Is it a condition to use dust to purify the bottom of a shoe if it has been soiled by impurity, or is it sufficient to walk on asphalt or tiles?

Firstly: The majority of scholars hold that impurity can only be removed by water, whereas the Hanafis and Ibn Taymiyyah are of the view that if the impurity is removed by any purifying agent, the place becomes pure, and this is the more preponderant opinion.

Secondly: It is not a valid condition to walk on dust to purify the sole of a shoe. The shoe is purified by rubbing it on the ground, and this includes sand and other parts of the earth, as indicated by the hadith of Abu Sa'id al-Khudri that the Prophet, peace and blessings be upon him, commanded that shoes be wiped on the ground if there was any filth on them.

Thirdly: If someone prays with an impurity while unaware of it, their prayer is valid and they are not required to repeat it. The same applies to circumambulation (Tawaf), as the matter is easier than prayer.