^{(1/h)} \bmod pq.$
Where $p$ and $q$ are prime numbers. I have $F$ and $h$. But how can I calculate $b$?
I tried to do this:
$\text{...
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0answers
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Efficient fields over arithmetic circuits
What sort of fields is efficient over an arithmetic circuit?
Efficient meaning that given a field $\mathbb{F}_p$, reduction (modulo) does not require many multiplications and preferably inversion was ...
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1answer
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Finding a base of an exponent in mod field
I have the following equation:
$$base^{exp} = number \pmod n : base,exp \in \mathbb{N}$$ and another known fact is:
$$gcd(base, number) = 1$$
$$n = 2^{2048}$$ base=...
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1answer
123 views
find $a$ and $k$ for a given el gamal cryptosystem
I am given this question:
Suppose Alice is using the ElGamal Signature scheme with parameters
$p = 31847$, $\alpha = 5$, and $\beta = 25703$
Assuming that we have received signed messages
$(x_1,(\...
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1answer
96 views
How does the squeeze function in the NaCl Poly1305 implementation work?
The NaCl ref implementation of the Poly1305 algorithm uses the following reduction function (which is called squeeze()):
...
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1answer
80 views
How can I exploit the structure of the secp256k1 prime for fast arithmetic?
I'm implementing logic on an FPGA (programmable chip) that does the key verification part of ECDSA on the curve secpk256k1, in which all operations are mod p where $p = 2^{256} - 2^{32} - 2^9 - 2^8 - ...
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Additive Differential of XOR
I'm studying about Additive differential of XOR
I saw two papers that are "H.Lipmaa et al., On the Additive Differential Probability of Exclusive-or" and "V.Velichkov et al., The Additive ...
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1answer
374 views
Salsa20 Implementation: Sum of 2 Words with Carries Suppressed
I'm working on the 2nd part of the Salsa20 spec, and I want to implement a closure for the exclusive-or of two words(u32). The author defines the operation as the sum of two words with carries ...
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1answer
152 views
Factoring RSA weak modulus
Given a public key for RSA, I have extracted the modulus which looks like this :
...
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0answers
33 views
Speeding up quotient determination in high-radix montgomery modular multiplication
In this paper Simplifying Quotient Determination in High-Radix Modular Multiplication, the authors have proposed to replace the original modulus $M$ in Montgomery Multiplication $ABR^{-1} \bmod{M}$ to ...
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1answer
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$m^b = z^b \pmod{n}$ - single solution $m = z$?
If i have the following equation:
$$m^b = z^b \pmod{n}$$
Where all numbers are taken from the multiplicative group $\mathbb{Z}_n^*$.
Suppose I know the values $z$ and $b$. Can I state that $z = m$, ...
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0answers
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Finding generator element for multiplicative group of order $n$ - feasible?
Given a multiplicative group of order $n$, how hard is it to find a generator element (such that all other elements can be expressed as powers of that generator)?
What would be the complexity?
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1answer
224 views
Diffie-Hellman private key recover with non-prime modulus
Say we have a classic Diffie-Hellman key exchange. We have the following parameters of a public key:
p,g,y
Where $p$ is the modulus, $g$ is the base, $y$ is the ...
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0answers
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Hamming Weight of Solutions to the Frobenius Equation
My area of research is algorithms for calculating optimal addition chains.
I have a new high performance algorithm that at its core benefits from estimating the number of set bits ($v(n)$ Hamming ...
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1answer
92 views
Diffie hellman: fixed modulus a risk?
In DH, are algorithms with fixed modulus p (large safe prime) to be avoided?
What if it was changed every x minutes?
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1answer
84 views
What is the difference between subtracting the modulus from a scalar field element and reducing it?
When implementing a Field element, we define the necessary operations on the data structure.
One function that I see is a "scalar reduce" function, which effectively reduces a random scalar so that ...
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1answer
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find additive inverse of modular arithmetic [closed]
For a set to be called as a ring, it should have the following properties
closed
commutative
associative
Identity existence
Inverse existence
but how is Z7 a ring, as there aren't any inverse ...
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2answers
174 views
What prevents the successful use of the Giant-step/Baby-step algorithm solving a discrete log problem implemented with modulo arithmetic?
Does the size of base, exponent, and modulus thwart the Giant-step/Baby-step algorithm in solving DLP using modular arithmetic or is it the use of a property of a particular prime as the modulus, or ...
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1answer
123 views
Determine RSA modulus from encryption oracle
Suppose we have an RSA encryption oracle $E(m)$ which basically just calculates $m^e \mod n$ for a given message $m$. Here $e=65537$ is known but $n$ is not. Can we determine the value of $n$ without ...
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1answer
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Can we find the exact number given remainder of the numbers with mod m?
I have around 1500 numbers. The numbers $x_i$ are calculated as
$x_i$=($p*t_i$) mod m. $p$ constant and same for all the numbers while $t_i$ are chosen randomly everytime. For example the given ...
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1answer
50 views
Collisions in Diffie-Hellman private keys
Given a generator $g$, a large, safe prime $P$ and a result of the DH key exchange $g^{xy} \mod P$, how would I come up with two different $x', y'$ s.t. $g^{x'y'} = g^{xy} \mod P$
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3answers
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Mixing x into permutation through XOR vs. modular addition
Block ciphers mix key material into the permutation through XOR. Also do pre and post whitening this way.
Chacha/Salsa finishes by 32-bit modular adding key and iv material (among other bits but ...
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3answers
120 views
In modular arithmetic, why is $\left(g^{k_1} \bmod n\right)^{k_2} \bmod n = \left(g^{k_1}\right)^{k_2} \bmod n$? [duplicate]
Why is $\left(g^{k_1} \bmod n\right)^{k_2} \bmod n = \left(g^{k_1}\right)^{k_2} \bmod n$ ?
Note of the editor: I used $=$ where the original question used == because $=$ in a cryptographic context is ...
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4answers
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Diffie-Hellman key exchange: Why is $(g^{k_1} \bmod n )^{k_2} \bmod n \equiv (g^{k_2} \mod n)^{k_1} \bmod n$
In a Diffie-Hellman key exchange, with a generator $g$ and a modulo $n$, and two keys $k_1$ and $k_2$, why is
$(g^{k_1} \bmod n )^{k_2} \bmod n \equiv (g^{k_2} \mod n)^{k_1} \bmod n$
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How to avoid side channel attacks when handling large numbers?
For cryptography, the platforms have limited size as 32 or 64-bit operations. We definitely need big numbers to implement the encryption/decryption and digital signatures for cryptosystems like RSA, ...
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2answers
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Regular and Efficient doubling in $GF(2^n)$
A naive approach to compute a multiplication $a \cdot b$ in a Galois field of the form $GF(2^n)$ is the russian peasant algorithm. However, it does not run in constant time and therefore is not safe ...
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1answer
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Can modular multiplicative inverse be used to create a secure cipher?
Let $N$ be a large number; and $(e, d, N)$ be a secret-key; where $e$ is a one-time random factor, and large enough (say the more or less the same number of bits as $N$), where $e < N$ and $e * d \...
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One Time Pad OTP: with modular math, ciphertext reveals modulus always, yes or no?
Assuming that in the OTP scheme, the key has more values than the alphabet, then: using modular math predetermines that the highest possible value in the ciphertext will reveal the modulus used in ...
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2answers
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OTP One Time Pad : How does modulus size greater than alphabet size eliminate perfect secrecy?
If my plaintext alphabet is {0,1,2}, it has three characters, and I understand why I cannot use a modulus less than 3 (decryption won't work).
My key has 100 different values, {0,1,2 .....99} . ...
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2answers
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One Time Pad OTP : Optimal modular base per given plaintext space?
Background: my question at :
My pencil and paper One Time Pad works fine without modular math ...... or does it?
I am trying to understand, from a layman's point of view, how to make the best one ...
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1answer
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choices for k in binary finite field modular reduction algorithm
In the Guide to Elliptic Curve Cryptography there's this algorithm:
My question is... what is $k$? Is it just some random value we pick? If so are some numbers better than others?
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1answer
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RSA given d and d = p
Assume we have a somehow unorthodox implementation of RSA whereby:
$p$ and $q$ are chosen primes of length $n/2$ where $n$ is the number of bits desired in $N=p\,q$ and
$$\begin{align}
\phi &= (p-...
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1answer
49 views
Additive homomorphic encryption: strict equality - handling congruence
Is there an additive homomorphic encryption scheme which guarantees that if provided with
$E(v)$, $E(m_1)$ and $E(m_2)$
and $E(v)=E(m_1).E(m_2)$ then $v=m_1+m_2$
Please note this is not $v \equiv ...
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2answers
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My pencil and paper One Time Pad works fine without modular math … or does it?
With no background in higher math or computer science, I could not quite grasp the value of modular math for simple OTP systems as described in other posts here.
I have an alphabet size of 40 ...
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2answers
55 views
Break large exponent calculation into smaller calculations?
I am implementing SRP on an embedded platform. It has crypto acceleration and I am using its built in modular exponentiation function, however it doesn't seem to be able to handle an exponent over 32 ...
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1answer
62 views
Computations in extended finite field p^2
I would like to construct a distortion map from a point $\in \mathbb{F}_p$ to $\mathbb{F}_{p^2}$.
If I have an elliptic curve $Y^2 = X^3 + 1$ over $\mathbb{F}_p$ and a distortion map $\phi(x,y) \...
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0answers
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Geometry of the outputs of linear congruential random number generators
I learned that possible $m$ long sequences produced by linear congruential random number generators(of the form $r_{i+1}\equiv ar_i+b \mod n$) fall on hyperplanes. Using this fact I have come to think ...
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Indistinguishability vs fixed bit-length
Suppose there is a cyclic group $\mathbb{G}$ of prime order $q$ of elements in $Z^{∗}_p$ with a generator $g$ and values $a,b,c,d \in Z_q$.
There is also an equation $g^{ab} = g^{cd}$, where $a,b,c$ ...
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2answers
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Public-key generation - primes reuse
Given the following Trap-door Commitment scheme
Secret key receiver: $x_B \in_u Z_q$
Public key receiver: $y_B = g^{x_B} \mod p$
Here, $p=q*k+1$ for two primes $p,q$ and $k \in Z$. And $g$ is the ...
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1answer
349 views
Paillier: guessing the message when knowing the cipher and the random number
I cannot get my head around this.
In Paillier, the ciphertext is calculated using
$c = g^m.r^n\ mod\ n^2$
where $(n,g)$ forms the public key and $r$ is a random number $0<r<n$.
Assuming an ...
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2answers
197 views
RSA: finding e given n, m and c
So I'm not sure if this is possible but assume I have
$$c \equiv m^e \pmod n$$
I have $c$, $m$ and $n$ but no $e$, which in my case I a value that can be up to 15 digits. Is there a way to do this?
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1answer
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Is there a formula for calculating the inverse of a sum in modular arithmetic?
Suppose there is a cyclic group $G$ of prime order $q$ of elements in $Z_p^*$ with a generator $g$ and a value $x \in Z_q$.
$h_1 = g^{x+1}$
$h_2 = g^{x}$
Is it possible to write $h_2^{((x+1)^{-1})}...
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1answer
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Why FHE takes modulo space $\Bbb Z_p$ as $[−p/2,p/2)$ rather than $[0,p)$?
Why doesn't FHE scheme see a modulus space $\Bbb Z_p$ as $[0,p)$ ?
Instead, it consider $\Bbb Z_p$ as $\left[-\frac{p}{2},\frac{p}{2}\right)$.
What's the concrete reason?
What happens if I use $[0,p)$...
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2answers
166 views
A quick way to justify whether $a^m \equiv 1 \pmod{n}$?
Say I would like to justify whether $10^{28} \equiv 1 \pmod{29}$. I know according to Fermat's little theorem that $a^{p-1} \equiv 1 \pmod{p}$ when $a$ is a primitive root modulo $p$. What about ...
4
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1answer
479 views
Solving the discrete logarithm problem for a weak group
I was reading an answer about an attack on a weak group for the discrete logarithm problem and wanted to formalize and verify that the attack was correct. That is, that it was guaranteed to always ...
