This contains algorithms for calculating square roots modulo a prime p using methods described in the paper "On Calculating Square Roots in GF(p)"
modsqrt.py
This contains two different methods of calculating square roots in GF(p): the Cipolla-Lehmer algorithm and the GF(p3) square root algorithm. The four main functions in modsqrt.py are the following:
CL(c,b,p)
This function is the Cipolla-Lehmer algorithm for calculating the square root of c mod p assuming that c is a quadratic residue, that p is any prime > 3 and that 0 < b < p or it returns the value of 0. If the quadratic polynomial selected by the algorithm is irreducible, it will return a square root of c mod p otherwise it will return 0.
S(d,b,p)
This is the GF(p3) square root algorithm that calculates the square root of d mod p assuming that d is a quadratic residue, that p is prime = 5 mod 6, and that 0 < b < p and that the cubic polynomial selected by the algorithm is irreducible, otherwise it returns the value of 0. This algorithm is based on exponentiating in GF(p3) using the irreducible cubic polynomial x3 + ax + b where a is chosen such that d = -(4a3 + 27b2) mod p.
sqrt1(c,p)
This function calculates the square root of c mod p assuming p > 3 by calling the function CL(c,b,p) until an irreducible quadratic polynomial is found.
sqrt2(d,p)
This function calculates a square root of d mod p assuming p = 5 mod 6 by calling S(d,b,p) until an irreducible cubic polynomial is found.
cubicsqrt.py
This is a deterministic algorithm for calculating the square root of the discriminant of a cubic polynomial that is irreducible mod p. The two main functions in cubicsqrt.py are:
ds(q,p)
This determines the discriminant d mod p of the cubic polynomial q(x) = x3 + q[0]x2 + q[1]x + q[2].
S3(q,p)
This function uses the GF(p3) square root algorithm to deterministically calculate the square root of d mod p where d is the discriminant of the cubic polynomial q(x) = x3 + q[0]x2 + q[1]x + q[2] in O(log3p) time for p = 5 mod 6 where p is the prime modulus.
cubicreciprocity.py
This program can be used to verify the correctness of the cubic reciprocity identity in the paper "On Calculating Square Roots in GF(p)" which is listed as conjecture 7. The main function is CubicReciprocity(x,y,q) which checks the correctness of a cubic reciprocity identity for two primes p and q where p = x2 + xy + y2 for positive integers x and y and where q = 5 mod 6.
cubicreciprocity2.py
This is an improved version of cubicreciprocity.py since the value of the prime q in the function CubicReciprocity(x,y,q) can be equal to 5 mod 6 as in cubicreciprocity.py but also generalizes to the case where q = 1 mod 6.
modsqrt21.py
This is the newest version of modsqrt.py and has the following added functions:
(1) verify(a,b,p) This function can be used to verify the correctness of conjecture 4 in the paper "On Calculating Square Roots in GF(p)"
(2) sqrt2(d,p) and S(d,b,p) have been improved to work in the cases p = 4 mod 9 or p = 7 mod 9 in addition to working in the case that p = 5 mod 6 where p refers to the prime modulus.
(3) sqrt4(d,p) and S4(d,b,p) have been added. These are improved versions of sqrt2(d,p) and S(d,b,p) respectively and work in all cases for p = 1 mod 6 as well as p = 5 mod 6. Unlike sqrt2(d,p), sqrt4(d,p) while calculating the modular square root of a quadratic residue d mod p avoids having to calculate cube roots which improves the efficiency if p = 1 mod 6.
SquareRoot03.tex and SquareRoot03.pdf
These are the TeX and pdf files for the third version of the paper "On Calculating Square Roots in GF(p)"