Greatest Common Divisor

Description

The Greatest Common Divisor (GCD), sometimes known as the highest common factor, is the largest number which divides two positive integers (a,b).

For a = 12, b = 8 we can calculate the divisors of a: {1,2,3,4,6,12} and the divisors of b: {1,2,4,8}. Comparing these two, we see that gcd(a,b) = 4.

Now imagine we take a = 11, b = 17. Both a and b are prime numbers. As a prime number has only itself and 1 as divisors, gcd(a,b) = 1.

We say that for any two integers a,b, if gcd(a,b) = 1 then a and b are coprime integers.

If a and b are prime, they are also coprime. If a is prime and b < a then a and b are coprime.

Think about the case for a prime and b > a, why are these not necessarily coprime?

There are many tools to calculate the GCD of two integers, but for this task we recommend looking up Euclid’s Algorithm.

Try coding it up; it’s only a couple of lines. Use a = 12, b = 8 to test it.

Now calculate gcd(a,b) for a = 66528, b = 52920 and enter it below.

Extended GCD

Description

Let a and b be positive integers.

The extended Euclidean algorithm is an efficient way to find integers u,v such that

Later, when we learn to decrypt RSA, we will need this algorithm to calculate the modular inverse of the public exponent.

Using the two primes p = 26513, q = 32321, find the integers u,v such that

Enter whichever of u and v is the lower number as the flag.

Knowing that p,q are prime, what would you expect gcd(p,q) to be? For more details on the extended Euclidean algorithm, check out this page.

Modular Arithmetic 1

Description

Imagine you lean over and look at a cryptographer’s notebook. You see some notes in the margin:

At first you might think they’ve gone mad. Maybe this is why there are so many data leaks nowadays you’d think, but this is nothing more than modular arithmetic modulo 12 (albeit with some sloppy notation).

You may not have been calling it modular arithmetic, but you’ve been doing these kinds of calculations since you learnt to tell the time (look again at those equations and think about adding hours).

Formally, “calculating time” is described by the theory of congruences. We say that two integers are congruent modulo m if a ≡ b mod m.

Another way of saying this, is that when we divide the integer a by m, the remainder is b. This tells you that if m divides a (this can be written as m | a) then a ≡ 0 mod m.

Calculate the following integers:

The solution is the smaller of the two integers.

Modular Arithmetic 2

Description

We’ll pick up from the last challenge and imagine we’ve picked a modulus p, and we will restrict ourselves to the case when p is prime.

The integers modulo p define a field, denoted Fp.

If the modulus is not prime, the set of integers modulo n define a ring.

A finite field Fp is the set of integers {0,1,...,p-1}, and under both addition and multiplication there is an inverse element b for every element a in the set, such that a + b = 0 and a * b = 1.

Note that the identity element for addition and multiplication is different! This is because the identity when acted with the operator should do nothing: a + 0 = a and a * 1 = a.

Lets say we pick p = 17. Calculate 317 mod 17. Now do the same but with 517 mod 17.

What would you expect to get for 716 mod 17? Try calculating that.

This interesting fact is known as Fermat’s little theorem. We’ll be needing this (and its generalisations) when we look at RSA cryptography.

Now take the prime p = 65537. Calculate 27324678765465536 mod 65537.

Did you need a calculator?

Analyze

$a^{p} \equiv a \mod p$，浅浅变形一下：$a^{p-1} \equiv 1 \mod p$

Modular Inverting

Description

As we’ve seen, we can work within a finite field Fp, adding and multiplying elements, and always obtain another element of the field.

For all elements g in the field, there exists a unique integer d such that g * d ≡ 1 mod p.

This is the multiplicative inverse of g.

Example: 7 * 8 = 56 ≡ 1 mod 11

What is the inverse element: 3 * d ≡ 1 mod 13?

Think about the little theorem we just worked with. How does this help you find the inverse of an element?

Analyze

python 的 pow() 函数真的很好用！

SumUp

$φ(n)$ 即 1 ~ (n-1) 范围内与 n 互素的数的个数，若 n 本身为素数，则 $φ(n)=n-1$ 。

Description

We’ve looked at multiplication and division in modular arithmetic, but what does it mean to take the square root modulo an integer?

For the following discussion, let’s work modulo p = 29. We can take the integer a = 11 and calculate a² = 5 mod 29.

As a = 11, a² = 5, we say the square root of 5 is 11.

This feels good, but now let’s think about the square root of 18. From the above, we know we need to find some integer a such that a² = 18

Your first idea might be to start with a = 1 and loop to a = p-1. In this discussion p isn’t too large and we can quickly look.

Have a go, try coding this and see what you find. If you’ve coded it right, you’ll find that for all a ∈ Fp* you never find an a such that a² = 18.

What we are seeing, is that for the elements of F*p, not every element has a square root. In fact, what we find is that for roughly one half of the elements of Fp*, there is no square root.

We say that an integer x is a Quadratic Residue if there exists an a such that a² = x mod p. If there is no such solution, then the integer is a Quadratic Non-Residue.

In other words, x is a quadratic residue when it is possible to take the square root of x modulo an integer p.

In the below list there are two non-quadratic residues and one quadratic residue.

Find the quadratic residue and then calculate its square root. Of the two possible roots, submit the smaller one as the flag.

If a² = x then (-a)² = x. So if x is a quadratic residue in some finite field, then there are always two solutions for a.

Analyze

Legendre 符号（欧拉判别法）的内容，二次剩余。

Given: p, x[]

Legendre Symbol

Description

In Quadratic Residues we learnt what it means to take the square root modulo an integer. We also saw that taking a root isn’t always possible.

In the previous case when p = 29, even the simplest method of calculating the square root was fast enough, but as p gets larger, this method becomes wildly unreasonable.

Lucky for us, we have a way to check whether an integer is a quadratic residue with a single calculation thanks to Legendre. In the following, we will assume we are working modulo a prime p.

Before looking at Legendre’s symbol, let’s take a brief detour to see an interesting property of quadratic (non-)residues.

Want an easy way to remember this? Replace “Quadratic Residue” with +1 and “Quadratic Non-residue” with -1, all three results are the same!

So what’s the trick? The Legendre Symbol gives an efficient way to determine whether an integer is a quadratic residue modulo an odd prime p.

Legendre’s Symbol: (a / p) ≡ a(p-1)/2 mod p obeys:

Which means given any integer a, calculating pow(a,(p-1)/2,p) is enough to determine if a is a quadratic residue.

Now for the flag. Given the following 1024 bit prime and 10 integers, find the quadratic residue and then calculate its square root; the square root is your flag. Of the two possible roots, submit the larger one as your answer.

So Legendre’s symbol tells us which integer is a quadratic residue, but how do we find the square root?! The prime supplied obeys p = 3 mod 4, which allows us easily compute the square root. The answer is online, but you can figure it out yourself if you think about Fermat’s little theorem.

Analyze

Legendre 符号的欧拉判别法。

Modular Square Root

Description

In Legendre Symbol we introduced a fast way to determine whether a number is a square root modulo a prime. We can go further: there are algorithms for efficiently calculating such roots. The best one in practice is called Tonelli-Shanks, which gets its funny name from the fact that it was first described by an Italian in the 19th century and rediscovered independently by Daniel Shanks in the 1970s.

All primes that aren’t 2 are of the form p ≡ 1 mod 4 or p ≡ 3 mod 4, since all odd numbers obey these congruences. As the previous challenge hinted, in the p ≡ 3 mod 4 case, a really simple formula for computing square roots can be derived directly from Fermat’s little theorem. That leaves us still with the p ≡ 1 mod 4 case, so a more general algorithm is required.

In a congruence of the form r² ≡ a mod p, Tonelli-Shanks calculates r.

Tonelli-Shanks doesn’t work for composite (non-prime) moduli. Finding square roots modulo composites is computationally equivalent to integer factorization.

The main use-case for this algorithm is finding elliptic curve co-ordinates. Its operation is somewhat complex so we’re not going to discuss the details, however, implementations are easy to find and Sage has one built-in.

Find the square root of a modulo the 2048-bit prime p. Give the smaller of the two roots as your answer.

Analyze

sympy 模块内有封装好的函数可以计算二次剩余。

SumUp

$x^{n} \equiv a\ mod\ p$，求 x

Chinese Remainder Theorem

Description

The Chinese Remainder Theorem gives a unique solution to a set of linear congruences if their moduli are coprime.

This means, that given a set of arbitrary integers ai, and pairwise coprime integers ni, such that the following linear congruences hold:

Note “pairwise coprime integers” means that if we have a set of integers {n1, n2, ..., ni}, all pairs of integers selected from the set are coprime: gcd(ni, nj) = 1.

There is a unique solution x ≡ a mod N where N = n1 * n2 * ... * nn.

In cryptography, we commonly use the Chinese Remainder Theorem to help us reduce a problem of very large integers into a set of several, easier problems.

Given the following set of linear congruences:

Find the integer a such that x ≡ a mod 935

Starting with the congruence with the largest modulus, use that for x ≡ a mod p we can write x = a + k*p for arbitrary integer k.

Description

Adrien’s been looking at ways to encrypt his messages with the help of symbols and minus signs. Can you find a way to recover the flag?

Analyze

python 学得不好，加密函数理解了好久。

（最后一行的代码简直写得太优雅咯

Modular Binomials

Description

Rearrange the following equations to get p,q