The authorities' "power of the sword" is well-established throughout the Bible:

Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God. 2 Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment. 3 For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval, 4 for he is God’s servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God’s wrath on the wrongdoer. (Romans 13)

I'm not sure anyone is still reading this question, but the Hebrew word used in Exodus 20 and Deuteronomy 5 means "murder," not "kill." It's so translated in the Latin Vulgate, the most commonly-used translation in the Middle Ages. When Wycliffe rendered the commandment as "Thou schalt not sle," the word meant "murder" and had been used in that sense in secular laws. Similarly, the King James Bible's "Thou shalt not kill" had similar implications at the time. Since those words have changed since, more modern translations typically render the verse as "Thou shalt not murder."