Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value… the modulus.

learn more… | top users | synonyms

13
votes
1answer
15k views

Calculating RSA private exponent when given public exponent and the modulus factors using extended euclid

When given $p = 5, q = 11, N = 55$ and $e = 17$, I'm trying to compute the RSA private key $d$. I can calculate $\varphi(N) = 40$, but my lecturer then says to use the extended Euclidean algorithm to ...
10
votes
1answer
226 views

Shadowed identity in cryptography

I was trying to implement zero knowledge protocol for authentication based on the paper "A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing Both Transmission and Memory". ...
7
votes
3answers
3k views

Is sharing the modulus for multiple RSA key pairs secure?

In the public-key system RSA scheme, each user holds beyond a public modulus $m$ a public exponent, $e$, and a private exponent, $d$. Suppose that Bob's private exponent is learned by other users. ...
7
votes
2answers
704 views

How does Clifford Cocks 'Non-Secret Encryption' work?

I have read Clifford Cocks "A Note on 'Non-secret Encryption'" and thought I would try to implement this, but I don't seem to be able to get it to work. I'm obviously missing something. From the ...
6
votes
3answers
191 views

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: ...
6
votes
2answers
3k views

In RSA, why is it important choosing e so that it is coprime to φ(n)?

When choosing the public exponent e, it is stressed that $e$ must be coprime to $\phi(n)$, i.e. $\gcd(\phi(n), e) = 1$. I know that a common choice is to have $e = 3$ (which requires a good padding ...
6
votes
1answer
352 views

How does NaCL Poly1305 implementation do modular multiplication?

The NaCL ref implementation of Poly1305 performs modular multiplication to calculate a polynomial $\mod 2^{130} - 5$ using the following modular multiplication ...
6
votes
1answer
114 views

What are the computational benefits of primes close to the power of 2?

Recently I was reading some article about the Bernstein's Curve25519. This is a particular Montgomery curve over $\mathbb{F}_q$ where $q = {2^{255}-19}$. What I missed or was unable to understand is ...
6
votes
1answer
265 views

Why does this square root algorithm work?

I've been doing some elliptic curve cryptography, and a library I'm using has this slightly bizarre algorithm for computing modular square roots: Let $x$ be some quadratic residue modulo $p$ for some ...
5
votes
3answers
711 views

RSA with modulus n=p²q

I've been wondering what would happen if the RSA modulus would be $n=p^2q$ instead of $n=pq$. I feel like there is an obvious security flaw (as there is with most modifications). Would this choise of ...
5
votes
1answer
2k views

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 ...
5
votes
2answers
181 views

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 ...
5
votes
2answers
785 views

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 = ...
5
votes
1answer
214 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 ...
4
votes
4answers
440 views

Where does the $\varphi(n)$ part of RSA come from?

$e d \equiv 1 \pmod{\varphi(n)}$ Where does the $\varphi(n)$ part come from? How did the inventors of RSA arrive at $\varphi(n)$?
4
votes
2answers
404 views

Exactly two of the four roots must be greater than N/2

Theorem: Let $y$ be a quadratic residue in $\mathbb{Z}_N$* where $N=pq$. There are exactly four integers $x_1, x_2, x_3, x_4$ where $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ ...
4
votes
1answer
148 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?
4
votes
2answers
271 views

how to iteratively calculate a^emod n with modulus n sized 4096 bits

In most sites the exponent of the RSA public key is 24 bits. But the modulus can get to 4096 bits size. I have an accelerator that can get max. 2112 bit size modulus. It calculates ...
4
votes
1answer
242 views

Solve a Modular Exponentiation

It might be common, but if we had to solve an equation like this $m=s^{e}$ mod $n$ where $m,e,n$ are known. How can we find $s$. What optimisations could be applied? And what would the complexity of ...
4
votes
2answers
184 views

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 + ...
4
votes
4answers
872 views

How do institutions like banks do RSA with big primes?

When encrypting with RSA it is often infeasible to decrypt by just doing c^d mod n, because for example when using the primes $(p,q)=(12553,1233)$, which are small ...
4
votes
3answers
135 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 ...
3
votes
1answer
249 views

How is information disclosed by modular multiplication?

Consider the case that $c = a \cdot b \mod p$ where $p$ is a known prime and $0 < a < p$ and $0 < b < p$ are unknown integers numbers. Furthermore, some bits on the value of $c$ are ...
3
votes
2answers
366 views

Avoiding overflow when encrypting with RSA

When encrypting with RSA one calculates $ m^e \pmod n $ by doing the following: m^e % n Where $m$ is what we encrypt. Often $e$ is a very big number to make it ...
3
votes
3answers
217 views

Attack against modular inversion operation using side-channels?

I'm building a device that performs a modular inversions using a secret modulus. I would like to know if it is possible to recover all or part of this modulus by side-channels (timing, power, EMR, ...
3
votes
1answer
84 views

For which RSA moduli, precisely, is $e=d$ for all $e$?

This question shows that there are at least two valid RSA moduli $n$, namely $35$ and $91$, such that for any $e$ coprime to $\lambda(n)$, $$e^2\equiv1\mod\lambda(n)\text.$$ Reading the linked ...
3
votes
2answers
117 views

What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$?

The symbol $\mathbb Z/q\mathbb Z$ (given that $q$ is prime) represents the prime field $\mathbb Z_q$. Basically, the elements of this field are represented by $\{0, 1, \dots, q-1\}$, let's call this ...
3
votes
2answers
132 views

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?
3
votes
2answers
1k views

Reversing DJB2 Hashes

I have some spare time, and a few hundred DJB2-hashed values sitting around. I thought I'd try to do something "useful" and invert DJB2, such that I could calculate the plaintext of the hashes (which ...
3
votes
1answer
233 views

ECDSA: How to retrieve a non-random k

I have a question regarding the random $k$ number of ECDSA encryption. As far as I know, it is possible to retrieve $k$ (and thus the private key) from two signed messages if both used the same $k$. ...
3
votes
1answer
87 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 $m^{ed}$ ...
3
votes
1answer
105 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 ...
3
votes
4answers
1k views

Pseudocode for constant time modular exponentiation

I'm looking to implement modular exponentiation (for RSA) in constant time, but most of the examples I've found are more mathematical descriptions of the operations. Are there any references with ...
3
votes
1answer
640 views

How difficult is it to brute force d in RSA: d = (1/e) mod φ in a CPT attack?

Given that RSA key generation works by computing: n = pq φ = (p-1)(q-1) d = (1/e) mod φ If I was an attacker who wanted to brute force d, could I brute force d given just the public key, the ...
3
votes
2answers
111 views

Security concern about reducing hash value using modulo operation

As stated in the title, what I am looking for is information about a "technique" that I would like to use in some of my algorithms. Sometimes I need to map a hash function's result into a range of ...
3
votes
1answer
249 views

Decode message $m = p * q$, where $p ^ 5 \bmod N$, $q ^ 5 \bmod N$ and $N$ are known

I been going through some cryptography exercises and stumbled across this problem. Discover message $m$. You know that $m = p * q$. Also $p ^ 5 \bmod N$, $q ^ 5 \bmod N$ and $N$ are known ...
3
votes
0answers
85 views

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 $0 <= A,B < N$ in the Montgomery representation ...
3
votes
3answers
117 views

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 ...
2
votes
3answers
143 views

How does $g$ being a generator imply Diffie-Hellman's correctness?

In the Diffie-Hellman protocol for key exchange over an unsecured channel, we choose $p$ and $g$, where $g$ is a generator of $\mathbf{Z}_p^*$. However I want to know why this assumption makes the ...
2
votes
3answers
421 views

In RSA, why does $p$ have to be bigger than $q$ where $n=p \times q$?

In openSSL – during RSA key generation – if $q$ is bigger than $p$, they exchange them. Why is that?
2
votes
2answers
520 views

How does this happen in RSA malleability?

I don't understand how the $E(m)$ turns into $E(mt)$. I mean, I don't know how does that transformation happen and how does the equation occur. $$E(m) \cdot t^e \bmod n = (mt)^e \bmod n = E(mt)$$ ...
2
votes
2answers
3k views

How to find generator $g$ in a cyclic group?

As generator $g$ is use in DH how to find combination of prime $p$ and $g$? eg: if we choose $p=23$ its generator is $7$ (given in book) how to find it?
2
votes
2answers
160 views

Speed up modular exponentation with fixed base and modulus

Can someone explain, how $a^x \mod N$ can be speeded up, when $a$ and $N$ are known constants? How big is the gain and what resources are needed? ...
2
votes
2answers
89 views

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, ...
2
votes
1answer
79 views

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 ...
2
votes
2answers
263 views

PRP, PRF and modular arithmetic

Are there any arithmetic or mathematical functions that can be used as PRPs or PRFs ? Since, Conventional block ciphers like AES are that are proven to be good PRPs are not based on mathematics but ...
2
votes
2answers
43 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 ...
2
votes
1answer
98 views

Implementation of modular arithmetic?

In FIPS 186-3 appendix D.2 "implementation of Modular Arithmetic", they show shortcuts for solving $$B = A \mod m$$ for select Curves. How would you go about determining a valid short cuts for the ...
2
votes
1answer
1k views

ECC - Point Addition/Point Multiplication

So I have a very beginner-esque knowledge of ECDSA and I'm trying to write something in python to take a private key and output the public key (Basically from what I understand just trying to do the ...
2
votes
1answer
180 views

Why encrypting with private and public keys produce the same result?

Let say $e = 5$, $n = 119$ and $d = 77$. If I encrypt, for example, $m=15$ I get: $$m_1 = m^e=15 ^ 5 \mod 119 = 36\qquad\text{and}\qquad m_2 =m^d= 15 ^ {77} \mod 119 = 36$$. Why? Is it always like ...