## Legendre’s Symbol

### May 1, 2012

The Legendre Symbol and its cousin the Jacobi Symbol are used in modular arithmetic to determine if a number a is a quadratic residue to the modulus m. A number a is a quadratic residue if there exists a number x such that x2a (mod m) and is defined only when a and m are co-prime. For instance, a2 (mod 7) for a from 0 to 6 is the list 02 (mod 7) = 0, 12 (mod 7) = 1, 22 (mod 7) = 4, 32 (mod 7) = 2, 42 (mod 7) = 2, 52 (mod 7) = 4, and 62 (mod 7) = 1, so the quadratic residues of 7 are 1, 2 and 4 (0 is excluded because it isn’t co-prime to 7). The jacobi symbol considers any odd modulus; the legendre symbol is limited to odd prime moduli. The symbols are usually written in parentheses with a over m, like this: $\left( a \atop m \right)$. Sometimes the symbol is written with a horizontal rule between the a and m, and sometimes it is written on a single line as (a / m).

The legendre/jacobi symbol can be calculated according to the following three termination rules:

1. (0 / m) = 0

2. (1 / m) = 1

3. (2 / m) = −1 if m mod 8 ∈ {3, 5} or 1 if m mod 8 ∈ {1, 7}

and the following three reduction rules:

4. [reducing factors of 2] (2a / m) = (2 / m) × (a / m)

5. [reducing modulo m] (a / m) = (a (mod m) / m) if am or a < 0

6. [reducing odd co-primes a and m] (a /m) = −(m / a) if am ≡ 3 (mod 4), or (m / a) otherwise

Thus, the legendre/jacobi symbol is 1 if a is a quadratic residue, -1 if a is not a quadratic residue, and 0 if a and m are not co-prime.

Our various prime-number programs have used a definition of the legendre/jacobi symbol that doesn’t work; for some values of a and m it returned wrong results, and for other values of a and m it entered an infinite loop. This exercise fixes the problem.

Your task is to write a function that calculates the legendre/jacobi symbol using the rules given above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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### 13 Responses to “Legendre’s Symbol”

1. ardnew said

I first tried translating this to a completely iterative algorithm, but it was becoming a mess.. I also couldn’t think of a better way to reduce the rule 3 logic.

use strict;
use warnings;

sub reduce
{
my ($p,$a, $m) = (1, @_); if ($a > 2)
{
# rule 4
$p *= reduce(2,$m) * reduce($a / 2,$m) unless $a & 1; # rule 6$p *= -reduce(-$m,$a) if $a % 4 == 3 and$m % 4 == 3;
}

# rules 1 and 2
$p *=$a unless $a >> 1; # rule 3 if (2 ==$a)
{
my $mod8 =$m % 8;
$p *= -1 if 3 ==$mod8 or 5 == $mod8;$p *=  1 if 1 == $mod8 or 7 ==$mod8;
}

return $p; } sub legendre($)
{
my ($a,$m) = @{(shift)};

die "modulus must be prime positive integer$/" unless int($m) == $m and$m > 0 and $m & 1; # need to also verify$m is prime, but meh

# rule 5
my $n =$a % $m; print "($a / $m) = ".reduce($n, $m).$/;
}

legendre(\@ARGV);

2. programmingpraxis said

Ardnew: the usual iterative solution looks like this:

a = a mod m; t = 1; while (a != 0) {     while (a even) {         a = a / 2;         if (m mod 8 in {3,5}) t = -t;     }     (a,m) = (m,a) // swap variables     if (a === m === 3 (mod 4)) t = -t     a = a mod m; } if (m == 1) return t; return 0;

3. ardnew said

very nice! always seems so simple once you see the answer…

also, there’s a bug in my code that doesn’t reapply rule 5 when rule 6 is evaluated. I’m not familiar enough with the math to prove this behavior is incorrect though

4. programmingpraxis said

In a private email, Mike pointed out that I missed the else clause in Rule 6. It is now fixed.

5. Mike said

An iterative Python version:

def jacobi(n, m):
j = 1

# rule 5
n %= m

while n:
# rules 3 and 4
t = 0
while not n & 1:
n /= 2
t += 1
if t&1 and m%8 in (3, 5):
j = -j

# rule 6
if (n % 4 == m % 4 == 3):
j = -j

# rules 5 and 6
n, m = m % n, n

return j if m == 1 else 0


6. ardnew said

a) I love that 3-way equality condition in rule 6. Never seen that before :)
b) Can you explain the behavior of the expression: “if t&1 and m%8 in (3, 5)”

And Mr. programmingpraxis (i assume that is your real name), a question:
Is rule 5 even a necessary reduction? I’m beginning to think its there merely for computational optimization

7. ardnew said

Oops, I was caught up in that equality expression, disregard comment b), Mike.

8. Mike said

@ardnew,

The ability to chain relational operators is a nice feature of Python syntax. It compiles to a series of 2-way relational tests ‘and’ed together. “if n%4 == m%4 == 3:” compiles like “if n%4 == m%4 and m%4 == 3:”, but the m%4 is only evaluated once. It is handy when you want to check if a value is between two limits, like “if lowerbound <= x = y <= z:" in a previous exercise to determine that y was no larger than either x or z. You can also chain any number together: "a c == d …”

Rule 5 is the only way to reduce ‘a’ when it’s not even.

9. Mike said

Oops, wordpress ate some of my relational operators as html tags. Should be:

The ability to chain relational operators is a nice feature of Python syntax. It compiles to a series of 2-way relational tests ‘and’ed together. “if n%4 == m%4 == 3:” compiles like “if n%4 == m%4 and m%4 == 3:”, but the m%4 is only evaluated once. It is handy when you want to check if a value is between two limits, like “if lowerbound <= x < upperbound:". I used something like "if x >= y <= z:" in a previous exercise to determine that y was no larger than either x or z. You can also chain any number together: "a < b > c == d …”

Rule 5 is the only way to reduce ‘a’ when it’s not even.

10. Graham said

Since Mike already did an iterative Python version, here’s a straight translation of the rules to Haskell:

jacobi :: Integer -> Integer -> Integer
jacobi a m
| even m        = error "m must be odd."
| a == 0        = 0
| a == 1        = 1
| a == 2        = if (m mod 8) elem [3, 5] then -1 else 1
| even a        = jacobi 2 m * jacobi (a div 2) m
| a >= m        = jacobi (a mod m) m
| otherwise     = let j = jacobi m a in
if (a mod 4 == 3) && (m mod 4 == 3) then -j else j


Not the most efficient solution, probably. Happy to see another math exercise, though! (even though it should be titled “Jacobi’s Symbol,” since we don’t know m is prime)

11. Mike said

As a preprocessing step, I think rule 5 needs to be applied if a (7 / -1) by rule 6 (the only rule that could apply
(7 / -1) => (0 / -1) by rule 5
(0 / -1) => 0 by rule 1

However, (-1 / 7) = -1 according to the programmingpraxis solution.

If rule 5 is applied first, then:
(-1 / 7) => (6 / 7) by rule 5
(6 / 7) => (2 / 7) * (3 / 7) => (3 / 7) => -(7 / 3) => -(1 / 3) => -1 by rules 4, 3, 6, 5, and 2, respectively

12. Mike said

I did it again. (Maybe I need an option to preview a post)

The previous post should be:

As a preprocessing step, I think rule 5 needs to be applied if a < 0.

For example:
(-1 / 7) => (7 / -1) by rule 6 (the only rule that could apply)
(7 / -1) => (0 / -1) by rule 5
(0 / -1) => 0 by rule 1

However, (-1 / 7) = -1 according to the programmingpraxis solution.

If rule 5 is applied first, then:
(-1 / 7) => (6 / 7) by rule 5
(6 / 7) => (2 / 7) * (3 / 7) => (3 / 7) => -(7 / 3) => -(1 / 3) => -1; by rules 4, 3, 6, 5, and 2, respectively.

13. programmingpraxis said

The jacobi symbol for (-1 7) is -1. Wolfram Alpha agrees.

In arithmetic mod m, the only numbers allowed are 0 to m-1 inclusive. There are no negative numbers. So any time you see a negative number you have to add m repeatedly until it is in the range 0 to m-1 inclusive. So yes, Rule 5 applies whenever a<0. I'll edit the rule.