*e* = 2 is not a permissible exponent in RSA, as the mapping **E**_{e} is *not* one-to-one. The ambiguity in decipherment might be resolved by using a standard format for the plaintext *x*. It is unlikely, but *not* proved that only one of the (four) square roots would be in the standard format.

Hugh Williams [1980] found a very clever way around this difficulty. He transforms the plaintext *x* into **E**_{1}[*x*] where **E**_{1} is an invertible mapping with inverse **D**_{1}. For primes with appropriate restrictions, there exist transformations **E**_{2} and **D**_{2} that correspond to encipherment and encipherment such that

The *Jacobi symbol J*(*p/q*) when *q* is a prime is defined by

*J*(*p/q*) can be extended for *q* not being a prime.

**Proposition 12.8:** (Properties of the Jacobi Symbol):

**12.8a** If *p*_{1} = *p*_{2} (modulo *q*), then *J*(*p*_{1}/*q*) = *J*(*p*_{2}/*q*);

**12.8b** *J*(*p*_{1}*p*_{2}/*q*) = *J*(*p*_{1}/*q*)*J*(*p*_{2}/*q*);

**12.8c** *J*(*p*/*q*_{1}*q*_{2}) = *J*(*p*/*q*_{2})*J*(*p*/*q*_{2});

**12.8d** if *p*, *q* are odd; and

**12.8e** If *q* is a prime, then .

**Proposition 12.9:** If *p*, *q* are primes such that 3 = *p* (modulo 4) = *q* (modulo 4) and *J*(*x/pq*) = 1, then

*Proof:* Using Proposition **12.8c** and the hypothesis ...

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