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 ...

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Is it possible to determine the group order by knowing the “public” and “private” key exponents in an RSA group?

I have an RSA group with modulus $n = p \cdot q$, two safe primes $p=2p'+1$ and $q=2q'+1$ and the "public" and "private" key exponents $d$ and $e$. $\phi(n) = 4p'q'$ is the order of the RSA group. If ...
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93 views

How difficult is it to check if a group element is in a sub group?

I am just curious. We have a group $G$ and its subgroup $H$ with a generator element $h \in H$. How difficult is it to check for $x \in G$ that $x \in \langle h \rangle$? Is there a better way than ...
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144 views

Efficiently computing the neutral element in a ring isomorphic to Z/NZ?

Edited to clarify question So my question is whether anyone knows of an efficient way to compute the neutral element (I'm gonna call it 1, but the operation doesn't have to be multiplication) in an ...
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175 views

Need 32-bit mixing function that has perfect avalanche between octets

for my hobby tinkering project, I need a mixing function that takes 32-bit input and has 32-bit output (and will, most likely, run in a 32-bit C environment) and the following property (independent of ...
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65 views

Comprehension question on a signature protocol based on the RSA assumption

We have the following two-party protocol between Alice and Bob. Alice sends messages $m_1, m_2, \ldots \in_R \mathbb{Z}_n^*$ to Bob and Bob signs these values by calculating $v_1, v_2, \ldots \in_R ...
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138 views

Discrete log analog of ECM factoring algorithm?

Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas. Is there an analog of the elliptic curve (ECM) ...
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438 views

Understanding elliptic curve encryption

I'm having a hard time understanding the elliptic curve encryption. One thing thing I don't understand is listing all the points on the curve (mod p). Suppose I have the following elliptic curve: y^2 ...
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415 views

Is there an algorithm for factoring N, which is just as simple as this one, but faster?

I found a simple algorithm for factoring semiprime numbers, you can read about it in Factoring Semiprimes and Possible Implications for RSA. It basically works like this: You reverse the digits in ...
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206 views

Which area of Maths should I pursue?

I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to ...
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190 views

What does modular inversion mean?

I'm trying to implement an e-voting algorithm, which is described at the paper "Internet Voting Protocol Based on Improved Implicit Security" by Abhishek Parakh & Subhash Kak. At the Example 1 ...
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104 views

Question about Fermat's little theorem

Why is $g^e \mod p = g^{e \mod (p-1)} \mod p$ if p is prime. I don't get it. It follows from Fermat's little theorem.
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110 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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159 views

How to solve the reverse of an equation that uses MOD?

I've been tasked with reverse engineering an unknown crypto function. The function uses the following constants: $a=380951$: I noticed that this is a prime number $b=3182$: I noted that this is a ...
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1answer
44 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
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249 views

How costly is to find millions of large prime numbers for RSA?

Consider I need to assign a large distinct prime number to each element in a large set. This must be deterministic so the function always gives me the same prime to the same value. What is the most ...
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346 views

What does $(\mathbb{Z}_n^*)^2$ mean?

In a paper they write once, $(\mathbb{Z}_n^*)^2$. Is this the group of quadratic residues or is it something else? Here the theorem: Under the strong RSA assumption, given a modulus $n$, along with ...
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350 views

Montgomery Exponentiation - selecting input value R for a given BigInteger

I have Montgomery exponentiation working, but it's working quite slow. I suspect there are two reasons for this - I implemented it bit size instead of word size (I didn't realize at the time that ...
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161 views

Equal length of primes in paillier cryptosystem

In the key generation step of paillier cryptosystem , In order to satisfy $\gcd(pq,(p-1)(q-1))=1$ , we can take equal length primes. Instead of taking(length as parameter to generate $p,q$) equal ...
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77 views

Non-commutitive and nonassociative algebraic structures in cryptography

Are there any cryptographic algorithms or primitives that have been developed and studied that make use of non-commutative or non-associative algebraic structures such as quaternion integers or ...
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509 views

Algorithm for proving Carmichael numbers

I have an application for determining if a number is a prime or not, currently I'm getting a random number, then doing the Fermat primality testing to find out if the number is probably prime (so this ...
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1answer
318 views

What are some different cryptography methods?

Some of the most effective cryptography methods and algorithms are based of factoring large prime numbers (e.g. RSA). I'm curious whether there are some other cryptography methods. Somethings that is ...
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35 views

What constitutes a “description of B” for probabilistic encryption as defined in Cryptology 6.3.4?

On page 21 of the Rivest's Cryptology chapter, he defines a trapdoor predicate as a boolean function for which it is easy to choose an x such that ...
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102 views

$\phi$ function in Dual_EC_DRBG

I am trying to understand the operation of the Dual_EC_DRBG. I'm reading the formal specification (SP 800-90) and can't seem to find a definition of the $\phi$ function used throughout the definition ...
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139 views

Key Refresh in Diffie-Hellman

Assume the case, that two participants have agreed on a key $K=g^{ab} \mod p$ via Diffie Hellman. I have the need to change the key every now and then. The first idea I had was to simply initiate a ...
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44 views

Applications of GF(p) polynomials

A Galois field of the type $GF(p)$, where $p$ is prime, is normally expressed as the ring of integers modulo $p$. If my understanding is correct, it is also possible to represent its elements as a ...
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91 views

Prime Numbers in Discrete Log

I am implementing a security protocol based on discrete log. I came across the equation $p = kq + 1$. Understand that based on number theories that both $p$ and $q$ should be large enough to be ...
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100 views

Quadratic Residues in identification setting

Suppose Alice has two primes $p$ and $q$ such that $p\equiv q \equiv 3 \pmod 4$. $n=p\cdot q$ is part of her certificate to identify her. A party say Bob, sends her a random quadratic residue ${}\bmod ...
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92 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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112 views

What are alternatives to number theory based crypto? [closed]

Quantum crypto,lattice based crypto, Neurocryptography and cellular automata based cryptography are alternatives to number theory based crypto. I need to know what are the other hard problems like ...
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156 views

RSA, finding p,q [duplicate]

If the public key $(e,n)$ and the private key $(d,n)$ are known, what is the easiest way to find the primes $p$ and $q$? When $n$ and $\phi(n)$ are given this is easy to solve. But I can't manage it ...
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80 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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78 views

What does $\mathbb Z_p^*$ contain?

I have a prime $p = 7$ and was tasked to select a random value in $\mathbb Z_p^*$ in my signature scheme. What does the full range of $\mathbb Z_p^*$ contain in this case? Is it $\{0...7\}$ or ...
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111 views

What parts of number theory does the RSA algorithm use?

It is said that the RSA algorithm uses number theory. What parts of number theory does it use? I know it uses modular arithmetic and Euler's totient theorem and function. Is that all?