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Twisted-edwards

Twisted-edwards-curve-completep

Check if a twisted Edwards curve is complete.

Signature
(twisted-edwards-curve-completep curve) → yes/no
Arguments: curve — Guard (twisted-edwards-curvep curve).
Returns: yes/no — Type (booleanp yes/no).

According to the paper on twisted Edwards curves referenced in twisted-edwards, this is the case when a is a square and d is a non-square. Completeness means that the addition formula (see twisted-edwards-add) works for every pair of point on the curve. In particular, the denominators of the coordinates of the sum are both always different from 0 when the curve is complete.

Definitions and Theorems

Function: twisted-edwards-curve-completep

(defun twisted-edwards-curve-completep (curve)
  (declare (xargs :guard (twisted-edwards-curvep curve)))
  (let ((acl2::__function__ 'twisted-edwards-curve-completep))
    (declare (ignorable acl2::__function__))
    (and (pfield-squarep (twisted-edwards-curve->a curve)
                         (twisted-edwards-curve->p curve))
         (not (pfield-squarep (twisted-edwards-curve->d curve)
                              (twisted-edwards-curve->p curve))))))

Theorem: booleanp-of-twisted-edwards-curve-completep

(defthm booleanp-of-twisted-edwards-curve-completep
  (b* ((yes/no (twisted-edwards-curve-completep curve)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: twisted-edwards-curve-completep-of-twisted-edwards-curve-fix-curve

(defthm
 twisted-edwards-curve-completep-of-twisted-edwards-curve-fix-curve
 (equal (twisted-edwards-curve-completep
             (twisted-edwards-curve-fix curve))
        (twisted-edwards-curve-completep curve)))

Theorem: twisted-edwards-curve-completep-twisted-edwards-curve-equiv-congruence-on-curve

(defthm
 twisted-edwards-curve-completep-twisted-edwards-curve-equiv-congruence-on-curve
 (implies (twisted-edwards-curve-equiv curve curve-equiv)
          (equal (twisted-edwards-curve-completep curve)
                 (twisted-edwards-curve-completep curve-equiv)))
 :rule-classes :congruence)