Moses gave the Law

Moses gave the Law in the Old Covenant. The Law demands righteousness from mankind. 

All these curses shall come upon you and pursue you and overtake you till you are destroyed, because you did not obey the voice of the Lord your God, to keep his commandments and his statutes that he commanded you. Deuteronomy 28:45

To be righteous, a person had to keep the Ten Commandments and all the other laws. An impossible task for fallen humans, not one person ever fulfilled the demands of the law to be declared righteous, right with God.
None is righteous, no, not one. Romans 3:10b

Then Jesus, grace, and truth came, fulfilling the demands of the law, providing righteousness to all who would believe. 
For Christ is the end of the law for righteousness to everyone who believes
. Romans 10:4  
So by the one man’s obedience the many will be made righteous. Romans 5:19
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God. 2 Corinthians 5:21

God’s own righteousness:
…..their righteousness is of me, says the Lord. Isaiah 54:17b
And this righteousness from God comes through faith in Jesus Christ to all who believe. Rom 3:22a

Everyone who believes in him is declared right with God–something the Law of Moses could never do. Acts 13:39
What shall we say, then? That Gentiles who did not pursue righteousness have attained it, that is, a righteousness that is by faith. Romans 9:30

But now the righteousness of God has been manifested apart from the Law, although the Law and the Prophets bear witness to it.
Romans 3:21
And because of him you are in Christ Jesus, who became to us wisdom from God, righteousness and sanctification and redemption 1 Corinthians 1:30

Faithful, born again Christians are the righteousness of God in Christ!