Discrete Root

The problem of finding a discrete root is defined as follows. Given a prime $n$ and two integers $a$ and $k$, find all $x$ for which:

$x^k \equiv a \pmod n$

The algorithm

We will solve this problem by reducing it to the discrete logarithm problem.

Let's apply the concept of a primitive root modulo $n$. Let $g$ be a primitive root modulo $n$. Note that since $n$ is prime, it must exist, and it can be found in $O(Ans \cdot \log \phi (n) \cdot \log n) = O(Ans \cdot \log^2 n)$ plus time of factoring $\phi (n)$.

We can easily discard the case where $a = 0$. In this case, obviously there is only one answer: $x = 0$.

Since we know that $n$ is a prime and any number between 1 and $n-1$ can be represented as a power of the primitive root, we can represent the discrete root problem as follows:

$(g^y)^k \equiv a \pmod n$

where

$x \equiv g^y \pmod n$

This, in turn, can be rewritten as

$(g^k)^y \equiv a \pmod n$

Now we have one unknown $y$, which is a discrete logarithm problem. The solution can be found using Shanks' baby-step giant-step algorithm in $O(\sqrt {n} \log n)$ (or we can verify that there are no solutions).

Having found one solution $y_0$, one of solutions of discrete root problem will be $x_0 = g^{y_0} \pmod n$.

Finding all solutions from one known solution

To solve the given problem in full, we need to find all solutions knowing one of them: $x_0 = g^{y_0} \pmod n$.

Let's recall the fact that a primitive root always has order of $\phi (n)$, i.e. the smallest power of $g$ which gives 1 is $\phi (n)$. Therefore, if we add the term $\phi (n)$ to the exponential, we still get the same value:

$x^k \equiv g^{ y_0 \cdot k + l \cdot \phi (n)} \equiv a \pmod n \forall l \in Z$

Hence, all the solutions are of the form:

$x = g^{y_0 + \frac {l \cdot \phi (n)}{k}} \pmod n \forall l \in Z$.

where $l$ is chosen such that the fraction must be an integer. For this to be true, the numerator has to be divisible by the least common multiple of $\phi (n)$ and $k$. Remember that least common multiple of two numbers $lcm(a, b) = \frac{a \cdot b}{gcd(a, b)}$; we'll get

$x = g^{y_0 + i \frac {\phi (n)}{gcd(k, \phi (n))}} \pmod n \forall i \in Z$.

This is the final formula for all solutions of the discrete root problem.

Implementation

Here is a full implementation, including procedures for finding the primitive root, discrete log and finding and printing all solutions.

int gcd(int a, int b) {
    return a ? gcd(b % a, a) : b;
}

int powmod(int a, int b, int p) {
    int res = 1;
    while (b > 0) {
        if (b & 1) {
            res = res * a % p;
        }
        a = a * a % p;
        b >>= 1;
    }
    return res;
}

// Finds the primitive root modulo p
int generator(int p) {
    vector<int> fact;
    int phi = p-1, n = phi;
    for (int i = 2; i * i <= n; ++i) {
        if (n % i == 0) {
            fact.push_back(i);
            while (n % i == 0)
                n /= i;
        }
    }
    if (n > 1)
        fact.push_back(n);

    for (int res = 2; res <= p; ++res) {
        bool ok = true;
        for (int factor : fact) {
            if (powmod(res, phi / factor, p) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return res;
    }
    return -1;
}

// This program finds all numbers x such that x^k = a (mod n)
int main() {
    int n, k, a;
    scanf("%d %d %d", &n, &k, &a);
    if (a == 0) {
        puts("1\n0");
        return 0;
    }

    int g = generator(n);

    // Baby-step giant-step discrete logarithm algorithm
    int sq = (int) sqrt (n + .0) + 1;
    vector<pair<int, int>> dec(sq);
    for (int i = 1; i <= sq; ++i)
        dec[i-1] = {powmod(g, i * sq * k % (n - 1), n), i};
    sort(dec.begin(), dec.end());
    int any_ans = -1;
    for (int i = 0; i < sq; ++i) {
        int my = powmod(g, i * k % (n - 1), n) * a % n;
        auto it = lower_bound(dec.begin(), dec.end(), make_pair(my, 0));
        if (it != dec.end() && it->first == my) {
            any_ans = it->second * sq - i;
            break;
        }
    }
    if (any_ans == -1) {
        puts("0");
        return 0;
    }

    // Print all possible answers
    int delta = (n-1) / gcd(k, n-1);
    vector<int> ans;
    for (int cur = any_ans % delta; cur < n-1; cur += delta)
        ans.push_back(powmod(g, cur, n));
    sort(ans.begin(), ans.end());
    printf("%d\n", ans.size());
    for (int answer : ans)
        printf("%d ", answer);
}

Practice problems