Prime Numbers

542 Appendix BOOK PSEUDOCODE
Because of this relegation of English statements to their own lines, the
interpretation that equation (Y) is to be invoked once, after the pseudoprime
loop, is immediate. Accordingly, when an English statement is sufficiently long
that it wraps around, we have adopted reverse indentation, like so:
Find a random t ∈ [0,p− 1] such that t 2 − a is a quadratic nonresidue
(mod p), via Algorithm 2.3.5;
x =(t + √ t 2 − a) (p+1)/2 ;
...;
In this last example, one continually chooses random integers t in the stated
range until one is found with the required condition, and then one goes to the
next step, which calls for a single calculation and the assignment of letter x
to the result of the calculation.
Also in English will be comments throughout the book pseudocode. These
take the following form (and are right-justified, unlike pseudocode itself):
x =(t + √ t 2 − a) (p+1)/2 ; // Use F p 2 arithmetic.
The point is, a comment prefaced with “//” is not to be executed as
pseudocode. For example, the above comment is given as a helpful hint,
indicating perhaps that to execute the instruction one would first want to have
a subroutine to do F p 2 arithmetic. Other comments clarify the pseudocode’s
nomenclature, or provide further information on how actually to carry out the
executable statement.
Assignment of variables, and conditionals
We have elected not to use the somewhat popular assignment syntax x := y,
rather, we set x equal to y via the simple expedient x = y. (Note that in
this notation for assignment used in our pseudocode, the symbol “=” does
not signify a symmetric relation: The assignment x = y is not the same
instruction as the assignment y = x.) Because assignment appears on the face
of it like equality, the conditional equality x == y means we are not assigning,
merely testing whether x and y are equal. (In this case of testing conditional
equality, the symbol “==” is indeed symmetric.) Here are some examples of
our typical assignments:
x =2; // Variable x gets the value 2.
x = y =2; // Both x and y get the value 2.
F = {}; // F becomes the empty set.
(a, b, c) =(3, 4, 5); // Variable a becomes 3, b becomes 4, c becomes 5.
Note the important rule that simultaneous (vector) assignment assumes first
the full evaluation of the vector on the right side of the equation, followed by
the forcing of values on the left-hand side. For example, the assignment
(x, y) =(y 2 , 2x);
means that the right-hand vector is evaluated for all components, then the
left-hand vector is forced in all components. That is, the example is equivalent
to the chain (technically, we assume neither of x, y invokes hidden functions)