This does NOT affect all players. For some reason, it only occurs for some players. It cannot be resolved by any means that I can come up with (Restarting server, deleting the affected player's player file, making the player reinstall minecraft, reinstalling java on server).

Here is the error that is churned out when a player is kicked from the server immediately upon joining:

Connection #118 read thread/ERROR: Exception on tcp connection
java.net.SocketException: Connection reset
at java.net.SocketInputStream.read(Unknown Source) ~[?:1.7.0_21]
at java.net.SocketInputStream.read(Unknown Source) ~[?:1.7.0_21]
at java.io.FilterInputStream.read(Unknown Source) ~[?:1.7.0_21]
at org.bouncycastle.crypto.io.CipherInputStream.nextChunk(Unknown Source) ~[13w39b.jar:?]
at org.bouncycastle.crypto.io.CipherInputStream.read(Unknown Source) ~[13w39b.jar:?]
at java.io.DataInputStream.readUnsignedByte(Unknown Source) ~[?:1.7.0_21]
at gw.a(SourceFile:180) ~[13w39b.jar:?]
at ec.j(SourceFile:251) [13w39b.jar:?]
at ec.c(SourceFile:19) [13w39b.jar:?]
at ed.run(SourceFile:96) [13w39b.jar:?]
[01:19:19] [Server thread/INFO]: SirGregorius lost connection: disconnect.genericReason
[01:19:19] [Server thread/INFO]: SirGregorius left the game

That's about all the information I've been able to gather on this bug.