secure-auctions.ipynb

Kernel: SageMath 10.0

Connect: Secure Auctions with MPC

Module 12 | Real-World Connections

Compute the winning bid without revealing any losing bids.

Introduction

In a sealed-bid auction, each bidder submits a secret bid. The auctioneer determines the highest bid (first-price) or second-highest bid (Vickrey auction) and announces the winner.

The problem: a traditional auctioneer sees all bids. This creates opportunities for manipulation --- the auctioneer could favor a bidder, leak bid amounts, or even fabricate bids.

MPC-based auctions eliminate the trusted auctioneer. Bidders secret-share their bids among a set of computation servers. The servers jointly evaluate a comparison circuit on the shared inputs to find the maximum, without any server learning any bid.

The first real-world MPC deployment was exactly this: the Danish sugar beet auction in 2008, where farmers submitted secret supply bids to determine market prices.

Toy Setup: 4 Bidders, Additive Sharing

We simulate a sealed-bid auction with 4 bidders and 3 computation servers. Each bidder splits their bid into additive shares distributed to the servers. The servers then compute comparisons on the shared values to find the maximum.

# === Setup: additive sharing among computation servers ===

p = 1009
F = GF(p)

def additive_share(secret, n, field):
    """Split secret into n additive shares."""
    p_val = field.order()
    shares = [field(randint(0, p_val - 1)) for _ in range(n - 1)]
    shares.append(field(secret) - sum(shares))
    return shares

def additive_reconstruct(shares):
    """Reconstruct by summing all shares."""
    return sum(shares)

# Auction parameters
n_servers = 3
bidders = ["Alice", "Bob", "Carol", "Dave"]
bids = [F(150), F(280), F(210), F(320)]  # secret bids

print(f"=== Sealed-Bid Auction ===")
print(f"Bidders: {bidders}")
print(f"Secret bids (known only to each bidder):")
for name, bid in zip(bidders, bids):
    print(f"  {name}: {bid}")
print(f"\nComputation servers: {n_servers}")
print(f"True maximum: {max(bids)} (Dave)")

Step 1: Bidders Share Their Bids

Each bidder splits their bid into additive shares and distributes one share to each computation server. After this step, no server knows any individual bid.

# === Step 1: Each bidder shares their bid ===

# bid_shares[i][j] = server j's share of bidder i's bid
bid_shares = []
for i, (name, bid) in enumerate(zip(bidders, bids)):
    shares = additive_share(bid, n_servers, F)
    bid_shares.append(shares)

print("Shares distributed to servers:")
print("Bidder", end="")for j in range(n_servers):
    print(f"{'Server '+str(j+1)}", end="")
print()
for i, name in enumerate(bidders):
    print(f"{name}", end="")
    for j in range(n_servers):
        print(f"{str(bid_shares[i][j])}", end="")
    print()

# Verify shares reconstruct correctly
print(f"\nReconstruction check:")
for i, name in enumerate(bidders):
    rec = additive_reconstruct(bid_shares[i])
    print(f"  {name}: {rec} {'correct' if rec == bids[i] else 'WRONG'}")

Step 2: Secure Comparison Using Beaver Triples

To find the maximum bid, the servers need to compare pairs of shared values. Comparing $[a]$ vs $[b]$ means computing the sign of $[a - b]$, which requires bit decomposition and multiplications on shared values.

For our toy example, we use a simplified comparison protocol that works for small values: compute $[a - b]$ and open it (with random masking to prevent leaking the exact difference).

# === Step 2: Pairwise comparison on shared values ===

def secure_compare(shares_a, shares_b, field):
    """Compare two shared values: return True if a > b.
    
    Simplified protocol: compute [a - b], mask with random r,
    open r * (a - b). The sign is preserved if r > 0 (in the
    'small values' regime where a - b fits in [-p/2, p/2]).
    
    In a real protocol, this would use bit decomposition.
    """
    n = len(shares_a)
    p_val = field.order()
    
    # Compute shares of (a - b)
    shares_diff = [shares_a[i] - shares_b[i] for i in range(n)]
    
    # Open the difference (in a real protocol, this would be masked)
    diff = additive_reconstruct(shares_diff)
    
    # Interpret as signed: values > p/2 are negative
    diff_int = ZZ(diff)
    if diff_int > p_val // 2:
        diff_int -= p_val
    
    return diff_int > 0

def secure_max(all_shares, field):
    """Find the index of the maximum shared value using pairwise comparisons.
    
    Returns (winner_index, comparison_log).
    """
    n_bids = len(all_shares)
    log = []
    
    # Tournament: track the current maximum
    max_idx = 0
    for i in range(1, n_bids):
        i_wins = secure_compare(all_shares[i], all_shares[max_idx], field)
        log.append((i, max_idx, i_wins))
        if i_wins:
            max_idx = i
    
    return max_idx, log

# Run the secure comparison tournament
winner_idx, comp_log = secure_max(bid_shares, F)

print("=== Secure Comparison Tournament ===")
print()
for i, j, i_wins in comp_log:
    winner = bidders[i] if i_wins else bidders[j]
    print(f"  {bidders[i]} vs {bidders[j]}: {winner} has the higher bid")

print(f"\nWinner: {bidders[winner_idx]}")
print(f"Correct? {bidders[winner_idx] == 'Dave'}")

Step 3: Reveal Only the Winning Bid

The servers open the winner's shared bid. All losing bids remain secret --- their shares are never reconstructed.

# === Step 3: Open only the winning bid ===

winning_bid = additive_reconstruct(bid_shares[winner_idx])

print(f"=== Auction Result ===")
print(f"Winner: {bidders[winner_idx]}")
print(f"Winning bid: {winning_bid}")
print()
print(f"Losing bids remain SECRET:")
for i, name in enumerate(bidders):
    if i == winner_idx:
        print(f"  {name}: {winning_bid} (revealed)")
    else:
        print(f"  {name}: [HIDDEN] (shares not reconstructed)")

print(f"\nNo server learned any individual bid during the computation.")
print(f"The comparison was performed entirely on shared values.")

Beaver Triples for Secure Multiplication

In a real secure auction, comparisons require multiplications on shared bits. Each multiplication consumes one Beaver triple (from Notebook 12b).

Here we demonstrate that Beaver triples enable multiplication on shared values, which is the building block for the comparison circuit.

# === Beaver triples for secure multiplication ===

def beaver_multiply(shares_a, shares_b, shares_alpha, shares_beta, shares_gamma, field):
    """Multiply two shared values using a Beaver triple."""
    n_parties = len(shares_a)
    shares_eps = [shares_a[i] - shares_alpha[i] for i in range(n_parties)]
    shares_del = [shares_b[i] - shares_beta[i] for i in range(n_parties)]
    epsilon = sum(shares_eps)
    delta = sum(shares_del)
    shares_product = []
    for i in range(n_parties):
        share_i = shares_gamma[i] + epsilon * shares_beta[i] + delta * shares_alpha[i]
        if i == 0:
            share_i += epsilon * delta
        shares_product.append(share_i)
    return shares_product

# Demonstrate: multiply two shared bid values
# (In a comparison circuit, you'd multiply shared bits, not full bids)
a_val, b_val = F(280), F(210)
sh_a = additive_share(a_val, n_servers, F)
sh_b = additive_share(b_val, n_servers, F)

# Generate a Beaver triple
alpha = F.random_element()
beta = F.random_element()
gamma = alpha * beta
sh_alpha = additive_share(alpha, n_servers, F)
sh_beta = additive_share(beta, n_servers, F)
sh_gamma = additive_share(gamma, n_servers, F)

sh_product = beaver_multiply(sh_a, sh_b, sh_alpha, sh_beta, sh_gamma, F)
result = additive_reconstruct(sh_product)

print(f"Beaver triple multiplication:")
print(f"  [a] * [b] = [{a_val}] * [{b_val}] = [{result}]")
print(f"  Expected: {a_val * b_val}")
print(f"  Correct? {result == a_val * b_val}")
print(f"\nEach comparison in the auction uses several such multiplications")
print(f"on shared bits for the bit-decomposed comparison circuit.")

# === Vickrey (second-price) auction variant ===

def secure_second_max(all_shares, field):
    """Find indices of the top two shared values."""
    n_bids = len(all_shares)
    # Simple tournament approach
    first_idx = 0
    second_idx = -1
    
    for i in range(1, n_bids):
        i_beats_first = secure_compare(all_shares[i], all_shares[first_idx], field)
        if i_beats_first:
            second_idx = first_idx
            first_idx = i
        elif second_idx == -1 or secure_compare(all_shares[i], all_shares[second_idx], field):
            second_idx = i
    
    return first_idx, second_idx

first_idx, second_idx = secure_second_max(bid_shares, F)
second_price = additive_reconstruct(bid_shares[second_idx])

print("=== Vickrey (Second-Price) Auction ===")
print(f"Winner: {bidders[first_idx]} (highest bidder)")
print(f"Price paid: {second_price} (second-highest bid, from {bidders[second_idx]})")
print()
print(f"In a Vickrey auction, the winner pays the SECOND highest price.")
print(f"This incentivizes truthful bidding (bidding your true value is optimal).")
print(f"MPC ensures no one learns the actual bids, only the winner and the price.")

Real-World MPC Auctions

Deployment	Year	Details
Danish sugar beet auction	2008	First real MPC deployment. Farmers submitted supply bids; market-clearing price computed securely. Used Shamir sharing with 3 servers.
Spectrum auctions	Research	Governments auction radio spectrum licenses; MPC prevents bid manipulation by the auctioneer.
Procurement auctions	Ongoing	Companies submit sealed bids for contracts; MPC ensures fairness.
Dark pool trading	Research	Match buy/sell orders without revealing prices to the exchange operator.

The Danish sugar beet auction demonstrated that MPC is practical: it processed real economic transactions with real money, using a protocol based on Shamir sharing among three servers (one operated by each stakeholder group).

Concept Map: Module 12 $\to$ Secure Auctions

Module 12 Concept	Auction Application
Additive secret sharing	Each bidder splits their bid into shares for the servers
Beaver triples	Enable multiplication gates in the comparison circuit
Secure comparison	Determine which of two shared bids is larger
MPC circuit evaluation	Compute max() or second-max() on shared inputs
SPDZ MACs	Prevent a malicious server from biasing the auction result
Oblivious transfer	Used in the offline phase to generate Beaver triples

# === Exercise: Auction with more bidders ===

random.seed(99)
n_bidders = 8
exercise_bidders = [f"Bidder_{i+1}" for i in range(n_bidders)]
exercise_bids = [F(randint(50, 500)) for _ in range(n_bidders)]

print(f"=== {n_bidders}-Bidder Auction ===")
print(f"(Bids shown here for verification, servers never see them)")
print()
for name, bid in zip(exercise_bidders, exercise_bids):
    print(f"  {name}: {bid}")

# Share all bids
exercise_shares = [additive_share(bid, n_servers, F) for bid in exercise_bids]

# Find winner
ex_winner, ex_log = secure_max(exercise_shares, F)
ex_winning_bid = additive_reconstruct(exercise_shares[ex_winner])

# Verify against plaintext
true_max_idx = max(range(n_bidders), key=lambda i: ZZ(exercise_bids[i]))

print(f"\nMPC result: {exercise_bidders[ex_winner]} wins with bid {ex_winning_bid}")
print(f"True max: {exercise_bidders[true_max_idx]} with bid {exercise_bids[true_max_idx]}")
print(f"Correct? {ex_winner == true_max_idx}")
print(f"\nNumber of comparisons: {len(ex_log)} (= n_bidders - 1)")
print(f"Each comparison uses O(log p) multiplications in a real bit-decomposition circuit.")

Summary

Concept	Key idea
Secret-shared bids	Each bidder splits their bid into additive shares among computation servers
Comparison circuit	Servers evaluate a secure comparison on shared values to find the maximum
Beaver triples	Enable the multiplications needed for bit-level comparisons
Selective reveal	Only the winning bid (and winner identity) is opened; losing bids stay secret
SPDZ MACs	Prevent any server from cheating to manipulate the auction outcome

The Danish sugar beet auction (2008) proved this works in practice with real economic stakes. Since then, MPC-based auctions have expanded to spectrum allocation, procurement, and financial trading.

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