The difficult task of redrawing State congressional districts, known as redistricting, is given to which legislative body?

Redistricting may follow other criteria depending on State and local laws:

compactness.

contiguity.

equal population.

preservation of existing political communities.

partisan fairness.

racial fairness.

found this on google

The majority of states assigned the state legislatures to be responsible for reapportionment and redistricting but over the last decades, this responsibility was delegated to commissions.

Explanation:

The origins for each of the nation's redistricting commissions that holds primary responsibility for drawing plans differ from each other.

Commissions in 13 states have primary responsibility for drawing legislative district lines. In 6 states, commissions have primary responsibility for drawing congressional lines. Arkansas in 1956 was the first commission to be established with one or two more states switching every ten years.

Nine states created their commissions through legislative referrals responsible for legislative redistricting. Four other commissions were created through a citizens' initiatives.

For congressional redistricting, four states created their commissions through legislative referrals, and the other two commissions were created through citizens' initiatives. Here too, constitutional amendments were required.

Some states have an advisory commissions or backup commissions that comes in if legislators fail to redistrict by themselves.