Wash and drain rice 4 to 5 times in a medium sized pot. Cover rice with water and allow it to soak for 20 minutes. Drain rice.
Pour in 1¼ cups (300ml) water. Drop the kombu into the pot and place pot on stove over medium heat. Just as water is about to boil, remove kombu.
Place the lid back on and allow water to come to a boil. Reduce heat to medium low and let rice cook for about 8 to 10 minutes or until all water is absorbed.
Turn off heat and remove pot without opening the lid. Allow rice to sit for 10 minutes.
Combine vinegar, sugar, and salt in a small bowl. Stir until sugar and salt dissolve.
Transfer rice to a moistened sushi barrel or large shallow bowl. Pour vinegar mixture evenly over the rice. Gently mix in the vinegar mixture with a rice paddle in a cutting motion. Do not mash. Turn the rice with the paddle and fan as you go along. This will help the rice absorb the vinegar and give it a glossy look.
Making the rolls and serving
Place a sushi mat on your work surface. Then place a sheet of nori with shiny side down onto mat, lining with the edge of the mat closest to you.
Moistened your hands and spread an even layer of vinegared rice with the help of a rice paddle onto nori, leaving about 1 inch of nori at the far end.
Line up one asparagus and several strips of lox about ¼ way up the length of nori.
Holding down the filling with your fingers, lift the mat with your thumbs and start rolling forward while retracting the mat backwards. Use the mat to help shape the roll. Repeat until all rice is used up.
Moisten knife and slice each sushi roll about ¾ inch thick.
Serve with a small dollop of wasabi and pickled ginger.