The Jacobi symbol

The Jacobi symbol is a generalization of the Legendre symbol. In this post, we define the Jacobi symbol and discuss some of its important properties. The Legendre symbol (coupled with the law of reciprocity) is a useful tool in determining whether a number is a quadratic residue modulo an odd prime. However, the Legendre symbol is also restrictive since the bottom argument must be a prime. The Jacobi symbol generalizes the Legendre symbol and in many cases is easier to evaluate than the Legendre symbol. This is demonstrated in several examples.

____________________________________________________________________________

Defining The Jacobi Symbol

The Legendre symbol is \displaystyle  \biggl(\frac{a}{p}\biggr) where the bottom argument is always an odd prime and the top argument can be any integer. When a is an integer that is relatively prime to the odd prime p, the Legendre symbol \displaystyle  \biggl(\frac{a}{p}\biggr) carries information on whether the integer a is quadratic residue modulo p. The following gives the definition of the Legendre symbol.

    \displaystyle  \biggl(\frac{a}{p}\biggr) = \left\{ \begin{array}{rl}           0 &\ \text{if } a \equiv 0 \ (\text{mod} \ p) \\           1 &\ \text{if } a \not \equiv 0 \ (\text{mod} \ p) \text{ and } a  \text{ is a quadratic residue modulo }p\\          -1 &\ \text{if } a \not \equiv 0 \ (\text{mod} \ p) \text{ and } a  \text{ is a quadratic nonresidue modulo }p \end{array} \right.

The Jacobi symbol \displaystyle  \biggl(\frac{a}{n}\biggr) has the same look as the Legendre symbol. The top argument can be any integer. The bottom argument is an odd positive integer. If the odd positive integer n>1 has the prime factorization n=p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_t^{e_t}, the Jacobi symbol is defined as follows:

    \displaystyle  \biggl(\frac{a}{n}\biggr)=\biggl(\frac{a}{p_1}\biggr)^{e_1} \times \biggl(\frac{a}{p_2}\biggr)^{e_2} \times \cdots \times \biggl(\frac{a}{p_t}\biggr)^{e_t}

For convenience, we define \displaystyle  \biggl(\frac{a}{1}\biggr)=1. A slightly different, but equivalent, definition is that whenever the odd integer n>1 has the prime factorization n=q_1 \times q_2 \times \cdots \times q_k (the prime numbers q_j are not necessarily distinct), the symbol \displaystyle  \biggl(\frac{a}{n}\biggr) is defined as follows:

    \displaystyle  \biggl(\frac{a}{n}\biggr)=\biggl(\frac{a}{q_1}\biggr) \times \biggl(\frac{a}{q_2}\biggr) \times \cdots \times \biggl(\frac{a}{q_k}\biggr)

The Jacobi symbol is defined using the Legendre symbol according to the prime factorization of the bottom argument.

____________________________________________________________________________

Properties of The Jacobi Symbol

Now that the Jacobi symbol has been defined based on the Legendre symbol, there are two natural questions.

  • Does the value of \displaystyle  \biggl(\frac{a}{n}\biggr)= \pm 1 indicates that a is a quadratic residue or nonresidue modulo n?
  • The Legendre symbol follows a set of rules (e.g. the law of quadratic reciprocity) that make the Legendre symbol a versatile tool. Does the Jacobi symbol follow these rules? Conceptually, evaluating the Jacobi symbol requires factoring the bottom argument of the symbol. On the surface, this appears to be a limitation. However, if the Jacobi symbol follows the rules for the Legendre symbol including the law of quadratic reciprocity, it will be just as easy to evaluate the Jacobi symbol.

For the first question, the following is true: If a is a quadratic residue modulo n, then \displaystyle  \biggl(\frac{a}{n}\biggr)=1. The contrapositive of this statement is that if \displaystyle  \biggl(\frac{a}{n}\biggr)=-1, then the number a is a quadratic nonresidue modulo the odd integer n, i.e. the equation x^2 \equiv a \ (\text{mod} \ n) has no solutions. However, the value of \displaystyle  \biggl(\frac{a}{n}\biggr)=1 does not imply that a is a quadratic residue modulo n. For example, \displaystyle  \biggl(\frac{2}{15}\biggr)=1 but the equation x^2 \equiv 2 \ (\text{mod} \ 15) has no solutions. Note that both equations x^2 \equiv 2 \ (\text{mod} \ 3) and x^2 \equiv 2 \ (\text{mod} \ 5) have no solutions. This means that the value of \displaystyle  \biggl(\frac{2}{15}\biggr)=1 is the product of two Legendre symbols of -1.

Theorem 1
Let n be an odd positive integer. Let a be such that \text{GCD}(a,n)=1. Then If a is a quadratic residue modulo n, then \displaystyle  \biggl(\frac{a}{n}\biggr)=1. The converse does not hold if n is not a prime.

Proof
Suppose that a is a quadratic residue modulo n, i.e. the equation x^2 \equiv a \ (\text{mod} \ n) has a solution. Let n=p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_t^{e_t} be the prime factorization of n. Let m_i=p_i^{e_i} for each i. By the Chinese remainder theorem, the equation x^2 \equiv a \ (\text{mod} \ m_i) has a solution for each i=1,2,\cdots,t. it then follows that the equation x^2 \equiv a \ (\text{mod} \ p_i) has a solution too for each i=1,2,\cdots,t. Hence the Legendre symbol \displaystyle  \biggl(\frac{a}{p_i}\biggr)=1 for each i. It follows that the Jacobi symbol \displaystyle  \biggl(\frac{a}{n}\biggr)=1. \square

For the second question, the rules that are obeyed by the Legendre symbol are also obeyed by the Jacobi symbol. Some of the rules are easily seen to carry over to the Jacobi symbol. The fact that the law of quadratic reciprocity and the two supplements are obeyed by the Jacobi symbol is not entirely obvious from the definition. However the proof is not difficult. We give a complete proof here.

Lemma 2
Let a and b be odd positive integers. The following two conditions hold.

    \displaystyle \frac{a-1}{2} \times \frac{b-1}{2} \equiv \frac{ab-1}{2} \ (\text{mod} \ 2)

    \displaystyle \frac{a^2-1}{8} \times \frac{b^2-1}{8} \equiv \frac{(ab)^2-1}{8} \ (\text{mod} \ 2)

Proof
The lemma is established by the following derivation:

    \displaystyle \frac{ab-1}{2}-\biggl[\frac{a-1}{2}+\frac{b-1}{2} \biggr]=\frac{ab-a-b+1}{2}=\frac{(a-1)(b-1)}{2}

    \displaystyle \frac{(ab)^2-1}{8}-\biggl[\frac{a^2-1}{8}+\frac{b^2-1}{8} \biggr]=\frac{(ab)^2-a^2-b^2+1}{8}=\frac{(a^2-1)(b^2-1)}{8}

Because both a and b are odd integers, the right-hand-side of both equations are even integers and thus \equiv 0 \ (\text{mod} \ 2). \square

Theorem 3 (Properties of Jacobi Symbol)

  1. \displaystyle  \biggl(\frac{a}{n}\biggr) = \left\{ \begin{array}{rl}           0 &\ \text{if } \text{GCD}(a,n) >1 \\          \pm 1 &\ \text{if } \text{GCD}(a,n)=1 \end{array} \right.
  2. \text{ }

  3. If n is a prime, then the Jacobi symbol \displaystyle  \biggl(\frac{a}{n}\biggr) coincides with the Legendre symbol \displaystyle  \biggl(\frac{a}{n}\biggr).
  4. \text{ }

  5. If a \equiv b \ (\text{mod} \ n), then \displaystyle  \biggl(\frac{a}{n}\biggr)=\biggl(\frac{b}{n}\biggr).
  6. \text{ }

  7. The Jacobi symbol is multiplicative when the bottom argument is fixed, i.e. \displaystyle  \biggl(\frac{ab}{n}\biggr)=\biggl(\frac{a}{n}\biggr) \biggl(\frac{b}{n}\biggr).
  8. \text{ }

  9. The Jacobi symbol is multiplicative when the upper argument is fixed, i.e. \displaystyle  \biggl(\frac{a}{mn}\biggr)=\biggl(\frac{a}{m}\biggr) \biggl(\frac{a}{n}\biggr).
  10. \text{ }

  11. (The law of quadratic reciprocity)
    Let a and b be relatively prime odd positive integers. Then

      \displaystyle \biggl(\frac{a}{b}\biggr) \biggl(\frac{b}{a}\biggr)=(-1)^{(a-1)/2 \times (b-1)/2}

      Equivalently, \displaystyle  \biggl(\frac{a}{b}\biggr) = \left\{ \begin{array}{rl}           \displaystyle  \biggl(\frac{b}{a}\biggr) &\ \text{if } a \equiv 1 \ (\text{mod} \ 4) \text{ or }  b \equiv 1 \ (\text{mod} \ 4) \\         \displaystyle  -\biggl(\frac{b}{a}\biggr) &\ \text{if } a \equiv b \equiv 3 \ (\text{mod} \ 4) \end{array} \right.

  12. (First supplement to the law of quadratic reciprocity)
    Let n be an odd positive integer. Then

      \displaystyle  \biggl(\frac{-1}{n}\biggr)=(-1)^{(n-1)/2}.

      Equivalently, \displaystyle  \biggl(\frac{-1}{n}\biggr) = \left\{ \begin{array}{rl}           1 &\ \text{if } n \equiv 1 \ (\text{mod} \ 4) \\         -1 &\ \text{if } n \equiv 3 \ (\text{mod} \ 4) \end{array} \right.

  13. (Second supplement to the law of quadratic reciprocity)
    Let n be an odd positive integer. Then

      \displaystyle  \biggl(\frac{2}{n}\biggr)=(-1)^{(n^2-1)/8}.

      Equivalently, \displaystyle  \biggl(\frac{2}{n}\biggr) = \left\{ \begin{array}{rl}           1 &\ \text{if } n \equiv 1 \ (\text{mod} \ 8) \text{ or } n \equiv 7 \ (\text{mod} \ 8) \\         -1 &\ \text{if } n \equiv 3 \ (\text{mod} \ 8) \text{ or } n \equiv 5 \ (\text{mod} \ 8) \end{array} \right.

Proof
For the first bullet point, if the top argument and the bottom argument have a prime factor in common, then one of the Legendre symbols that make up the Jacobi symbol is zero. If the top argument and the bottom argument have no prime factor in common, then all the Legendre symbols that make up the Jacobi symbol is either 1 or -1.

The second point is clear. If the bottom argument is an odd prime, the definition of Jacobi symbol leads to the Legendre symbol.

For bullet points 3, 4 and 5, simply write out the Jacobi symbol and then use the corresponding properties of the Legendre symbol.

Now the proof of bullet point 6 (the law of quadratic reciprocity). Let a and b be relatively prime. Let a=p_1 \times p_2 \times \cdots \times p_w and b=q_1 \times q_2 \times \cdots \times q_t be their prime factorizations. Note that the primes p_i are not necessarily distinct and the primes q_i are not necessarily distinct. However, p_i \ne q_j since a and b are relatively prime. Consider the following derivation of the product \displaystyle \biggl(\frac{a}{b}\biggr) \biggl(\frac{b}{a}\biggr).

    \displaystyle \begin{aligned}\biggl(\frac{a}{b}\biggr) \biggl(\frac{b}{a}\biggr)&=\prod \limits_{i=1}^t \biggl(\frac{a}{q_i}\biggr) \ \prod \limits_{j=1}^w \biggl(\frac{b}{p_j}\biggr) \\&=\prod \limits_{i=1}^t \prod \limits_{j=1}^w\biggl(\frac{p_j}{q_i}\biggr) \ \prod \limits_{j=1}^w \prod \limits_{i=1}^t \biggl(\frac{q_i}{p_j}\biggr) \\&=\prod \limits_{i=1}^t \prod \limits_{j=1}^w\biggl(\frac{p_j}{q_i}\biggr) \ \prod \limits_{i=1}^t \prod \limits_{j=1}^w \biggl(\frac{q_i}{p_j}\biggr) \\&=\prod \limits_{i=1}^t \prod \limits_{j=1}^w\biggl(\frac{p_j}{q_i}\biggr) \biggl(\frac{q_i}{p_j}\biggr) \\&=\prod \limits_{i=1}^t \prod \limits_{j=1}^w (-1)^{(p_j-1)/2 \times (q_i-1)/2} \\&=(-1)^E \end{aligned}

    where

    \displaystyle E=\sum \limits_{i,j} \biggl[\frac{p_j-1}{2} \times \frac{q_i-1}{2} \biggr]

The bullet point 6 is established after the quantity E is simplified as follows:

    \displaystyle \begin{aligned} \sum \limits_{i,j} \biggl[\frac{p_j-1}{2} \times \frac{q_i-1}{2} \biggr]&=\sum_j \biggl[\sum_i \frac{q_i-1}{2}\biggr] \frac{p_j-1}{2} \\&\equiv \sum_j \biggl[ \frac{b-1}{2}\biggr] \frac{p_j-1}{2} \ \ \ \text{ Use Lemma 2}\\&\equiv \frac{b-1}{2} \sum_j  \frac{p_j-1}{2} \\&\equiv \frac{b-1}{2} \frac{a-1}{2} \ (\text{mod} \ 2) \ \ \ \text{ Use Lemma 2}  \end{aligned}

Now consider the bullet point #7 (the first supplement to the law of quadratic reciprocity). The proof is an induction proof on the number of prime factors of n. If n is a prime, then it is done. So we assume n=p_1 \times p_2 (not necessarily distinct). Consider the following derivation:

    \displaystyle \begin{aligned} \biggl(\frac{-1}{n}\biggr)&=\biggl(\frac{-1}{p_1}\biggr) \biggl(\frac{-1}{p_2}\biggr) \\&=(-1)^{(p_1-1)/2} \ (-1)^{(p_2-1)/2} \\&=(-1)^{(p_1-1)/2+(p_2-1)/2} \\&\equiv (-1)^{(p_1 \times p_2 -1)/2} \ \ \ \ \ \text{Use Lemma 2} \\&\equiv (-1)^{(n -1)/2} \ (\text{mod} \ 2) \end{aligned}

It is a straightforward argument that whenever the property is true for n being a product of k primes, the property is true for n being a product of k+1 primes. Thus bullet 7 is established.

The proof of bullet point 8 (the second supplement to the law of quadratic reciprocity) is also an induction proof on the number of prime factors of n. The most important case is the of two prime factors.

    \displaystyle \begin{aligned} \biggl(\frac{2}{n}\biggr)&=\biggl(\frac{2}{p_1}\biggr) \biggl(\frac{2}{p_2}\biggr) \\&=(-1)^{(p_1^2-1)/8} \ (-1)^{(p_2^2-1)/8} \\&=(-1)^{(p_1^2-1)/8+(p_2^2-1)/8} \\&\equiv (-1)^{((p_1 \times p_2)^2 -1)/8} \ \ \ \ \ \text{Use Lemma 2} \\&\equiv (-1)^{(n^2 -1)/8} \ (\text{mod} \ 2) \end{aligned}

With the 2-case established, it is straightforward to carry out the remainder of the induction proof of bullet 8. \square

____________________________________________________________________________

Examples

Example 1
Evaluate \displaystyle  \biggl(\frac{1783}{7523}\biggr), an example where both the upper and lower arguments are primes. In the previous post, the example is calculated using only Legendre symbol. Here we work it in Jacobi symbol. As much as possible, we flip the symbol without factoring.

    \displaystyle \begin{aligned} \biggl(\frac{1783}{7523}\biggr)&=-\biggl(\frac{7523}{1783}\biggr) \\&=-\biggl(\frac{391}{1783}\biggr) \ \ *\\&=\biggl(\frac{1783}{391}\biggr) \\&=\biggl(\frac{219}{391}\biggr) \ \ *\\&=-\biggl(\frac{391}{219}\biggr) \\&=-\biggl(\frac{172}{219}\biggr) \\&=-\biggl(\frac{2}{219}\biggr)^2 \biggl(\frac{43}{219}\biggr) \\&=-\biggl(-1\biggr)^2 (-1) \biggl(\frac{219}{43}\biggr) \\&=\biggl(\frac{4}{43}\biggr) \\&=1 \end{aligned}

One advantage of using Jacobi symbol is that when the upper argument is an odd integer that is not prime, we can still flip it (the steps with *). Then reduce and then flip again until we reach a point with small numbers. Thinking in Legendre symbol only, the steps with * cannot be flipped. As Jacobi symbols, they can be flipped freely as long as the arguments are odd integers. The advantage may not be apparent in this example. When the composite integers are large to the point that their factorizations are in effect not obtainable, this advantage is huge. \square

Example 2
Evaluate \displaystyle  \biggl(\frac{a}{p}\biggr) where p= 1298351, an odd prime and a= 756479. This is Example 2 in this previous post. The number a is not a prime. Here we use Jacobi symbol to demonstrate that factoring is not needed. We can start by flipping. We have the following derivation.

    \displaystyle \begin{aligned} \biggl(\frac{756479}{1298351}\biggr)&=-\biggl(\frac{1298351}{756479}\biggr) \\&=-\biggl(\frac{541872}{756479}\biggr) \\&=-\biggl(\frac{2}{756479}\biggr)^4 \times \biggl(\frac{33867}{756479}\biggr) \\&=-\biggl(1\biggr)^4 \times (-1) \biggl(\frac{756479}{33867}\biggr)  \\&=\biggl(\frac{11405}{33867}\biggr) \\&=\biggl(\frac{33867}{11405}\biggr) \\&=\biggl(\frac{11057}{11405}\biggr) \\&=\biggl(\frac{11405}{11057}\biggr) \\&=\biggl(\frac{348}{11057}\biggr) \\&=\biggl(\frac{2}{11057}\biggr)^2 \biggl(\frac{87}{11057} \biggr) \\&=\biggl(1\biggr)^2  \biggl(\frac{11057}{87} \biggr) \\&=\biggl(\frac{8}{87}\biggr) \\&=\biggl(\frac{2}{87} \biggr)^3 \\&=1 \end{aligned}

Note that the original symbol to evaluate is a Legendre symbol. The interim steps are done with Jacobi symbol as much as possible. With the Jacobi way, the symbol is flipped, reduced and flipped again multiple times. The derivation seems long. However, the flip-reduce-flip is much faster than factoring when the numbers involved are large. \square

Example 3
Evaluate \displaystyle  \biggl(\frac{a}{n}\biggr) where n= 12408107 and a= 4852777. Both numbers are not prime. The symbol has to be evaluated as a Jacobi symbol.

    \displaystyle \begin{aligned} \biggl(\frac{4852777}{12408107}\biggr)&=\biggl(\frac{12408107}{4852777}\biggr)  \\&= \biggl(\frac{2702553}{4852777}\biggr) \\&= \biggl(\frac{4852777}{2702553}\biggr) \\&= \biggl(\frac{2150224}{2702553}\biggr) \\&= \biggl(\frac{2}{2702553}\biggr)^4 \biggl(\frac{134389}{2702553}\biggr) \\&= \biggl(1\biggr)^4 \biggl(\frac{2702553}{134389}\biggr) \\&= \biggl(\frac{14773}{134389}\biggr) \\&= \biggl(\frac{134389}{14773}\biggr) \\&= \biggl(\frac{1432}{14773}\biggr) \\&= \biggl(\frac{2}{14773}\biggr)^3 \biggl(\frac{179}{14773}\biggr) \\&= \biggl(-1\biggr)^3 \biggl(\frac{14773}{179}\biggr) \\&= - \biggl(\frac{95}{179}\biggr) \\&= - (-1) \biggl(\frac{179}{95}\biggr) \\&=\biggl(\frac{84}{95}\biggr) \\&=\biggl(\frac{2}{95}\biggr)^2 \biggl(\frac{21}{95}\biggr) \\&=\biggl(1\biggr)^2 \biggl(\frac{95}{21}\biggr) \\&=\biggl(\frac{11}{21}\biggr) \\&=\biggl(\frac{21}{11}\biggr) \\&=\biggl(\frac{10}{11}\biggr) \\&=\biggl(\frac{2}{11}\biggr) \biggl(\frac{5}{11}\biggr) \\&=\biggl(-1\biggr) \biggl(\frac{11}{5}\biggr) \\&=-\biggl(\frac{6}{5}\biggr) \\&=-\biggl(\frac{2}{5}\biggr) \biggl(\frac{3}{5}\biggr) \\&=-\biggl(-1\biggr) \biggl(\frac{5}{3}\biggr) \\&=\biggl(\frac{2}{3}\biggr) \\&=-1 \end{aligned}

The above derivation seems long in that there are quite a few steps. But many of the steps are flip-reduce-flip-reduce that are easy to do. The steps are all done without factoring, except when the top argument is an even number. The Jacobi symbol \displaystyle  \biggl(\frac{a}{n}\biggr) is -1. In this case, it gives definite information about the status of quadratic nonresidues. We know for sure that a is a quadratic nonresidue modulo the odd integer n, i.e. the equation x^2 \equiv a \ (\text{mod} \ n) has no solutions. \square

Example 4
Evaluate \displaystyle  \biggl(\frac{a}{n}\biggr) where n= 48746413 and a= 17327467. Both numbers are not prime. As in Example 3, the symbol is evaluated using Jacobi symbol.

    \displaystyle \begin{aligned} \biggl(\frac{17327467}{48746413}\biggr) &=\biggl(\frac{48746413}{17327467}\biggr)  \\&=\biggl(\frac{14091479}{17327467}\biggr) \\&=-\biggl(\frac{17327467}{14091479}\biggr) \\&=-\biggl(\frac{3236168}{14091479}\biggr) \\&=-\biggl(\frac{2}{14091479}\biggr)^3 \biggl(\frac{404521}{14091479}\biggr) \\&=-\biggl(1\biggr)^3 \biggl(\frac{14091479}{404521}\biggr) \\&=- \biggl(\frac{337765}{404521}\biggr) \\&=- \biggl(\frac{404521}{337765}\biggr) \\&=- \biggl(\frac{66756}{337765}\biggr) \\&=- \biggl(\frac{2}{337765}\biggr)^2 \biggl(\frac{16689}{337765}\biggr) \\&=- \biggl(-1\biggr)^2 \biggl(\frac{337765}{16689}\biggr) \\&=-\biggl(\frac{3985}{16689}\biggr) \\&=-\biggl(\frac{16689}{3985}\biggr) \\&=-\biggl(\frac{749}{3985}\biggr) \\&=-\biggl(\frac{3985}{749}\biggr) \\&=-\biggl(\frac{240}{749}\biggr) \\&=-\biggl(\frac{2}{749}\biggr)^4 \biggl(\frac{15}{749}\biggr) \\&=-\biggl(-1\biggr)^4 \biggl(\frac{749}{15}\biggr) \\&=-\biggl(\frac{14}{15}\biggr) \\&=-\biggl(\frac{-1}{15}\biggr)\\&=-(-1)\\&=1 \end{aligned}

The result of \displaystyle  \biggl(\frac{a}{n}\biggr)=1 gives ambiguous information about the status of a= 17327467 being a quadratic residue or nonresidue modulo n= 48746413. In this case, the Jacobi symbol cannot tell us whether the equation x^2 \equiv a \ (\text{mod} \ n) has a solution. In such a case, the only way to know the status of quadratic residue is to know some additional information, namely the factoring of the numbers. This example is small enough that a= 3049 x 5683 and n= 7109 x 6857. With this information, the Jacobi symbol can be set up as follows:

    \displaystyle \begin{aligned} \biggl(\frac{17327467}{48746413}\biggr) &=\biggl(\frac{17327467}{7109}\biggr) \biggl(\frac{17327467}{6857}\biggr)=(-1) (-1)=1   \end{aligned}

By knowing the factoring of n, the original Jacobi symbol is a product of two Legendre symbols. It can be shown that both of these Legendre symbols are -1. These two Legendre symbols corresponds to these two equations: x^2 \equiv a \ (\text{mod} \ 7109) and x^2 \equiv a \ (\text{mod} \ 6857). The Legendre values of -1 indicate that these two equations have no solutions. By the Chinese remainder theorem, the equation x^2 \equiv a \ (\text{mod} \ 7109 \times 6857) has no solutions. Hence a= 17327467 is a quadratic nonresidue modulo n= 48746413=7109 x 6857. \square

____________________________________________________________________________

Comments

One versatile feature of the Jacobi symbol is that it can be used in evaluating the Legendre symbol, as shown in Example 1 and Example 2. The original Legendre symbol in Example 1 have arguments that are both odd prime and thus can be flipped right away. However, there are numbers in the interim steps that are odd and not prime. With the use of the Jacobi symbol, these interim numbers do have to be factored. The original Legendre symbol in Example 2 has a top argument that is odd and not prime. Treating it as a Jacobi symbol, it can be flipped right away. When numbers involved in Legendre symbol evaluation are large such that the factoring is not obtainable, the Jacobi symbol can be flipped and reduced and flipped again to obtain the answer of the Legendre symbol. The Jacobi symbol plays an outsize role in the evaluation of the Legendre symbol and thus is a pivotal tool in determining whether a number is a quadratic residue modulo an odd prime p or the solvability of the equation x^2 \equiv a \ (\text{mod} \ p).

Example 3 shows that when the Jacobi symbol produces the value of -1, the top argument a is a quadratic nonresidue modulo the lower argument n. Example 4 shows that when the Jacobi symbol produces the value of 1, the picture is murky. In such a case, settling the question of whether the top argument a is a quadratic nonresidue modulo the lower argument n requires additional information. Example 4 shows that if we can factor the lower argument n, then evaluating the individual Legendre symbols (based on the prime factors) will help answer the original question. This points to the possibility that solving the equation x^2 \equiv a \ (\text{mod} \ n) is not feasible when the factorization of a large composite modulus n is not obtainable.

____________________________________________________________________________
\copyright \ 2015 \text{ by Dan Ma}

Advertisements

The Legendre symbol

The law of quadratic reciprocity is a beautiful result. It is also an excellent tool to answer the question: given an odd prime p, and given that a is an integer such that a and p are relatively prime, is a is a quadratic residue modulo p, in other words, is the equation x^2 \equiv a \ (\text{mod} \ p) solvable? Using the law of quadratic reciprocity requires the evaluation of the Legendre symbol. The focus of this post is on the Legendre symbol as well as related concepts of quadratic residues and the law of quadratic residues. The versatility of the law of quadratic reciprocity is demonstrated with examples.

____________________________________________________________________________

Quadratic Residues

The setting is that the modulus in question is an odd prime p. Consider an integer a such that a and p are relatively prime, i.e. having no common prime factor. The number a is said to be a quadratic residue modulo p if the equation x^2 \equiv a \ (\text{mod} \ p) has an integer solution in x. Another way to say this is that a is a quadratic residue modulo p if there exists a square root of a modulo p (a square root would be a solution to the equation). If the integer a is not a quadratic residue modulo p, we say that a is a quadratic nonresidue modulo p. If the context is clear, the word quadratic can be omitted. We can then say a is a residue modulo p or a is a nonresidue modulo p.

Since every integer is congruent modulo p to one of the numbers in the set \mathbb{Z}_p=\left\{0,1,2,\cdots,p-1 \right\}, the integers a can be considered from the set \mathbb{Z}_p^*=\left\{1,2,\cdots,p-1 \right\} (the non-zero elements of \mathbb{Z}_p).

Let’s look at a quick example. Let p=11. Squaring each number in \mathbb{Z}_{11} produces the set \left\{0^2,1^2,2^2,\cdots,10^2 \right\}. Reducing modulo 11 produces the set \left\{0,1,4,9,5,3 \right\}. Thus the integers 1, 3, 4, 5 and 9 are quadratic residues modulo 11. The square roots of each residue are the solutions to the equation x^2 \equiv a \ (\text{mod} \ 11). For a=3, the solutions are x = 5 and 6. There are two square roots of 3 modulo 11, namely 5 and 6.

When the modulus is small, it is easy to find all quadratic residues simply by squaring all the numbers in \mathbb{Z}_p^*. The focus in this post is on how to use the law of quadratic reciprocity to determine whether a given a is a quadratic residue modulo a large odd prime p.

To check whether the integer a is a quadratic residue modulo an odd prime p, the most important idea, before considering the law of reciprocity, is to check the modular exponentiation a^{(p-1)/2} \ (\text{mod} \ p). If the result is congruent to 1 modulo p, then a is a quadratic residue modulo p. If the result is congruent to -1 modulo p, then a is a quadratic nonresidue modulo p. This is called Euler’s Criterion (proved here). For clarity, it is stated below.

Theorem 1 (Euler’s Criterion)
Let p be an odd prime. Let a be an integer that is relatively prime to p. Then the integer a is a quadratic residue modulo p if and only if \displaystyle a^{(p-1)/2} \equiv 1 \ (\text{mod} \ p). On the other hand, the integer a is a quadratic nonresidue modulo p if and only if \displaystyle a^{(p-1)/2} \equiv -1 \ (\text{mod} \ p).

According to Euler’s Criterion, the task of checking for the status of quadratic residue is a matter of performing a modular exponentiation. This can be done using software or a calculator for modular arithmetic. If so desired, the exponentiation can also be programmed using the fast powering algorithm.

Though Euler’s Criterion (with a calculator for modular arithmetic) is a sure fire way for checking the status of quadratic residues, the law of quadratic reciprocity can simplify the task even further. As the examples below will show, checking the status of quadratic residues using the law of reciprocity may require no modular exponentiation at all and if exponentiation is required, it is of a much smaller size.

Euler’s Criterion also gives us several basic facts about quadratic residues.

Theorem 2
Let p be an odd prime. The following properties are true:

  • If a and b are both residues modulo p, then the product a \times b is a residue modulo p.
  • If If a and b are both nonresidues modulo p, then the product a \times b is a residue modulo p.
  • If a and b are such that one of them is a residue modulo p and the other is a nonresidue modulo p, then the product a \times b is a nonresidue modulo p.

Theorem 2 says that the product of two integers of the same types (both residues or both nonresidues) is always a residue modulo the odd prime p. The product of two integers of different types is always a nonresidue modulo the odd prime p. The theorem follows from Euler’s criterion and from the fact that (ab)^{(p-1)/2}=a^{(p-1)/2} \times b^{(p-1)/2}.

In arithmetic modulo a prime p, there exists a primitive root modulo p and that any primitive root generates by powering all the integers that are relatively prime to the modulus p. Let g be a primitive root modulo an odd prime p. It then follows that the quadratic residues modulo p are the even powers of g and the nonresidues are the odd powers of g. A related fact is that when p is an odd prime and when a and p are relatively prime, the equation x^2 \equiv a \ (\text{mod} \ p) either has two solutions or has no solutions.

Theorem 3
Let g be a primitive root modulo an odd prime p. For any a that is relatively prime to p, the following is true:

  • a is a quadratic residue modulo p if and only if a is of the form a \equiv g^{2k} \ (\text{mod} \ p) for some positive integer k,
  • a is a quadratic nonresidue modulo p if and only if a is of the form a \equiv g^{2k+1} \ (\text{mod} \ p) for some positive integer k.
  • (proved here)

Theorem 4
Let p be an odd prime. For any a that is relatively prime to p, the equation x^2 \equiv a \ (\text{mod} \ p) either has two solutions or has no solutions (proved here).

____________________________________________________________________________

Legendre Symbol

The law of quadratic reciprocity is stated using the Legendre symbol. For an odd prime p and for an integer a that is relatively prime to p, the symbol \displaystyle \biggl(\frac{a}{p}\biggr) is defined as follows:

    \displaystyle  \biggl(\frac{a}{p}\biggr) = \left\{ \begin{array}{rl}                      1 &\ \text{if }  a  \text{ is a quadratic residue modulo }p\\          -1 &\ \text{if }  a  \text{ is a quadratic nonresidue modulo }p \end{array} \right.

One obvious and important observation is that Euler’s Criterion can be restated as follows:

Theorem 1a (Euler’s Criterion)
Let p be an odd prime. Let a be an integer that is relatively prime to p. Then \displaystyle \biggl(\frac{a}{p}\biggr) \equiv \displaystyle a^{\frac{p-1}{2}} \ (\text{mod} \ p).

In the above definition, the lower argument of the Legendre symbol is always an odd prime and the upper argument is an integer that is relatively prime to the lower argument. To smooth out some statements involving the Legendre symbol and to make it easier to define the Jacobi symbol in the next post, we relax the Legendre symbol by making \displaystyle \biggl(\frac{a}{p}\biggr)=0 whenever a and p are not relatively prime, i.e. a \equiv 0 \ (\text{mod} \ p). The Legendre symbol can be broaden slightly by the following:

    \displaystyle  \biggl(\frac{a}{p}\biggr) = \left\{ \begin{array}{rl}           0 &\ \text{if } a \equiv 0 \ (\text{mod} \ p) \\           1 &\ \text{if } a \not \equiv 0 \ (\text{mod} \ p) \text{ and } a  \text{ is a quadratic residue modulo }p\\          -1 &\ \text{if } a \not \equiv 0 \ (\text{mod} \ p) \text{ and } a  \text{ is a quadratic nonresidue modulo }p \end{array} \right.

Here’s some useful basic facts about the Legendre symbol.

Theorem 5
Let p be an odd prime. The following properties hold:

  • The Legendre symbol is periodic with respect to the top argument, i.e. if a \equiv b \ (\text{mod} \ p), then \displaystyle \biggl(\frac{a}{p}\biggr)=\biggl(\frac{b}{p}\biggr).
  • The Legendre symbol is multiplicative with respect to the top argument, i.e. \displaystyle \biggl(\frac{ab}{p}\biggr)=\biggl(\frac{a}{p}\biggr) \times \biggl(\frac{b}{p}\biggr).
  • If a \equiv 0 \ (\text{mod} \ p), then \displaystyle  \biggl(\frac{a^2}{p}\biggr)=0. If a \not \equiv 0 \ (\text{mod} \ p), then \displaystyle  \biggl(\frac{a^2}{p}\biggr)=1.

The first bullet point in Theorem 5 is easily verified based on the definition of the Legendre symbol. For the case a \equiv 0 \ (\text{mod} \ p), the other facts in Theorem 5 are easily verified. For the case a \not \equiv 0 \ (\text{mod} \ p), the second bullet point follows from Theorem 2, i.e. the fact that the product of two residues and the product of two nonresidues are both residues modulo an odd prime and that the product of a residue and a nonresidue is a nonresidue modulo an odd prime. The second part of the third bullet point is true since any integer that is a square is a quadratic residue.

____________________________________________________________________________

Law of Quadratic Reciprocity

The law of reciprocity can ease the calculation of the Legendre symbol when both the upper and lower arguments are distinct odd primes. Theorem 6 is the law of reciprocity and Theorem 7 and Theorem 8 are two supplements to the law that can round out the calculation. After stating the theorems, we demonstrate with some examples.

Theorem 6 (the law of quadratic reciprocity)

    Let p and q be two distinct odd prime numbers. Both of the following statements hold.

      \displaystyle  \biggl(\frac{q}{p}\biggr) \displaystyle \biggl(\frac{p}{q}\biggr)=(-1)^{(p-1)/2 \times (q-1)/2}

      Equivalently and more explicitly,

      \displaystyle  \biggl(\frac{q}{p}\biggr) = \left\{ \begin{array}{rl}                      \displaystyle \biggl(\frac{p}{q}\biggr) &\ \text{if } p \equiv 1 \ (\text{mod} \ 4) \text{ or } q \equiv 1 \ (\text{mod} \ 4)\\          \displaystyle  -\biggl(\frac{p}{q}\biggr) &\ \text{if } p \equiv q \equiv 3 \ (\text{mod} \ 4) \end{array} \right.

Theorem 7 (first supplement to the law of quadratic reciprocity)

    Let p be an odd prime number. The following statement holds.

      \displaystyle  \biggl(\frac{-1}{p}\biggr) = (-1)^{(p-1)/2} = \left\{ \begin{array}{rl}                      1 &\ \text{if } p \equiv 1 \ (\text{mod} \ 4)\\          -1 &\ \text{if } p \equiv 3 \ (\text{mod} \ 4) \end{array} \right.

Theorem 8 (second supplement to the law of quadratic reciprocity)

    Let p be an odd prime number. The following statement holds.

      \displaystyle  \biggl(\frac{2}{p}\biggr) =(-1)^{(p^2-1)/8}= \left\{ \begin{array}{rl}                      1 &\ \text{if } p \equiv 1 \ (\text{mod} \ 8) \text{ or } p \equiv 7 \ (\text{mod} \ 8)\\          -1 &\ \text{if } p \equiv 3 \ (\text{mod} \ 8) \text{ or } p \equiv 5 \ (\text{mod} \ 8) \end{array} \right.

Theorem 6 shows how to flip the Legendre symbol if both the upper and lower arguments are distinct odd primes. As long as one of the primes is congruent to 1 modulo 4, we can safely flip the symbol. If both primes are not congruent to 1 modulo 4, we can still flip the symbol except that we have to attach a minus sign. Theorem 7 (the first supplement) indicates when -1 (or p-1) is a quadratic residue an odd prime p. In words, -1 is a residue modulo p if dividing p by 4 gives the remainder of 1. Otherwise -1 is a nonresidue modulo p. Theorem 8 (the second supplement) indicates when 2 is a residue modulo an odd prime p. In words, 2 is a residue modulo p if dividing p by 8 leaves a remainder of 1 or 7. Otherwise 2 is a nonresidue modulo p.

____________________________________________________________________________

Examples

Example 1
Evaluate \displaystyle  \biggl(\frac{1783}{7523}\biggr), an example where both the upper and lower arguments in the Legendre symbol are primes.

    \displaystyle \begin{aligned} \biggl(\frac{1783}{7523}\biggr)&=-\biggl(\frac{7523}{1783}\biggr) \\&=-\biggl(\frac{391}{1783}\biggr) \\&=-\biggl(\frac{17}{1783}\biggr) \times \biggl(\frac{23}{1783}\biggr) \\&=-\biggl(\frac{1783}{17}\biggr) \times (-1) \biggl(\frac{1783}{23}\biggr) \\&=\biggl(\frac{15}{17}\biggr) \times \biggl(\frac{12}{23}\biggr) \\&=\biggl(\frac{3}{17}\biggr) \times \biggl(\frac{5}{17}\biggr) \times \biggl(\frac{2}{23}\biggr)^2 \times \biggl(\frac{3}{23}\biggr) \\&=\biggl(\frac{17}{3}\biggr) \times \biggl(\frac{17}{5}\biggr) \times \biggl(1\biggr)^2 \times (-1)\biggl(\frac{23}{3}\biggr) \\&=-\biggl(\frac{2}{3}\biggr) \times \biggl(\frac{2}{5}\biggr) \times  \biggl(\frac{2}{3}\biggr) \\&=-\biggl(\frac{2}{5}\biggr) \\&=-(-1) \\&=1 \end{aligned}

The above derivation is a repeated use of Theorems 5, 6 and 8. The idea is to flip the Legendre symbols to make the lower arguments smaller. Then reduce the upper arguments and then factor the upper arguments. Then flip again until reaching a Legendre symbol of \displaystyle \biggl(\frac{2}{5}\biggr), which is easy to solve.

It then follows that 1783 is a quadratic residue modulo the odd prime 7523. The answer can be confirmed by using Euler’s Criterion. Note that 1783^{3761} \equiv 1 \ (\text{mod} \ 7523). \square

Example 2
Evaluate \displaystyle  \biggl(\frac{a}{p}\biggr) where p= 1298351, an odd prime and a= 756479.

The number a is not a prime and is factored as a=353 \times 2143. We have the following derivation.

    \displaystyle \begin{aligned} \biggl(\frac{756479}{1298351}\biggr)&=\biggl(\frac{353}{1298351}\biggr) \times \biggl(\frac{2143}{1298351}\biggr) \\&=\biggl(\frac{1298351}{353}\biggr) \times (-1)\biggl(\frac{1298351}{2143}\biggr) \\&=-\biggl(\frac{17}{353}\biggr) \times \biggl(\frac{1836}{2143}\biggr) \\&=-\biggl(\frac{17}{353}\biggr) \times \biggl(\frac{2}{2143}\biggr)^2 \times \biggl(\frac{3}{2143}\biggr)^3 \times \biggl(\frac{17}{2143}\biggr)  \\&=-\biggl(\frac{353}{17}\biggr) \times \biggl(1\biggr)^2 \times (-1) \biggl(\frac{2143}{3}\biggr)^3 \times \biggl(\frac{2143}{17}\biggr) \\&=\biggl(\frac{13}{17}\biggr) \times  \biggl(\frac{1}{3}\biggr)^3 \times \biggl(\frac{1}{17}\biggr) \\&=\biggl(\frac{17}{13}\biggr) \times  \biggl(1\biggr)^3 \times \biggl(1\biggr) \\&=\biggl(\frac{4}{13}\biggr) \\&=1 \end{aligned}

The evaluation of the Legendre symbols in this example does not start with a flipping since the upper argument 756479 is not a prime. Instead, we start with factoring the upper argument into prime factors and then proceed with a series of flipping, reducing and factoring. \square

Example 3
Evaluate \displaystyle  \biggl(\frac{a}{p}\biggr) where p= 569, an odd prime and a= 1610280.

    \displaystyle \begin{aligned} \biggl(\frac{1610280}{569}\biggr)&=\biggl(\frac{3}{569}\biggr)^4 \times \biggl(\frac{5}{569}\biggr) \times \biggl(\frac{7}{569}\biggr) \times \biggl(\frac{568}{569}\biggr) \\&= \biggl(\frac{569}{3}\biggr)^4 \times \biggl(\frac{569}{5}\biggr) \times \biggl(\frac{569}{7}\biggr) \times \biggl(\frac{-1}{569}\biggr) \\&= \biggl(\frac{2}{3}\biggr)^4 \times \biggl(\frac{4}{5}\biggr) \times \biggl(\frac{2}{7}\biggr) \times \biggl(1\biggr) \\&= \biggl(-1\biggr)^4 \times \biggl(1\biggr) \times \biggl(1\biggr) \\&=1 \end{aligned}

This example can also be evaluated by first reducing 1610280 modulo 569. \square

____________________________________________________________________________

Comment

The law of quadratic reciprocity is a deep and powerful result. It guides the evaluation of Legendre symbols in an attempt to answer whether a number is a quadratic residue modulo an odd prime. The law of reciprocity as represented above requires that the lower argument in the Legendre symbol is an odd prime. When the upper argument is an odd number that is not a prime, there is no way to flip it (based on the law of reciprocity using the Legendre symbol). In Example 2, the evaluation of the Legendre symbol cannot begin until the top argument is factored. The factoring in Example 2 is possible since the number is small. When the number is large, factoring may not always be feasible. It turns out that the Legendre symbol has a generalization, called the Jacobi symbol, that is even more versatile and is defined in the next post.

____________________________________________________________________________
\copyright \ 2015 \text{ by Dan Ma}

Quadratic Residues

In a previous post called Solving Quadratic Congruences, we discuss the solvability of the quadratic congruence

    x^2 \equiv a \ (\text{mod} \ p) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

where p is an odd prime and a is relatively prime to p. In this post, we continue to discuss the solvability of equation (1) from the view point of quadratic residues. In this subsequent post, we discuss specific algorithms that produce solutions to such equations.

____________________________________________________________________________

Definition

Let p be an odd prime. Let a be an integer that is not divisible by p (equivalently relatively prime to p). Whenever equation (1) has solutions, we say that the number a is a quadratic residue modulo p. Otherwise, we say that the number a is a quadratic nonresidue modulo p. When the context is clear, the word quadratic is sometimes omitted.

The term quadratic residues is more convenient to use. Instead of saying the equation x^2 \equiv a \ (\text{mod} \ p) has a solution, we say the number a is a quadratic residue for the modulus in question. The significance of the notion of quadratic residue extends beyond the convenience of having a shorter name. It and and the Legendre symbol lead to a large body of beautiful and deep results in number theory, the quadratic reciprocity theorem being one of them.

One property of the quadratic congruence equation (1) is that when equation (1) has solutions, it has exactly two solutions among the set \left\{1,2,3,\cdots,p-1 \right\} (see Lemma 1 in the post Solving Quadratic Congruences). Thus among the integers in the set \left\{1,2,3,\cdots,p-1 \right\}, \displaystyle \frac{p-1}{2} of them are quadratic residues and the other half are quadratic nonresidues modulo p.

For example, consider the modulus p=11. Among the numbers in the set \left\{1,2,3,\cdots,10 \right\}, the numbers 1,3,4,5,9 are quadratic residues and the numbers 2,6,7,8,10 are quadratic nonresidues. See the following two tables.

    \displaystyle \begin{bmatrix} x&\text{ }&x^2 \equiv \ (\text{mod} \ 11)  \\\text{ }&\text{ }&\text{ } \\ 1&\text{ }&1  \\ 2&\text{ }&4  \\ 3&\text{ }&9  \\ 4&\text{ }&5  \\ 5&\text{ }&3  \\ 6&\text{ }&3 \\ 7&\text{ }&5 \\ 8&\text{ }&9 \\ 9&\text{ }&4 \\ 10&\text{ }&1   \end{bmatrix}

The above table shows the least residues of x^2 for x \in \left\{1,2,3,\cdots,10 \right\}. It shows that there x^2 can only be 1,3,4,5,9. Thus these are the quadratic residues. The table below shows the status of residue/nonresidue among the integers in \left\{1,2,3,\cdots,10 \right\}.

    \displaystyle \begin{bmatrix} x&\text{ }&\text{residue or nonresidue mod } 11 \\\text{ }&\text{ }&\text{ } \\ 1&\text{ }&\text{residue}  \\ 2&\text{ }&\text{nonresidue}  \\ 3&\text{ }&\text{residue}  \\ 4&\text{ }&\text{residue}  \\ 5&\text{ }&\text{residue}  \\ 6&\text{ }&\text{nonresidue} \\ 7&\text{ }&\text{nonresidue} \\ 8&\text{ }&\text{nonresidue} \\ 9&\text{ }&\text{residue} \\ 10&\text{ }&\text{nonresidue}   \end{bmatrix}

____________________________________________________________________________

Legendre Symbol

The notion of quadratic residues is often expressed using the Legendre symbol, which is defined as follows:

    \displaystyle \biggl(\frac{a}{p}\biggr)=\left\{\begin{matrix}1&\ \text{if } a \text{ is a quadratic residue modulo }p \\{-1}&\ \text{if } a \text{ is a quadratic nonresidue modulo }p  \end{matrix}\right.

The bottom number p in the above notation is an odd prime. The top number a is an integer that is not divisible by p (equivalently relatively prime to p). Despite the appearance, the Legendre symbol is not the fraction of a over p. It follows from the definition that the symbol has the value of one if the equation x^2 \equiv a \ (\text{mod} \ p) has solutions. It has the value of negative one if the equation x^2 \equiv a \ (\text{mod} \ p) has no solutions.

For example, \displaystyle \biggl(\frac{a}{11}\biggr)=1 for a=1,3,4,5,9 and and \displaystyle \biggl(\frac{a}{11}\biggr)=-1 for a=2,6,7,8,10. To evaluate \displaystyle \biggl(\frac{11}{3}\biggr), consider the equation x^2 \equiv 11 \ (\text{mod} \ 3), which is equivalent to the equation x^2 \equiv 2 \ (\text{mod} \ 3). This last equation has no solutions. Thus \displaystyle \biggl(\frac{11}{3}\biggr)=-1.

The quadratic reciprocity law discussed below allows us to calculate \displaystyle \biggl(\frac{11}{3}\biggr) by flipping \displaystyle \biggl(\frac{3}{11}\biggr). In certain cases, flipping the symbol keeps the same sign. In other cases, flipping introduces a negative sign (as in this example).

____________________________________________________________________________

Basic Properties

Euler’s Criterion is a formula that determines whether an integer is a quadratic residue modulo an odd prime. We have the following theorem. A proof of Euler’s Criterion is found in this post.

    Theorem 1 (Euler’s Criterion)

      Let p be an odd prime number. Let a be a positive integer that is not divisible by p. Then the following property holds.

        \displaystyle \biggl(\frac{a}{p}\biggr) \equiv \displaystyle a^{\frac{p-1}{2}} \ (\text{mod} \ p)

The following lemma shows a connection between the notion of quadratic residue and the notion of primitive roots.

    Lemma 2

      Let p be an odd prime. Let g be a primitive root modulo p. Let a be a positive integer that is not divisible by p. Then we have the following equivalence.

      1. The number a is a quadratic residue modulo p if and only if a \equiv g^{2k} \ (\text{mod} \ p) for some integer k.
      2. Or equivalently, the number a is a quadratic nonresidue modulo p if and only if a \equiv g^{2k+1}  \ (\text{mod} \ p) for some integer k.

Proof of Lemma 2
A primitive root g exists since the modulus p is prime (see Theorem 1 in the post Primitive roots of prime moduli). Furthermore, any integer that is not divisible by p is congruent to a unique element of the set \left\{g^1,g^2,g^3,\cdots,g^{p-1} \right\}. Thus for the number a in question, either a \equiv g^{2k} or a \equiv g^{2k+1}. We can conclude that the first bullet point in the lemma is equivalent to the second bullet point.

We prove the first bullet point. First we show the direction \Longleftarrow. Suppose a \equiv g^{2k} \ (\text{mod} \ p). Clearly the equation x^2 \equiv a \ (\text{mod} \ p) has a solution since (g^k)^2 \equiv a \equiv g^{2k} \ (\text{mod} \ p).

Now we show the direction \Longrightarrow. We prove the contrapositive. Suppose that a \equiv g^{2k+1}  \ (\text{mod} \ p). We wish to show that a is a quadratic nonresidue modulo p. Suppose not. Then t^2 \equiv a \ (\text{mod} \ p) for some t. It follows that p \not \lvert \ t. Note that if p \ \lvert \ t, p \ \lvert \ a, which is not true. By Fermat’s little theorem, we have t^{p-1} \equiv 1 \ (\text{mod} \ p). We have the following derivation.

    \displaystyle (g^{2k+1})^{\frac{p-1}{2}} \equiv (t^{2})^{\frac{p-1}{2}} \equiv t^{p-1} \equiv 1 \ (\text{mod} \ p)

On the other hand, we can express \displaystyle (g^{2k+1})^{\frac{p-1}{2}} as follows:

    \displaystyle (g^{2k+1})^{\frac{p-1}{2}} \equiv g^{k(p-1)} \cdot g^{\frac{p-1}{2}} \equiv (g^{p-1})^k \cdot g^{\frac{p-1}{2}} \equiv 1^k \cdot g^{\frac{p-1}{2}} \equiv g^{\frac{p-1}{2}} \equiv 1 \ (\text{mod} \ p)

Note that the last congruence g^{\frac{p-1}{2}} \equiv 1 \ (\text{mod} \ p) contradicts the fact that g is a primitive root modulo p since p-1 is the least exponent that such that g^{p-1} \equiv 1 \ (\text{mod} \ p). So a cannot be a quadratic residue modulo p. We have proved that if a \equiv g^{2k+1}  \ (\text{mod} \ p), then a is a quadratic nonresidue modulo p. Equivalently, if a is a quadratic residue modulo p, then a \equiv g^{2k}  \ (\text{mod} \ p). Thus the lemma is established. \blacksquare

We can also obtain an alternative proof by using Theorem 1 (Euler’s Criterion). We show \Longleftarrow of both bullet points.

First, \Longleftarrow of the first bullet point. Suppose a \equiv g^{2k} \ (\text{mod} \ p). Then \displaystyle (g^{2k})^{\frac{p-1}{2}} \equiv (g^{p-1})^k \equiv 1^k \equiv 1 \ (\text{mod} \ p). Thus \displaystyle \biggl(\frac{a}{p}\biggr)=1 and a is a quadratic residue modulo p by Euler’s Criterion.

Now \Longleftarrow of the second bullet point. Suppose a \equiv g^{2k+1} \ (\text{mod} \ p). Then \displaystyle (g^{2k+1})^{\frac{p-1}{2}} \equiv (g^{p-1})^k \cdot g^{\frac{p-1}{2}} \equiv g^{\frac{p-1}{2}} \equiv -1 \ (\text{mod} \ p). The last congruence g^{\frac{p-1}{2}} \equiv -1 \ (\text{mod} \ p) because g is a primitive root. Thus \displaystyle \biggl(\frac{a}{p}\biggr)=-1 and a is a quadratic nonresidue modulo p by Euler’s Criterion. \blacksquare

Remark
Each number in the set \left\{1,2,3,\cdots,p-1 \right\} is congruent to a power of the primitive root g in question. Lemma 2 indicates that the even powers are the quadratic residues while the odd powers are the quadratic nonresidues. The following lemma is a corollary of Lemma 2.

    Lemma 3

      Let p be an odd prime. Then we have the following.

        1. If a and b are quadratic residues modulo p, then ab is a quadratic residue modulo p.
        2. If a is a quadratic residue and b is a quadratic nonresidue modulo p, then ab is a quadratic nonresidue modulo p.
        3. If a and b are quadratic nonresidues modulo p, then ab is a quadratic residue modulo p.

Proof of Lemma 3
Let g be a primitive root modulo p. Then we express each residue or nonresidue as a power of g and then multiply the two powers of g by adding the exponents as in the following.

    \displaystyle g^{2j} \cdot g^{2k}=g^{2(j+k)}

    \displaystyle g^{2j} \cdot g^{2k+1}=g^{2(j+k)+1}

    \displaystyle g^{2j+1} \cdot g^{2k+1}=g^{2(j+k+1)}

The first product above has an even exponent. Thus the product of two quadratic residues is a quadratic residue (the first bullet point). The second product above has an odd exponent. Thus the product of a quadratic residue and a quadratic nonresidue is a nonresidue (second bullet point). The third product above has an even exponent. Thus the product of two nonresidues is a residue. \blacksquare

One part of the following theorem is a corollary of Lemma 3.

    Theorem 4

      Let p be an odd prime. Then we have the following results.

        1. If p \not \lvert \ a and a \equiv b \ (\text{mod} \ p), then \displaystyle \biggl(\frac{a}{p}\biggr)=\biggl(\frac{b}{p}\biggr).
        2. If p \not \lvert \ a, then \displaystyle \biggl(\frac{a^2}{p}\biggr)=1.
        3. if p \not \lvert \ a and p \not \lvert \ b, then \displaystyle \biggl(\frac{a}{p}\biggr) \cdot \biggl(\frac{b}{p}\biggr)=\biggl(\frac{ab}{p}\biggr).

Proof of Theorem 4
The first and second bullets points are straightforward. We prove the third bullet point, which follows from Lemma 3. Given a and b, they would fall into one of the three cases of Lemma 3. Translating each case of Lemma 3 will give the correct statement in Legendre symbol. \blacksquare

____________________________________________________________________________

Quadratic Reciprocity

Quadratic reciprocity is a property that indicates how \displaystyle \biggl(\frac{p}{q}\biggr) and \displaystyle \biggl(\frac{q}{p}\biggr) are related when both p and q are odd prime. Even thought the statement of the theorem is easy to state and understand, it is an unexpected and profound result. Our goal here is quite simple – state the theorem and demonstrate how it can be used to simplify calculations. We have the following theorems.

    Theorem 5 (Quadratic Reciprocity)

      Let p and q be two distinct odd prime numbers. The following statement holds.

        \displaystyle \biggl(\frac{q}{p}\biggr)=\left\{\begin{matrix} \displaystyle \biggl(\frac{p}{q}\biggr) &\ \text{if } p \equiv 1 \ (\text{mod} \ 4) \text{ or } q \equiv 1 \ (\text{mod} \ 4) \\{\displaystyle  -\biggl(\frac{p}{q}\biggr)}&\ \text{if } p \equiv q \equiv 3 \ (\text{mod} \ 4)  \end{matrix}\right.

    Theorem 6

      Let p and q be two distinct odd prime numbers. The following statement holds.

        \displaystyle \biggl(\frac{2}{p}\biggr)=\left\{\begin{matrix} 1 &\ \text{if } p \equiv 1 \ (\text{mod} \ 8) \text{ or } p \equiv 7 \ (\text{mod} \ 8) \\{-1}&\ \text{if } p \equiv 3 \ (\text{mod} \ 8) \text{ or } p \equiv 5 \ (\text{mod} \ 8)  \end{matrix}\right.

    Theorem 7

      Let p and q be two distinct odd prime numbers. The following statement holds.

        \displaystyle \biggl(\frac{-1}{p}\biggr)=\left\{\begin{matrix} 1 &\ \text{if } p \equiv 1 \ (\text{mod} \ 4)  \\{-1}&\ \text{if } p \equiv 3 \ (\text{mod} \ 4)   \end{matrix}\right.

Theorems 4, 5, 6 and 7 are tools for evaluating Legendre symbols. We demonstrate with examples.

____________________________________________________________________________

Examples

Example 1
Is 1776 a quadratic residue modulo the prime 1777?

We evaluate the symbol \displaystyle \biggl(\frac{1776}{1777}\biggr)=\biggl(\frac{-1}{1777}\biggr). Note that 1777 \equiv 1 \ (\text{mod} \ 4). By Theorem 7, \displaystyle \biggl(\frac{-1}{1777}\biggr)=1. It follows that 1776 is a quadratic residue modulo the prime 1777. Furthermore, x^2 \equiv 1776 \ (\text{mod} \ 1777) has solutions.

Example 2
Solve x^2 \equiv 899 \ (\text{mod} \ 50261).

Note that 50261 is a prime while 899 is not since 899=29 \cdot 31. After applying Theorem 4, we have:

    \displaystyle \biggl(\frac{899}{50261}\biggr)=\displaystyle \biggl(\frac{29}{50261}\biggr) \displaystyle \biggl(\frac{31}{50261}\biggr).

Now we can start using quadratic reciprocity.

    \displaystyle \begin{aligned} \displaystyle \biggl(\frac{899}{50261}\biggr)&=\displaystyle \biggl(\frac{29}{50261}\biggr) \biggl(\frac{31}{50261}\biggr) \\&=\displaystyle \biggl(\frac{50261}{29}\biggr) \biggl(\frac{50261}{31}\biggr) \\&=\displaystyle \biggl(\frac{4}{29}\biggr) \biggl(\frac{10}{31}\biggr) \\&=\displaystyle \biggl(\frac{2}{29}\biggr)^2  \biggl(\frac{2}{31}\biggr) \biggl(\frac{5}{31}\biggr)  \\&=(-1)^2 \cdot 1 \cdot 1  \\&=1 \end{aligned}

The above derivation is the result of applying Theorems 4, 5 and 7. Of particular importance is the repeated applications of Theorem 5 (Quadratic Reciprocity) so that the numbers in the Legendre symbols are much smaller than the ones we start with.

As useful as it is, the theorem for quadratic reciprocity does not show us how to solve the equation x^2 \equiv 899 \ (\text{mod} \ 50261). See Example 2 in the post Solving Quadratic Congruences to see how it can be solved.

____________________________________________________________________________

\copyright \ 2013 - 2015 \text{ by Dan Ma}
Revised December 9, 2015