"But now..." v. 21 begins. But now something new has happened. But now there is a new possibility for righteousness. But now an entirely new epoch in history has dawned because of what God has done in Jesus Christ. But now a new strength has come to rescue our weak flesh. "But now apart from the law the righteousness of God has been revealed, being witnessed to by the law and the prophets - the righteousness of God through the faithfulness of Jesus Christ for all who believe."* This new righteousness, Paul claims, is for Jew and Gentile alike since there is no real difference between them since both have sinned. It is a gift freely given out of the redemption and atonement accomplished in Christ Jesus. As a result, there can be no boasting of Jewish Christians over Gentile Christians (or the other way around) since both are made righteous by the same God on account of the same faithfulness.
But just because God's righteousness has now been revealed apart from the law doesn't mean that God has now done away with law entirely. Rather this new righteousness revealed in Jesus requires a new kind of law. Paul says it is a law of faithfulness instead of the law of works. That is, rather than righteousness being determined by works of the Jewish law (things like circumcision, Sabbath observance, and food laws) the source of righteousness is now the faithfulness of Jesus Christ and living faithfully out of and in imitation of that faithfulness is the law which followers of Jesus are called to obey. Jesus is both the new righteousness revealed apart from the law and the new law of faithfulness.
But even to call this righteousness and this law "new" is a fairly serious misnomer if by that we mean that it has no connection to what is "old". To be sure, Paul's claim that Jesus is God's righteousness and God's law is radically new in some very substantial ways. It is no small thing that Paul claimed that circumcision, food laws, and Sabbath observance were not necessary for Gentiles to be considered righteous. These were central pieces of Jewish identity and themselves crucial parts of the law that God had given Israel. But, in other ways, Paul wants us to see that this new law and new righteousness in Jesus really aren't so new - at least not so new as to be entirely alien to God's history with Israel. After all, Paul says that this righteousness is witnessed to by the law and the prophets.
One of those prophets was Habakkuk whom Paul has already quoted in 1:17. Paul's language in these latter verses of chapter 3 echoes and expands on much of the language Paul use in 1:17 and by that connections helps us to see yet again why that quotation from Habakkuk is so critical to Paul's understanding of what God has done in Jesus Christ. In Habakkuk's opening verses we hear that the law is paralyzed and that justice goes forth perverted. This is precisely the same thing that Paul has been arguing in the opening chapters of Romans; that the law is paralyzed, weak, and powerless to produce true righteousness. In Habakkuk chapter 2, the prophet declares that "the righteous will live out of faithfulness." In other words, in a time when when the law is failing to serve its purpose and God is doing strange and unexpected things like using the Babylonians as his instrument to make Israel righteous, those who wish to be counted as righteous will still live faithfully before God trusting in God's faithfulness even its strange, new forms. Paul sees a parallel here as well; since the law has failed to produce righteousness a new righteousness must come through a life of faithfulness, even if it is the strange and unexpected faithfulness of a crucified messiah. Paul sees Habakkuk as a precedent for arguing that this is not the first time that God has called the righteous to live by a law of faithfulness as an alternative to a law of works. What is new is that this faithfulness has been embodied in the person of Jesus, the crucified messiah. He is the new law and Paul believes that the law of Christ can produce true righteousness in a way the law of works never could.
*You may have noticed that I translated this phrase in v.22 as "the faithfulness of Jesus Christ" rather than "faith in Jesus Christ." There are two issues that allow this phrase to be translated in these two ways. The first is that the word "pistis" in Greek can be translated either as "faith" or "faithfulness". It really means both intertwined together but one aspect of the word can be emphasized more than the other in certain contexts.
The second is an issue of Greek grammar. The words "of" or "in" are not actually in the Greek text of Romans 3:22. Instead, this phrase is in what is known as the genitive case. (In English, word order is an important part of determining the meaning of a sentence. So, for example, in the sentence "Dave teaches his class lots of crazy Greek stuff." we know that "Dave" is the subject of the sentence because it comes before the verb. Greek, on the other hand, uses a case system to indicate how a word functions in the sentence. A change in case is indicated by a slight change in the spelling of a word. So, for example, "Jesus Christ" which is "Iesous Christos" in the nominative case becomes "Iesou Christou" in the genitive case.) The most typical use of the genitive case is to indicate possession but it can have a whole range of meanings throughout the New Testament.
Two of those possibilities are known as the subjective genitive and objective genitive. You can see something similar to this in English in a phrase like the "the love of God." Does that phrase refer to God's love for us or our love for God? Is God doing the loving (so God is the subject, a subjective genitive) or is God receiving the love (so God is the object of the love, an objective genitive)? It can mean either or maybe even both at the same time but the only way you would be able to decide would be context. What makes the most sense with what is around it? If Paul said "Christ's death demonstrates the love of God" it would be clear that he was referring to God's love for us. If he said "Our love for others demonstrates the love of God" it might be more difficult to decide whose love Paul was talking about.
Something similar is at stake in our phrase in Romans 3:22. Is Jesus Christ the object of the faith(fulness)? That is, is Paul talking about Jesus receiving our faith? Or is Jesus the subject of the faith(fulness)? That is, is Jesus the one who has himself been faithful? Of course, both are true in the big picture! The issue here is not choosing one to the exclusion of the other. The question is a matter of emphasis. Is Paul's emphasis in Romans on our belief or on God's faithfulness through Jesus Christ?