Questions tagged [number-theory]

Number theory is the study of the properties and construction of numbers, particularly integers. Prime numbers are of particular interest to number theorists and consequently cryptographers as they are considered the "building blocks" of numbers and produce many interesting results which are useful in cryptography. Questions covering number theory and primes should use this tag; questions involving finite fields and groups might use this tag if relevant.

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23 views

Point-halving/solving quartic equations over the elliptic curve E(Z_N)/ring Z_N where N = pq

I am wondering whether there are any results/whether there is any knowledge about the following problem: Given a univariate polynomial (say, a quartic) equation defined over $\mathbb{Z}_N$, is it ...
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Trivariate Coppersmith Implementation

Bivariate Coppersmith is standard package in math software with number theory support. Bauer and Antoine Joux introduced trivariate Coppersmith in https://www.iacr.org/archive/eurocrypt2007/45150361/...
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Constructing an integer with specific totient without factorization

Given $N(=pq)$, but not its factorization. Somehow you manage to know, that $k$ divides $\phi(N)$. Is it possible to come up with an integer $N^{*}$, for which $\phi(N^{*})=\phi(N)/k$, without ...
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Why is Multiplicative group used in RSA or Euler's theorem but not additive?

I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $...
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FLT is partly applying to RSA equation, and also relation between ED mod phi and Phi + 1 mod N

After numerous attempts from myself and all of you guys, I finally came to understand RSA. I can now prove it and understand how I got there. But I still have some very few polishing questions. 1) We ...
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Protocol for proof of knowledge of $l$-th root

Assume we have Group G in which the adaptive root assumption holds. This assumption states that if we choose an element $w$ and after that, if we receive a prime value $l$ it is hard to find the $u$ ...
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Relation between $N = P \times Q$, and $\Phi(N)$

When studying RSA, and proving simple concepts to myself, I went and understood groups and rings, but I failed to understand Lagrange's theorem. I did understand how from invertible finite groups I ...
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30 views

Proof of two pairs with same exponent

Lets assume we have a group $G$ with unknown order. And we have a pair $(A_1,K_1), (A_2,K_2)$ in which all $A_1,K_1,A_2,K_2$ are group elements. The claim is $A_1= K_1 ^ x$ and $A_2 = K_2 ^ x$. or ...
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48 views

number theory question in a group with unknown order

I was reading a paper and I am struggling understanding one part of it. Lets say we have a group $G$ of an unknown order $n$. we know that $B<n<B+C$. both B and C are large values). we choose a ...
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A number theory problem in RSA

Let say we have $N=pq$ (Both $p$ and $q$ are safe primes, meaning $\gcd(p-1,q-1)=2$) let's assume $e$ is an odd number which can be efficiently factored and assume that we know the $e$'th root of 1 ...
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Schnorr proof of group with unknown order (but the prover know the order)

We assume that we have $n = pq$, where $p = 2p'+1$, $q = 2q'+1$, and all $p, q, p', q'$ are large primes. Pick $g' \gets_R Z_n^*$, and compute $g = (g')^2 \bmod n$. Then we have a generator $g$ for $...
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Distance between consecutive primes distribution

In several prime generation schemes I saw we pick a random number uniformly at random from a wide range and find the next prime after it. Obviously with such a scheme some primes are more likely than ...
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Help with next step in the Quadratic Sieve

So I am at the same step as someone from math.stackexchange but he never recieved an answer so I will copy-paste his question here: Say, for N = 90283, I compute bound 𝐵=𝑒(12+𝑜(1))(ln(𝑛)ln(ln𝑛√))...
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Sigma protocol when order is unknown

In the following paper (page 5) they have a proof that a triple $\left(g,b_{i-1},b_{i}\right)$ is of the form: $\left(g,g^{x},g^{x^{2}}\right)$ for some x. The relevant text from the paper is as ...
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Reciprocity Laws in Cryptography

Does anybody know applications of higher reciprocity laws (i.e. the reciprocity laws "higher" than the quadratic one, like the cubic or quartic reciprocity law) in cryptography? So far I have found ...
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Generalized Benaloh cryptosystem with $r=2$

Benaloh cryptosystem requires $\gcd(r, (q-1))=1$ which is impossible if $q>2$ (since it needs to be a large prime) and $r=2$. This confuses me, since Benaloh is referred to as an "extension" or "...
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Size of $E$ over $\mathbb{F}_p$ contains $p+1$ points

I am struggling to prove this claim: I proved that the map $x\mapsto x^3+1$ is a bijection from $\mathbb{F}_p$ to itself if we have that $p\equiv 2\bmod{3}$. We have to use this fact to prove that ...
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Why does a primitive root mod n gives a uniform distribution over n?

The result from a primitive root mod n seemly has a uniform distribution over n, is it true? For instance, 3 is a primitive root ...
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1answer
54 views

Paillier Complex Residuosity problem?

Paillier Cryptosystem depends on both the factorization where $n = p.q$ and the complex residuosity problem which is defined in the original paper as: The problem of deciding n-th residuosity, i.e. ...
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302 views

How to prove if a Hash Function is collision resistant

Say we have the following Hash Function, H(x) = 4x mod N where N is a number generated by multiplying two prime numbers and x={0,1,2,3,...,N-1}. So lets assume that ...
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How to compute ElGamal decryption by hand

Lets say that $u=3, x=5, v=2$, how do we work out $u^{-x}*v$, so $3^{-5} * 2$. I know how to work out the answer if it was $3^5 * 2$ but how do we do it with negative exponents?
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Why are some group representations much easier to compute discrete logarithm for? [duplicate]

The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative ...
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Why does square root $\pmod n$ find $p$ and $q$ (as $n = p \cdot q$)?

Let $n = p*q$, with $p \neq q$ and $x^2=1 \pmod n$, $x+1 \neq 0 \pmod n, x-1 \neq 0 \pmod n$ (So x is a non-trivial square root mod n.) I don't see how $\gcd(x+1,n) \in \{p,q\}$ follows. I ...
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Why can every prime number be written as 6k±1? [closed]

I want to know how to be sure that each prime number can be written in the form $6k±1$. How I can find the prime number that exists after a composite number with this property of prime or other ...
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Explain the meaning of “$E_i = g^{F_i} g^{G_i}$ is uniformly distributed in $G_q$”?

In this paper[1] Qiong et al.’s CRT-based VSS} {Dealer Phase} To share a secret $ d \epsilon Z_{m_0} $ among a group of $n$ users with verifiable shares, the dealer does the following: <...
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1answer
259 views

How should I address message size limits in RSA encryption?

I am making an end-to-end encryption software program in Java using RSA. I am using BigIntegers and its number theory methods. (I know this is a very slow approach, but I just want to learn to the ...
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RSA encryption, Number theory [duplicate]

In RSA algorithm we have the value of $e$ & $d$ exponent and also one of the prime numbers. My question is how to produce another prime number that is not equal to the first one and the digit of a ...
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Is it possibe to extend DGK method of comparion two integers to $d$-ary settings?

In [1], a method for comparing two integers is described by using polynomial operations related to the binary representations of the given two integers. The method is re-phrased as follows: Given the ...
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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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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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Difference between when select x from $\mathbb{Z}_{p-1}$ and $\mathbb{Z}_p$ in discrete logarithm Problem?

Reading "Security Arguments for Digital Signatures and Blind Signatures" paper, I confused by some questions. Q1. when it refers to "El Gamal signature scheme", The key generation algorithm: it ...
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Is it hard to compute $g^{ab}$ when given $(g, g^a, g^b, \frac{a}{b})$?

We know that the CDH problem, computing $g^{ab}$ from given $(g, g^a, g^b)\in\mathbb Z_p^3$, is hard. Is it still hard with an auxilary information $\frac{a}{b}\bmod q$ (where both $p$ and $q$ are ...
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87 views

What is the significance of $\prod \limits_{i=1}^{t}m_i > p \prod \limits_{i=1}^{t-1}m_{n-i+1}$ in Asmuth–Bloom threshold SSS?

In this paper by Asmuth–Bloom on threshold SSS, the algorithm is as follows: Shares Distribution To distribute $n$ shares of a secret $K$ among the set of participants $P = \{ p_i : 1 ≤ i ≤ n\}$, ...
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635 views

Which properties of a group are used in the steps of Diffie Hellman?

I’m trying to understand which properties of a group are used in DHKE at each step. For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e....
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Why is a number of shares that is greater than a threshold required to construct secret?

In this paper of Asmuth–Bloom threshold SSS, the algorithm is as follows: Shares Distribution To distribute n shares of a secret $K$ among the set of participants $P = \{ p_i : 1 ≤ i ≤ n\}$, the ...
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Random Primes Conjecture [closed]

I know it is believed that primes appear to be randomly distributed among the integers. Is there a formal conjecture or theorem that expressly states that the occurrence of the prime numbers is ...
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Parity of the order of a element

Given an element $g$ in a cyclic group $G$ of known order $m$ its easy to test if $m$ has even or odd order. In other words $\textrm{ord}(g) \pmod 2$ can be computed easily. In some cases where the ...
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1answer
217 views

Solving discrete log in partially known group

Suppose I have a group $G$ of unknown order $n$ where $n=p^k\cdot s$, $\gcd(p,s)=1$, $p$ is a known prime, $k,s$ are unknown positive integers and $k,s\ge1$. (Known - $p$ and $p\mid n$, Unknown - $n,k,...
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163 views

Upper bounds on difference of RSA primes

I was wondering whether given a concrete $N = p \cdot q$ whether we can find a upper bound on $\Delta = | p - q|$ as function of $N$ e.g, $N^\delta$, and thus test whether a given $N$ is vulnerable ...
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How we can calculate AES Inverse SBox?

I am trying to find out the inverse of and SBox, but in vain, I have seen multiple questions over StackExchange, But I cant be able to solve my issue. As in this question, How are the AES inverse S-...
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Why is Approximate GCD a hard problem?

There are many Fully Homomorphic Encryption over the Integers schemes whose security is based on the intractability of the Approximate GCD (AGCD) problem. The paper Algorithms for the Approximate ...
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How can process algebra help to design security protocols?

Would you please tell me how process algebra could help me to design security protocols? More specifically, can I use it for proving the security of protocols? Is process algebra used for any ...
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Why do one-way accumulators use rigid integers as the modulus?

In the paper that introduced one-way accumulators, the author's justify their use of rigid integers as the modulus with the following: The advantage of using a rigid integer $n = pq$ is that the ...
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Sum of two squares problem

I would like to know if there is any existing research on the following problem: $$\text{For }a, b \in \mathbb Z \text{, given }n = a^2 + b^2, \text{output }a, b$$. Searching for "sum of squares", "...
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1answer
224 views

Understanding the Pohlig-Hellman algorithm

The paper has the following relation: $$y^{(p-1)/p_i} \equiv \alpha^{x(p-1)/p_i} \equiv \gamma_i^x \equiv \gamma_i^{b_0} \pmod p$$ where $\gamma_i = \alpha^{(p-1)/p_i}$. I understand this relation ...
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78 views

Is this problem same as discrete logarithm?

Given $g,h\in\mathbb Z_p$ where $g$ generates $\mathbb Z_p^\star$ Discrete logarithm problem is to find $z$ such that $g^z\equiv h\bmod p$ holds. Take the problem given $g,g',h$ where $g^z\equiv h\...
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Why do algebraic proofs apply to cryptography?

How do we know that the number theoretic and algebraic results used in cryptography provide a perfect model for the behavior of integers as implemented in computers? Does there exist a bijection ...
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Explanation of a proof of one of Shoup's lemmas

In Lower Bounds for Discrete Logarithms and Related Problems, Victor Shoup states the following lemma: Lemma 1 Let $p$ be prime and let $t \ge 1$. Let $F(X_1, \dots, X_k) \in \mathbb{Z} / p^t[X_1, \...
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Using Euler's theorem.. how can I get appropriate x..?

The question is this... $x^{29}\equiv4\pmod{91}$. Note that $91 = 7*13$. Compute an integer $x$. Since 91 is not prime number, I know I need to use Euler's theorem, not Fermat's. So I get $a^{...
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Is the matrix step of GNFS still the hardest part?

When the factorization of RSA-768 was announced in December 2009: the sieving took about 24 months and the matrix step took 119 days (4 months). So sieving took about 6 times as long. This is despite ...