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A Game of Battleship in Leo

Contents​

Summary​

Battleship is a game where two players lay their ships into secret configurations on their respective 8x8 grids, and then take turns firing upon each other's board. The game ends when one player has sunk all of the other player's ships.

This application was translated into Leo from the zk-battleship example written by the Aleo community - show them some love!

How to Run​

Follow the Leo Installation Instructions.

This battleship program can be run using the following bash script. Locally, it will execute Leo program functions to create the board, place ships, and play a game of battleship.

cd battleship
./run.sh

The .env file contains a private key and address. This is the account that will be used to sign transactions and is checked for record ownership. When executing programs as different parties, be sure to set the private_key field in .env to the appropriate value. You can check out how we've set things up in ./run.sh for a full example of how to run the program as different parties.

1. Initializing the Players​

In order to play battleship, there must be two players with two boards. Players will be represented by their Aleo address.

We will be playing the role of these two parties:

The private key and address of player 1.
private_key: APrivateKey1zkpGKaJY47BXb6knSqmT3JZnBUEGBDFAWz2nMVSsjwYpJmm
address: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy

The private key and address of player 2.
private_key: APrivateKey1zkp86FNGdKxjgAdgQZ967bqBanjuHkAaoRe19RK24ZCGsHH
address: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry

2. Player 1 Places Ships on the Board​

Now, we need to make a board as Player 1. See the modeling the boards and ships section for information on valid ship bitstrings and placements on the board.

With player 1's private key, they initialize the board with the placement of 4 ships and the opponent's public address.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkpGKaJY47BXb6knSqmT3JZnBUEGBDFAWz2nMVSsjwYpJmm
" > .env

leo run initialize_board 34084860461056u64 551911718912u64 7u64 1157425104234217472u64 aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry
➡️  Output

 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: false.private,
  _nonce: 605849623036268790365773177565562473735086364071033205649960161942593750353group.public
}

Leo ✅ Finished 'battleship.aleo/initialize_board'

The output is a board_state record owned by Player 1. Notice that the game_started flag is false, as well as the composite ship configuration ships. 1157459741006397447u64 to a binary bitstring becomes 0001000000010000000111111000000010000000100000001000000000000111, or laid out in columns and rows:

0 0 0 1 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1

3: Player 1 Passes The Board To Player 2​

Now, we can offer a battleship game to player 2. Run offer_battleship with the record you just created:

leo run offer_battleship "{
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: false.private,
  _nonce: 605849623036268790365773177565562473735086364071033205649960161942593750353group.public
}"
➡️  Outputs

 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: true.private,
  _nonce: 5443521912126792569907060514335205174032013684291524549930033539632156136027group.public
}
 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 0u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 6986401140057557061321899375524513841643724821230599181456639629979203966487group.public
}

Leo ✅ Finished 'battleship.aleo/offer_battleship'

The first output record is the updated board_state.record. Notice the game_started flag is now true. This board cannot be used to offer any other battleship games or accept any battleship game offers. Player 1 would need to initialize a new board and use that instead. The second output record is a dummy move.record - there are no fire coordinates included to play on Player 2's board, and no information about any previous Player 2 moves (Player 2 has not made any moves yet). This move.record is owned by Player 2, who must use that in combination with their own board_state.record to accept the game. Let's do that now.

4: Player 2 Places Ships On The Board​

We switch our .env to player 2's private key and similarly run initialize_board to create a new and different board for player two.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkp86FNGdKxjgAdgQZ967bqBanjuHkAaoRe19RK24ZCGsHH
" > .env

leo run initialize_board 31u64 2207646875648u64 224u64 9042383626829824u64 aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy
➡️  Output

 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: false.private,
  _nonce: 677929557867990662961068737825412945684193990901139603462104629310061710321group.public
}

✅ Executed 'battleship.aleo/initialize_board'

Note, the output ships here is 9044591273705727u64, which in a bitstring is:

0 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1

5: Passing The Board Back To Player 1​

Now, we can accept Player 1's offer. Run start_battleship:

leo run start_battleship "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: false.private,
  _nonce: 677929557867990662961068737825412945684193990901139603462104629310061710321group.public
}" "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 0u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 6306786918362462465996698473371289503655844751914031374264794338640697795225group.public
}"
➡️  Outputs

 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: true.private,
  _nonce: 499506036017893504519951074816367233238764881167148207158107765834843789278group.public
}
 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  incoming_fire_coordinate: 0u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 7551593771072417773015833444631669906818701068612998340960968556531564726874group.public
}

✅ Executed 'battleship.aleo/start_battleship'

Notice the outputs here are similar to offer_battleship. A dummy move.record is owned by Player 1, and Player 2 gets a board_state.record with the game_started flag updated. However, now that Player 1 has a move.record and a started board, they can begin to play.

6: Player 1 Takes The 1st Turn​

We switch the .env back to player 1, and we run the transition function play.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkpGKaJY47BXb6knSqmT3JZnBUEGBDFAWz2nMVSsjwYpJmm
" > .env

leo run play "{
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: true.private,
  _nonce: 6313341191294792052861773157032837489809107102476040695601777954897783350080group.public
}" "{
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  incoming_fire_coordinate: 0u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 2798663115519921626400765401803177719929914180089719334947022448579691220488group.public
}" 1u64
➡️  Outputs

 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 1u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: true.private,
  _nonce: 5833516448655036599597838063894464861371198938108460526636526325286738488235group.public
}
 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 1u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 4383078917685812690935470339923943658033179718952229417171392956492546325808group.public
}

✅ Executed 'battleship.aleo/play'

Player 1 has an updated board_state.record - they have a new played_tiles bitstring, which corresponds to the fire coordinate they just sent to Player 2. You can see that the incoming_fire_coordinate in the move.record owned by Player 2 matches exactly the input given by Player 1. Player 2 can now play this move tile and respond with a fire coordinate of their own, and they will also let Player 1 know whether their fire coordinate hit or miss Player 2's ships.

7: Player 2 Takes The 2nd Turn​

We switch the .env back to player 2, and we run the transition function play.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkp86FNGdKxjgAdgQZ967bqBanjuHkAaoRe19RK24ZCGsHH
" > .env

leo run play "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 0u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: true.private,
  _nonce: 6864275139988909612799168784231775829713739147830284979332684562641318182923group.public
}" "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 1u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 8420474443174402614458578667801578345975509805478103542095622903412594983971group.public
}" 2048u64
➡️  Outputs

 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 2048u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: true.private,
  _nonce: 6284479302801058138006361960649628992876976428745392660731784830148359328839group.public
}
 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  incoming_fire_coordinate: 2048u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  prev_hit_or_miss: 1u64.private,
  _nonce: 8217837260140600949756911248177622179381338760298068527463640818659709985441group.public
}

✅ Executed 'battleship.aleo/play'

Player 2 now has an updated board_state.record which includes their newly updated played_tiles, only containing the fire coordinate they just sent to Player 1. Player 1 now owns a new move.record which includes the hits_and_misses field. This contains only the result of Player 1's previous fire coordinate they had sent to Player 2. It will always be a single coordinate on the 8x8 grid if it's a hit. A miss is 0u64 (8x8 grid of 0s), whereas a hit is the u64 equivalent of their previous fire coordinate in bitstring form.

If you check Player 2's ships configuration, you'll note their entire bottom row is covered by two ships, so sample valid hits on the bottom row would be: 1u64, 2u64, 4u64, 8u64, 16u64, 32u64, 64u64, and 128u64. Since Player 1's first fire coordinate (1u64) was a hit, the hits_and_misses field is also 1u64.

Player 1's next move will consume this move.record, which will update Player 1's board with the hit-or-miss, as well as figure out the result of Player 2's fire coordinate. Now that Player 1 has some played_tiles, they can no longer choose an alread-played fire coordinate. For example, running aleo run play 'board_state.record' 'move.record' 1u64 will fail, because 1u64 has already been played.

8: Player 1 Takes The 3rd Turn​

We switch the .env back to player 1, and we run the transition function play.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkpGKaJY47BXb6knSqmT3JZnBUEGBDFAWz2nMVSsjwYpJmm
" > .env

leo run play "{
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 0u64.private,
  played_tiles: 1u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: true.private,
  _nonce: 1962122153746742645258971561783872712461616481157617568489391338473028502271group.public
}" "{
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  incoming_fire_coordinate: 2048u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  prev_hit_or_miss: 1u64.private,
  _nonce: 1204008848449868423802652577996848559012797694551224583683080100053831915439group.public
}" 2u64
➡️  Outputs

 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  hits_and_misses: 1u64.private,
  played_tiles: 3u64.private,
  ships: 1157459741006397447u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  game_started: true.private,
  _nonce: 5338125050531864311985370830280952305688629865354830939402745656578990650505group.public
}
 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 2u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 7971995563631235472540847437984726419106193784727086463494463811056252801811group.public
}

✅ Executed 'battleship.aleo/play'

As before, both a board_state.record and move.record are created. The board_state.record now contains 3u64 as the played_tiles, which looks like this in bitstring form:

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1

The board_state.record hits_and_misses field has also been updated with the result of their previous move. The new move.record owned by Player 2 now contains information about whether Player 2's previous move was a hit or miss, as well as Player 1's new fire coordinate.

9: Player 2 Takes The 4th Turn​

We switch the .env back to player 2, and we run the transition function play.

echo "
NETWORK=testnet3
PRIVATE_KEY=APrivateKey1zkp86FNGdKxjgAdgQZ967bqBanjuHkAaoRe19RK24ZCGsHH
" > .env

leo run play "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 2048u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: true.private,
  _nonce: 591128247205636061702123861968396246163831838278146623498909560875485861872group.public
}" "{
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  incoming_fire_coordinate: 2u64.private,
  player_1: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  player_2: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  prev_hit_or_miss: 0u64.private,
  _nonce: 4871574741887919250014604645502780786361650856453535231083359604148337116539group.public
}" 4u64
➡️  Outputs

 • {
  owner: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  hits_and_misses: 0u64.private,
  played_tiles: 2052u64.private,
  ships: 9044591273705727u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  game_started: true.private,
  _nonce: 4866144015676673398767235148516158177034901439767024502676546368462039477864group.public
}
 • {
  owner: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  incoming_fire_coordinate: 4u64.private,
  player_1: aleo1wyvu96dvv0auq9e4qme54kjuhzglyfcf576h0g3nrrmrmr0505pqd6wnry.private,
  player_2: aleo15g9c69urtdhvfml0vjl8px07txmxsy454urhgzk57szmcuttpqgq5cvcdy.private,
  prev_hit_or_miss: 2u64.private,
  _nonce: 5304512645876453228434639693756897952439730718508628026257897445388710294282group.public
}

✅ Executed 'battleship.aleo/play'

10. Who Wins?​

Play continues back and forth between Player 1 and Player 2. When one player has a total of 14 flipped bits in their hits_and_misses field on their board_state.record, they have won the game.

ZK Battleship Privacy​

How can we ensure that the ship configurations of each player remains secret, while being able to trustlessly and fairly play with their opponent? By taking advantage of selective privacy powered by zero knowledge proofs on Aleo.

Broadly speaking, we can follow this general strategy:

  1. Create mathematical rules for placing the ships on the board, to ensure that neither player can cheat by stacking all their ships in one place, moving them off the board, or laying them across each other.

  2. Ensure that the players and boards that begin a game cannot be swapped out.

  3. Ensure that each player can only move once before the next player can move.

  4. Enforce constraints on valid moves, and force the player to give their opponent information about their opponent's previous move in order to continue playing.

Modeling the board and ships​

Most battleship representations in programs use a 64 character string or an array of arrays (8 arrays of 8 elements each) to model the board state. Unfortunately, Aleo instructions don't represent strings well yet, nor can we use for or while loops. Luckily for us, Aleo has the unsigned 64 bit integer type, or u64. To represent every space on a battleship board, from top left to bottom right, we can use each bit in a u64. For example, an empty board would be: 0u64 =

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0

Battleship is played with 4 different ship types - a ship of length 5, length 4, length 3, and length 2. Some versions of battleship have an extra length 3 ship or another extra ship type, however, we will stick to the most basic version for this project. In order to be a valid ship placement, a ship must be placed vertically or horizontally (no diagonals). On a physical board, a ship cannot break across rows or intersect with another ship, but ships are allowed to touch one another.

Similar to how we represent a board with a u64 bitstring, we can represent a ship horizontally as a bitstring. We "flip" the bits to represent a ship: | Length | Bitstring | u64 | | ------ | --------- | --- | | 5 | 11111 | 31u64| | 4 | 1111 | 15u64| | 3 | 111 | 7u64 | | 2 | 11 | 3u64 |

We can also represent a ship vertically as a bitstring. To show this, we need 7 "unflipped" bits (zeroes) in between the flipped bits so that the bits are adjacent vertically. | Length | Bitstring | u64 | | --- | --- | --- | | 5 | 1 00000001 00000001 00000001 00000001 | 4311810305u64 | | 4 | 1 00000001 00000001 00000001 | 16843009u64 | | 3 | 1 00000001 00000001 | 65793u64 | | 2 | 1 00000001 | 257u64 |

With a board model and ship bitstring models, we can now place ships on a board.

Examples of valid board configurations:​

17870284429256033024u64

1 1 1 1 1 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1  
1 1 1 1 0 0 0 1  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 1 1  
0 0 0 0 0 0 0 0

16383u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 1 1 1 1 1 1  
1 1 1 1 1 1 1 1

2157505700798988545u64

0 0 0 1 1 1 0 1  
1 1 1 1 0 0 0 1  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1

Examples of invalid board configurations:​

Ships overlapping the bottom ship:
67503903u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 1 0 0  
0 0 0 0 0 1 1 0  
0 0 0 0 0 1 1 1  
0 0 0 1 1 1 1 1

Diagonal ships:
9242549787790754436u64

1 0 0 0 0 0 0 0  
0 1 0 0 0 1 0 0  
0 0 1 0 0 0 1 0  
0 0 0 1 0 0 0 0  
0 0 0 1 1 0 0 0  
0 0 1 0 0 0 0 1  
0 1 0 0 0 0 1 0  
1 0 0 0 0 1 0 0

Ships splitting across rows and columns:
1297811850814034450u64

0 0 0 1 0 0 1 0  
0 0 0 0 0 0 1 0  
1 1 0 0 0 0 0 1  
0 0 0 0 0 0 0 0  
1 0 0 1 0 0 0 1  
0 0 0 1 0 0 0 0  
0 0 0 1 0 0 1 0  
0 0 0 1 0 0 1 0

Given these rules, our strategy will be to validate each individual ship bitstring placement on a board, and then, if all the ships are valid, compose all the positions onto a board and validate that the board with all ships are valid. If each individual ship's position is valid, then all the ships together should be valid unless any overlapping occurs.

Validating a single ship at a time​

To follow along with the code, all verification of ship bitstrings is done in verify.aleo. We know a ship is valid if all these conditions are met: If horizontal:

  1. The correct number of bits is flipped (a ship of length 5 should not have 6 flipped bits)
  2. All the bits are adjacent to each other.
  3. The bits do not split a row.

If vertical:

  1. The correct number of bits is flipped.
  2. All the bits are adjacent to each other, vertically. This means that each flipped bit should be separated by exactly 7 unflipped bits.
  3. The bits do not split a column.

If a ship is valid vertically or horizontally, then we know the ship is valid. We just need to check for the bit count, the adjacency of those bits, and make sure those bits do not split a row/column. However, we can't loop through the bit string to count bits, or to make sure those bits don't break across columns. We'll need to turn to special bitwise operations and hacks.

Bit Counting​

See the "c_bitcount" closure to follow along with the code. 50 years ago, MIT AI Laboratory published HAKMEM, which was a series of tricks and hacks to speed up processing for bitwise operations. https://w3.pppl.gov/~hammett/work/2009/AIM-239-ocr.pdf We turned to HAKMEM 169 for bitcounting inspiration, although we've tweaked our implementation to be (hopefully) easier to understand. Before diving into details, let's build some intuition.

Let a,b,c,d be either 0 or 1. Given a polynomial 8a + 4b + 2c + d, how do we find the summation of a + b + c + d? If we subtract subsets of this polynomial, we'll be left with the summation.

Step 1: 8a + 4b + 2c + d
Step 2: -4a - 2b - c
Step 3: -2a - b
Step 4: - a
Step 5: = a + b + c + d

This polynomial is basically a bitwise representation of a number, so given a 4 bit number, e.g. 1011 or 13u64, we can follow these instructions to get the bit count. Step 2 is just subtracting the starting number but bit shifted to the right (equivalent to dividing by 2). Step 3 bit shifts the starting number to the right twice and is subtracted, and Step 4 bit shifts thrice and is subtracted. Put another way: Start with a 4-digit binary number A. A - (A >> 1) - (A >> 2) - (A >> 3) = B.

Step 1: 1101 = 13u64
Step 2: -0110 = 6u64
Step 3: -0011 = 3u64
Step 4: -0001 = 1u64
Step 5: =0011 = 3u64

To make this process work for any bit-length number, where the sum of the bits is left in groups of 4 bits, we'll need to use some bit-masking, so that the sum of one group of 4 does not interfere with the next group of 4. With a larger starting number, like 1111 0001 0111 0110, we will need the following bit maskings:

For A >> 1, we'll use 0111 0111 0111 .... (in u64, this is 8608480567731124087u64)
For A >> 2, we'll use 0011 0011 0011 .... (in u64, this is 3689348814741910323u64)
For A >> 3, we'll use 0001 0001 0001 .... (in u64, this is 1229782938247303441u64)

For example, finding the sums of groups of 4 with a 16-bit number we'll call A to yield the bit sum number B:

A:    1111 0001 0111 0110  
A>>1: 0111 1000 1011 1011  
A>>2: 0011 1100 0101 1101  
A>>3: 0001 1110 0010 1110  

A>>1: 0111 1000 1011 1011  
    & 0111 0111 0111 0111:  
      0111 0000 0011 0011  

A>>2: 0011 1100 0101 1101  
    & 0011 0011 0011 0011:  
      0011 0000 0001 0001  

A>>3: 0001 1110 0010 1110  
    & 0001 0001 0001 0001:  
      0001 0000 0000 0000  

A - (A>>1 & 0111....) - (A>>2 & 0011....) - (A>>3 & 0001....):
B:    0100 0001 0011 0010  
      4    1    3    2

The next step is to combine the summation of each of those 4-bit groups into sums of 8-bit groups. To do this, we'll use another bit trick. We will shift this number B to the right by 4 (B >> 4), and add that back to B. Then, we'll apply a bit masking of 0000 1111 0000 1111 .... (in u64, this is 1085102592571150095u64) to yield the sums of bits in groups of 8, a number we'll call C.

B:    0100 0001 0011 0010  
B>>4: 0000 0100 0001 0011  
      0100 0101 0100 0101  
      4    5    4    5

apply the bit mask  
      0000 1111 0000 1111  

C:    0000 0101 0000 0101  
      0    5    0    5

At this point, we've gone from a bit sum in groups of 4 to bit sums in groups of 8. That's great, but ultimately we want the total sum of bits in the original binary number. The final bit trick is to modulo C by 255. This is 2^8 - 1. For a bit of intuition, consider the number 1 0000 0001. If we take 1 0000 0001 mod 256, we're left with 1. If we take 1 0000 0001 mod 255, we're left with 2. Modding by 255 gives us the amount of bits beyond the first 255 numbers, as 255 is the largest number that can be represented with 8 bits.

A full summary of abbreviated steps to get the bit count, starting with a 64 bit integer A (closely following the c_bitcount closure in the verify.aleo code):

let A = 64 unsigned bit integer
let B = A - (A>>1 & 8608480567731124087u64) - (A>>2 & 3689348814741910323u64) - (A>>3 & 1229782938247303441u64)
let C = (B - B>>4) & 1085102592571150095u64
bit count = C mod 255u64

Adjacency Check​

Given a ship's placement on the board and its bitstring representation (horizontally or vertically), we can determine if the bits are adjacent. Follow the c_adjacency_check closure in verify.aleo. Given the ship of length 2, we know it's horizontal bitstring is 11 (3u64) and it's vertical bitstring is 100000001 (257u64). If on the board, the ship starts at the bottom right corner, its horizontal ship placement string would be:
3u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 1 1

Vertical ship placement:
257u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1

If we move the ship to the left one column:
Horizontal 6u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 1 1 0

Vertical 514u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 1 0  
0 0 0 0 0 0 1 0

If we move the ship up one row:
Horizontal 768u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 1 1  
0 0 0 0 0 0 0 0

Vertical 65792u64

0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 1  
0 0 0 0 0 0 0 0

We can make the observation that the original bitstring is always shifted by a power of 2 to get to a new valid position on the board. Therefore, if we take the ship placement bitstring and divide by the ship bitstring (either horizontal or vertical), as long as the remaining number is a power of 2 (2^0, 2^1, 2^2, 2^3...), we know the ship's bits are adjacent.

To ensure that the remaining number is a power of 2, we can use a bit trick. See the bit trick for ensuring a bitstring is a power of 2 section.

In the code, you'll notice one extra step. Dividing a ship placement bitstring by a ship bitstring representation could result in 0, and then subtracting by 1 will result in an underflow. In that case, we know the ship placement is not valid, so we can set a number which is gauranteed to not be a power of 2.

Splitting a row or column​

Follow the c_horizontal_check closure in verify.aleo to follow the code. Assume all the bits are adjacent (see the adjacency check section). The column case is trivial. We can be certain that if a ship bitstring splits columns, the division of that ship placement bitstring by its ship bitstring representation will not yield a power of 2, and it would have failed the adjacency check.

The horizontal case must be checked because a split row bitstring could still contain a ship with adjacent bits. To make this check easier, we will condense the 64 bitstring into an 8 bitstring by taking it modulo 255. If we assume that a bitstring is not splitting a row, then taking the ship placement bitstring modulo 255 will yield an 8 bit valid bitstring. If the original ship placement bitstring is not valid, then we will have an invalid 8 bit bitstring. E.g.:

1 1 1 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0

mod 255 = 11100000 (valid)

0 0 0 0 0 0 0 1  
1 1 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0  
0 0 0 0 0 0 0 0

mod 255 = 11000001 (invalid)

How do we know the 8 bit bitstring is valid or not? We can simply do an adjacency check, as before.

Ensuring a bitstring is a power of 2​

Any power of 2 will have a single bit flipped. If we subtract 1 from that number, it will result in a complementary bitstring that, bitwise-anded with the original, will always result in 0.

E.g.

8:   1000
8-1: 0111
8&7: 0000 == 0

7:   0111
7-1: 0110
7&6: 0110 != 0

Validating all ships together in a single board​

Give individual valid ship position bitstrings, we can combine all these together into a single board using bitwise or operators. See the create_board function in verify.aleo to follow the code. Once all ships are on the board, we can count the total number of bits, which should be 14 exactly for a ship of length 5, 4, 3, and 2.

Ensure that players and boards cannot swap mid-game​

Board states are represented with the board_state record. Each board has a flag indicating whether a game has been started with the board. This flag is set when offering a battleship game to an opponent, or accepting a battleship game from an opponent. Move records are created only in 3 ways:

  1. Offering a battleship game creates a dummy move record that sets the two players to the addresses set in the board state record.
  2. Accepting a battleship game consumes the first dummy move record and checks that the move record contains the same two players as the board of the player accepting the game. Then, a new dummy move record is created and keeps the same two players.
  3. A move record must be consumed in order to play and create the next move record. There's a check to ensure the players in the move record matches the players in the board, and the players in the next move record are automatically set.

The only way moves not matching a board can be combined is if the players begin multiple games with each other. As long as one player is honest and only accepts a single game with a particular opponent, only one set of moves can be played on one board between them.

Ensure that each player can only move once before the next player can move​

A move record must be consumed in order to create the next move record. The owner of the move record changes with each play. Player A must spend a move record in order to create a move record containing their fire coordinate, and that move record will be owned by Player B. Player B must spend that move record in order to create the next move record, which will belong to Player A.

Enforce constraints on valid moves, and force the player to give their opponent information about their opponent's previous move in order to continue playing​

A valid move for a player is a fire coordinate that has only one flipped bit in a u64. We can make sure only one bit is flipped with the powers of 2 bit trick. That single bit must be a coordinate that has not been played by that player before, which we check in board.aleo/update_played_tiles.

In order to give their next move to their opponent, a player must call the main.aleo/play function, which checks the opponent's fire coordinate on the current player's board. The move record being created is updated with whether that fire coordinate was a hit or a miss for the opponent.

Winning the game​

Right now, the way to check when a game has been won is to count the number of hits on your hits_and_misses field on your board_state record. Once you have 14 hits, you've won the game.