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Finding a Primitive p-th Root of Unity in a Finite Field, with C++ Code

To this day, no method of finding a generator of Z_p^* is known to be more efficient than essentially trying 2, then 3, and so on. Who cares? Well, the difficulty of breaking a certain public key cryptosystem (due to El Gamal) depends on the difficulty of working with generators of Z_p^*.Keith Conrad

An nth root of unity in a finite field F is an element r \in F satisfying r^n=1, where n is an integer. If n is the smallest positive integer with this property, r is called a primitive nth root of unity. If r is a primitive nth root of unity, then all elements in the set \mu_n = \{1, r, r^2, \cdots, r^{n-1}\} are also roots of unity. Actually, the set \mu_n form a cyclic group of order n under multiplication, with generator r.

Problem: Suppose you are given a finite field F=GF(2^d) of degree d, and you are promised that there indeed exists a primitive pth root of unity r\in F for p prime. Find r, and in particular, produce a C++code that finds it.

In what follows, we talk about how to find such a root and provide my C++ code; the code uses the awesome NTL library.

Continue reading “Finding a Primitive p-th Root of Unity in a Finite Field, with C++ Code”

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Const-Pointer and Pointer-to-Const in C / C++

You must have known it already, but I can’t stop myself from showing off the¬†interplay of const¬†with pointers in C/C++.

Here goes the code. It’s simple.


void main() {

	int arr[]={1,2,3,4}; // data

	const int *p1 = &arr[0]; // non-const ptr to const data
	int const *p2 = &arr[0]; // non-const ptr to const data
	int* const p3 = &arr[0]; // const ptr to non-const data
	const int* const p4 = &arr[0]; // const ptr to const data

	p1++; // ok
	p1[0]++; // compile error: modifying const data

	p2++; // ok
	p2[0]++; // compile error: modifying const data

	p3++; // compile error: modifying const ptr
	p3[0]++; // ok 

	p4++; // compile error: modifying const ptr
	p4[0]++; // compile error: modifying const data

}

Continue reading “Const-Pointer and Pointer-to-Const in C / C++”