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 happens if no final subtraction is done in Montgomery multiplication?

I'm doing Montgomery arithmetic modulo $N = 2^{255}-19$ for the Curve25519, picking $R = 2^{256}$ for Montgomery. After multiplying two numbers $0 \leq A,B < N$ in the Montgomery representation ...
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How to find the time complexity of modular multiplication? [duplicate]

There are two number of length m bits. How do I prove that the complexity of modular multiplication of these two numbers is $O(m^2)$.
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Understanding elliptic curve encryption [closed]

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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156 views

Modular Multiplicative Inverse and RSA?

In trying to understand this specific part of the RSA algorithm, I found this online: $$e \cdot d = 1 \pmod{(p-1)\cdot(q-1)}$$ Therefore: $$e \cdot d \cdot d^{-1}= d^{-1} \cdot 1 ...
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Why does the modulus of Diffie–Hellman need to be a prime?

I read a lot about Diffie-Hellman, but there is one thing I dont understand: why does the modulus p need to be a prime? What if it would not be a prime?
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102 views

How do I create a secure encryption scheme using addition?

You have a 4 number long PIN code with each number ranging from 0 to 9 which you wish to encrypt. You are then given a random 4 digit number ranging from 0 to 9999, which when added to the original ...
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1answer
73 views

Clifford Cocks Interview: “Raising numbers to a power with the product as modulus”

Disclaimer: I'm new to cryptography Background: Clifford Cocks, the former GCHQ mathemetician and crytologist, gave an interview where he said: "My thinking was that you need something that is ...
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169 views

How to use the Extended Euclidean algorithm to invert a finite field element?

I was doing some practice on cryptography (I'm new to this topic) and was wondering what the following question even means or what it is asking me to find. I do know how to do Extended Euclidean ...
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How can I find a key matrix to decode a Hill Cipher?

For an assignment, I've been given a 2x2 cipher and a 3x3 cipher and I have to decode both of them. I have no key, so I really don't know where to start. Also, this is all modulo 27, because my prof ...
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1answer
68 views

How is it possible that $g^q \equiv 1 \pmod p$ for a generator g?

The context of this question is coming up with the parameters for the ElGamal encryption scheme. One of the requirements for the parameters for ElGamal is that we have primes $p$ and $q$ such that $p ...
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69 views

Factoring large $N$ given oracle to find square roots modulo $N$

When $p$ and $q$ are distinct odd primes and $N = pq$, the points in $\mathbb Z_N^\ast$ have either zero or four square roots. A quarter of the points have four square roots; the rest have no ...
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57 views

multiplicative inverse in galois field $2^8$

I am trying to compute the multiplicative inverse in galois field $2^8$.The question is to find the multiplicative inverse of the polynomial $x^5+x^4+x^3$ in galois field $2^8$ with the irreducible ...
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1answer
56 views

Asmuth-Bloom's threshold secret sharing scheme

I was learning Asmuth-Bloom's threshold secret sharing scheme. I was working out an example as given in this Wikipedia article. As per the example, the secret,d is 2, the number of shares, n is 4 and ...
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51 views

Fast algorithm for reduction modulo a prime [closed]

If the prime is $p=2^a\cdot3^b+1$ , is there any fast reduction technique modulo this prime?
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67 views

What is the fastest modular reduction algorithm available?

I have been browsing for the fastest and most efficient modular reduction algorithms and came across quite a few. But the one in A Fast Modular Reduction Method (2014) by Zhengjun Cao, Ruizhong Wei ...
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133 views

Does RSA have two trapdoors?

I watched some videos from Khan Academy explaining the algorithm for RSA encryption/decryption. They explained that there are two trapdoors, modular exponentiation being one and prime factorization ...
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219 views

Constant time multiplicative inverse within a word

I was playing with an algorithm which at one step, calculated $f(x) = x^{-1} \mod p$ for $0 < x < p = 2^{64}-59$ (note $p$ is a prime). I used Knuth's Vol 2 Algorithm X algorithm for calculating ...
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622 views

Public key crypto without modular arithmetic?

This comment from Reddit math, in response to a statement about how people can communicate secrets to each other with a third party listening, has a very small, simple example of public key ...
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59 views

RSA : Double-Encryption and order of Encryption

I'm studying cryptography and we're looking at RSA. We're required to do a double encryption from Alice to Bob and then Bob back to Alice. We've been told the order of encryption is important and it ...
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41 views

Recover secret $x$ when $c\equiv m^x \pmod p$ with public $p$ (modified)

Given an encryption system where $c\equiv m^x \pmod p$, $p$ is a known prime, 1. Is it possible to recover $x$ with a known plaintext attack? Given $(p,\text{factorization of }\varphi(p),m,c)$ 2. Is ...
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67 views

What can I do to improve my encyption method [closed]

I'm new to cryptography and I want to learn more. For me, the best way to learn about something that I have no experience with is to experiment and I've coded a very basic approach to a simple ...
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1answer
82 views

NIST PRIMES - cryptography

I have a problem, If I have $p = \text{P-192} = 2^{192} - 2^{64} - 1$ with the base $2^{64}$ so, $2^{192} \equiv 2^{64} + 1 \pmod p$ the same that $(p=\text{P-192})$ or $2^{256} \equiv 2^{128} + ...
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Is there any alternative for extended euclidean algorithm to perform modulo division?

I'm implementing point addition and point doubling operations for elliptic curve cryptography. I have implemented extended euclidean algorithm to perform modulo division. It appears the that ...
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1answer
210 views

Modular exponentiation on calculator for textbook RSA

How do you encrypt $51$ with public key $(n,e) = (91,23)$ I understand that $c = 51^{23} \bmod 91$. How can I calculate the result on a calculator?
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Clarification regarding multiple modular exponentiation

If the base is same and exponents are different, for example: $R_1=b^x\bmod{p}$; $R_2=b^y\bmod{p}$; $R_3=b^z\bmod{p}$; ($p$ is large prime (2048 bit); $x$, $y$ and $z$ - 160 bit integers)) To ...
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Why are some x-coordinates unsuitable for an ECDSA generator point?

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ...
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162 views

Modular Arithmetic in RSA

Consider the following the following RSA public key $pk = (N, e) = (1457, 1307)$. (a) Knowing that $187^2 \equiv 1 \pmod {1457}$ find the factorization of $N$. (b) Given the factorization of $N$ ...
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How to perfrom modular division while numerator is lesser than the denominator?

I'm implementing point addition and point doubling of elliptic curve cryptography. The formula that I'm using for slope is Point addition: $S = \frac{(P_y-Q_y)}{(P_x-Q_x)}$ where $P$ and $Q$ are the ...
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50 views

what is the fastest and efficient method to perform modular multiplication?

I'm working on a code that needs to perform modular multiplication of big numbers several times. Since the operation takes place several times, using division to find the remainder is very expensive. ...
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80 views

Does the RSA algorithm use repeated squaring?

Simple question: in order to reduce such huge exponents in modular arithmetic, is repeated squaring used in RSA or is there a better way to implement it?
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98 views

RSA: must $d$ be an integer?

I am only taking baby steps in RSA. If $p=11$, $q=7$ and $e=3$, $$\phi(n) = 10*6 = 60$$ Then: $$d = (2 (\phi(n)) + 1 ) / 3 = 121/3$$ Should $d$ be kept as a non-integer or is such a $d$ invalid? ...
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Which one is fastest? Karatsuba or Montgomery multiplication?

Is there any complexity analysis between Karatsuba and Montgomery multiplication algorithms? It seems that Karatsuba is more general in the sense that is not modulo tuned while Montgomery it is. Does ...
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Why do $\alpha$ and the private key value in diffie-hellman should be from $2$ to $p-2$

I just started learning Diffie-Hellman Key Exchange. I couldn't get the reason of making $\alpha$ and private key for Alice and Bob constrained between $2$ and the prime number generated minus $2$ ...
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Is it possible to get my x,y coordinates from my “secret exponent”?

...or is that possible? I am very new to cryptography! But, I think it's very interesting!! I have autism and numbers "are my thing" :) I already understand a couple of things. For example I know ...
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Modular arithmetics in diffie-hellman

Studying the basics of Diffie-Hellman key exchange, I'm stuck at a basic operation used in the end of the key exchange (where you show that both computations actually are the same). Can someone ...
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Is it possible to recover an RSA modulus from its signatures?

Let's say that you have some small number of RSA signatures of known data: you know some pairs $(m_k, c_k)$ such that ${c_k}^e \equiv m_k \pmod n$. If you know $e$, because probably it's one of $\{3, ...
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46 views

What is the correct way to notate elliptic curve properties

When looking through elliptic curve and modulus posts i can see many examples of people referencing $p$ and $n$, in upper and lower case, and sometimes $Q$ is written instead of $p$ or $P$, and I ...
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Discrete logarithms: large prime modulus vs. large semi prime modulus

I have a cryptography homework question, in the question it says a cryptographic hash function in the form of $f(x) = g^x \bmod n$ , where $n$ is a very large prime (1024bits and more), $g$ is a ...
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lcm versus phi in RSA

In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) := (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the ...
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53 views

How hard to solve the given mod problem

Let c = a.b mod p where p is n bit prime number (e.g. 128 or 160 bit prime number); a - random number between 1 and (p-1); b - random number between 1 and (p-1); Given c, a and p, how hard to ...
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Is there an upper bound to the private exponent in RSA?

In the RSA algorithm, we choose $p$ and $q$ as prime numbers and we select a value $e$ which is coprime to $\varphi(pq)=(p-1)(q-1)$. Then we calculate $d:=e^{-1}\bmod\varphi(pq)$. My question is: ...
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138 views

Calculating $\mathbb F_{p^2}$-rational points of an elliptic curve defined over $\mathbb F_p$

How can I calculate points on an elliptic curve defined over $\mathbb F_p$, for example $y^2 \equiv x^3 + 1 \pmod p$, with coordinates in $\mathbb F_{p^2}$? (points might have complex number format in ...
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1answer
90 views

How to understand RSA Proofs of correctness at Wikipedia

From Wikipedia(https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29#Proofs_of_correctness): $m^{ed}$ ≡ $m$ $\pmod{q}$ $m^{ed}$ ≡ $m$ $\pmod{p}$ then $m^{ed}$ ≡ $m$ $\pmod{pq}$ Question: if $m^{ed}$ ...
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257 views

Using same modulus for RSA

I know that there exist some attack when using same modulus. Can two different pairs of RSA key have the same modulus? RSA cracking: The same message is sent to two different people problem But ...
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44 views

Requirements for the modulus in the Massey-Omura three pass protocol

In the Massey-Omura three pass protocol: How many bits long should the prime modulus $M$ be in order to be secure? Should the $M$ be secret? Should the $M$ be generated every time or it could be ...
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1answer
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In $\mathbb Z/p\mathbb Z$, is $(a+b\cdot r)$ a random value for fixed $a,b$ and random $r$?

Let $p$ be a prime number. Consider two fixed values $a,b\in\mathbb Z/p\mathbb Z$, where $b\neq0$, and a uniformly random value $r\leftarrow \mathbb Z/p\mathbb Z$. Is $v=a+b\cdot r$ a uniformly ...
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129 views

Modular reduction for NIST prime P256— understanding the data

I am working on a project where I need to implement elliptic curve cryptography, I am struggling from a long time in order to understand the working and the process. Modular finite field arithmetic, ...
3
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1answer
136 views

Base point in Ed25519?

The paper "High-speed high-security signatures" by Bernstein et al. introduces the Edwards curve Ed25519. Concerning the base point $B$, it says that $B$ is the unique point $(x, 4/5)\in E$ for ...
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RSA Encryption: What happens if n is a factor of the message M? [duplicate]

From what I have learned about RSA encryption, the message M and the modulo n must be coprime because Euler's theorem only holds for coprime numbers? for example, what happens if I choose p = 3, q = ...
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232 views

How does knowing the factorization of N help to obtain the secret?

Assuming $x=a^2 \pmod n$ and knowing $x$, $p$, $q$ how is it possible to obtain $a$?