SECURE ONE-WAY INTERACTIVE COMMUNICATION

Abhinav Aggarwal Varsha Dani, Thomas Hayes, Jared Saia

PROBLEM STATEMENT

Alice Bob

Adversary

➤ Alice has a message of length L for Bob

➤ Adversary can flip T bits

➤ Adversary is oblivious

➤ Error tolerance 0 < ✏ < 1

Can the message be sent to Bob with :

➤ Probability of success , and

➤ small number of bits exchanged?

� 1� ✏

OUR ASSUMPTIONS

➤ Both Alice and Bob know L and

➤ T is unknown

➤ Private channel

➤ Individual computation is instantaneous

✏

OUR MAIN RESULT

Expected number of bits exchanged between Alice and Bob :

Probability that Bob has the correct guess of Alice’s message upon termination � 1� ✏

L+O (T +min{(T + 1), L/ logL} log(L/✏))

Much better then ECC for small T!!

For constant ε

L+

(O(logL) for fixed T

⇥(T ) otherwise

REED-SOLOMON CODES [RS]

➤ Degree d = ⌈ L / log L⌉ polynomial used to represent the message

➤ Bob needs at least (d+1) evaluations of this polynomial for reconstruction

➤ Polynomial constructed over field of size

(1/3)-ERROR CORRECTING CODES

➤ Corrects at most a third of total bits

➤ Multiplicative blowup of at most 2

➤ Forces the adversary to pay Θ(length of the message)

2dlogLe

ALGORITHM

Alice Bob

Message polynomial

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Echo of the fingerprint

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Terminate

Echo of the fingerprint

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Terminate

Echo of the fingerprint

Terminate

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Successful round…

Terminate

Echo of the fingerprint

Terminate

FINGERPRINTING [NAOR]➤ Let r = random binary string, m = message of length ℓ

➤ Produces randomized fingerprints (r,F(r,m))

➤ Given probability of collision p, produces hash of length Θ(ℓ/p)

ALGORITHM

Alice Bob

Message polynomial in plaintext

ALGORITHM

Alice Bob

Message polynomial in plaintext

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

String of all zeros

Fingerprint mismatch

Message polynomial in plaintext

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Noise

String of all zeros

Fingerprint mismatch

Message polynomial in plaintext

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Noise

String of all zeros

Fingerprint mismatch

Message polynomial in plaintext

ECC and AMD encoded fingerprint

ECC encoded evaluation

ECC encoded evaluation

ALGORITHM

Alice Bob

NoiseRe

peat

String of all zeros

Fingerprint mismatch

Message polynomial in plaintext

ECC and AMD encoded fingerprint

ECC encoded evaluation

ECC encoded evaluation

ALGEBRAIC MANIPULATION DETECTION CODES [AMD]

➤ Enable detection of bit corruption

➤ Work only for private channels

➤ Encode a message m into a value m’

➤ Any bit flipping of m is detected with probability ≥ 1-δ

➤ Produces codewords of length |m’| = |m| + O(1/δ)

ALGORITHM

Alice Bob

Message polynomial

ALGORITHM

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Message polynomial

Echo of the fingerprint

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Message polynomial

Noise

Echo mismatch

Echo of the fingerprint

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Message polynomial

ECC encoded evaluation

ECC encoded evaluation

Noise

Echo mismatch

Echo of the fingerprint

ECC and AMD encoded fingerprint

ALGORITHM

Alice Bob

Message polynomial

ECC encoded evaluation

ECC encoded evaluation

Repe

atNoise

Echo mismatch

Echo of the fingerprint

ECC and AMD encoded fingerprint

ALGORITHM (WORST CASE)

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Echo of the fingerprint

Terminate

ALGORITHM (WORST CASE)

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

Echo of the fingerprint

TerminateChannel not silent

ALGORITHM (WORST CASE)

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

ECC encoded evaluation

ECC encoded evaluation

Echo of the fingerprint

Terminate

Channel not silent

ALGORITHM (WORST CASE)

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

ECC encoded evaluation

ECC encoded evaluation

Echo of the fingerprint

Terminate

Channel not silent

String of all zeros

ALGORITHM (WORST CASE)

Alice Bob

Message polynomial

ECC and AMD encoded fingerprint

ECC encoded evaluation

ECC encoded evaluation

Repe

at

Echo of the fingerprint

Terminate

Channel not silent

String of all zeros

OUR NEW IDEAS➤ Handle unknown T

➤ Ensure high cost to adversary to delay termination

➤ Synchronization achieved implicitly

➤ Tradeoff between probability of failure and number of bits sent

➤ Distinguishing between “silence”, “noise” and codewords on the channel

➤ Ensuring a constant ratio of algorithmic cost vs. adversary’s cost for large T

CHALLENGES FACED

FUTURE WORK➤ Remove assumption of knowledge of L by Bob

➤ Extend results to multi-party case

➤ Establish lower bounds

32

OUR TEAM

Varsha Dani

Tom Hayes

Jared Saia

QUESTIONS??

RELATED WORK

SCHULMAN’ 96Tolerates noise rate with constant blowup in cost using non-constructive tree codes� <

1

240

BRAKERSKI AND KALAI, FOCS’12Made the upper bounds constructive for noise rate � <

1

32

KOL AND RAZ, FOCS’13Expected cost overhead of bits for stochastic noise rate � <<

1

2O⇣p

� log(1/�) L⌘

HAEUPLER’14O⇣p

� log log(1/�) L⌘

Expected cost overhead of bits for known and sufficiently small

adversarial noise rate �

ICALP'15O⇣T +

pLT + L

⌘Expected cost overhead of bits for unknown noise rate, private channels

and probability of success � 1� 1

L logL

ALICE’S ALGORITHM

BOB’S ALGORITHM