This paper provides a comprehensive cryptanalysis of the Dark Skippy attack , a specialized implementation of a fundamental vulnerability in the Elliptic Curve Digital Signature Algorithm (ECDSA) —a nonce reuse attack , also known as the Phantom Curve Attack . This vulnerability exploits the weak entropy of the k (nonce) parameter during the signing process of cryptographic transactions, leading to the complete compromise of Bitcoin wallet users’ private keys.
The study includes a detailed mathematical analysis of the attack mechanism, a description of the seed phrase recovery algorithm using the Pollard-Kangaroo algorithm , and a practical demonstration of the use of the KeySilentLeak cryptanalytic tool for recovering lost private keys based on identified vulnerabilities in real Bitcoin transactions.
It is demonstrated that the Dark Skippy attack allows an attacker to extract the full 16-byte entropy of a seed phrase from just two signed transactions observed in the Bitcoin network mempool. A critical factor is the use of malicious hardware wallet firmware that deliberately generates low-entropy nonces directly derived from the seed phrase. Quantitative estimates of the attack’s computational complexity are presented, demonstrating a reduction in the cost of cryptanalysis from a theoretical 2256 to a practically achievable 232 operations on modern hardware.
The paper presents recommendations for protecting against this class of attacks , including implementing the RFC 6979 standard for deterministic nonce generation, conducting regular audits of cryptographic pseudorandom number generators (PRNGs), and raising user awareness of the critical importance of verifying hardware wallet firmware.
- Tutorial: https://youtu.be/pM0fuUZk8p4
- Tutorial: https://cryptodeeptech.ru/phantom-curve-attack
- Tutorial: https://dzen.ru/video/watch/69c8603451b3e70d64f66471
- Google Colab: https://bitcolab.ru/keysilentleak-cryptanalytic-research-tool
In August 2024, the Dark Skippy attack , targeting Bitcoin hardware wallets, was unveiled . Its essence lies in the malicious firmware deliberately generating weak (low-entropy) nonce values when signing transactions . These nonces are directly derived from the wallet’s seed phrase: the first 8 bytes of the seed are used for the first signature, and the remaining 8 bytes for the second.
Thus, the two signatures encode the entire 16-byte entropy of the seed. An attacker, observing these signatures in the mempool, uses Pollard’s Kangaroo algorithm to recover the seed phrase.
With just two signed transactions, the attacker gains complete control over all of the user’s assets.
Dark Skippy isn’t a fundamentally new vulnerability in cryptographic protocols. It’s a highly effective implementation of a long-known problem: the repeated or predictable use of the nonce parameter in the ECDSA digital signature algorithm . Therefore, to fully understand the threat, it’s necessary to turn to a more general cryptographic tool: the ECDSA Nonce Reuse Attack .
Phantom Curve Attack: A deadly re-nonce vulnerability in ECDSA. This attack has been described as one of the most damaging to Bitcoin security . Its scientific name is “ECDSA Private Key Recovery Attack via Nonce Reuse .” The vulnerability occurs when the same (or predictably related) secret parameter k (the nonce) is used to generate two or more signatures.
In ECDSA, the signature for a message m is formed as follows:
- Let d be a private key, G be a base point of the elliptic curve secp256k1, and Q = d·G be a public key.
- A random number k (nonce) is selected in the range [1, n-1], where n is the order of the curve.
- The point R = k·G is calculated ; its x-coordinate r = Rx mod n is the first component of the signature.
- The second component is calculated s = k⁻¹·(H(m) + r·d) mod n , where H(m) is the hash of the message.
The signature is a pair (r, s) .
Critical weakness: If the same k is used for two different messages m₁ and m₂ , then the values of r will be the same . Knowing the two signatures (r, s₁) and (r, s₂) , an attacker can solve a system of equations for the private key d :
s₁ = k⁻¹·(H(m₁) + r·d) mod n
s₂ = k⁻¹·(H(m₂) + r·d) mod nBy subtracting one equation from the other, we can eliminate d and find k :
k = (H(m₁) – H(m₂))·(s₁ – s₂)⁻¹ mod nThen the private key is calculated trivially:
d = (s₁·k – H(m₁))·r⁻¹ mod nThus, reusing a nonce results in direct disclosure of the private key .
Dark Skippy uses a more sophisticated scenario: the nonce is not repeated exactly, but is weak (low-entropy) and is directly derived from the seed. This turns the key recovery problem into a discrete logarithm problem on an elliptic curve: knowing the public nonce R = k G and knowing that k belongs to an extremely small subset of possible values (only 2⁶⁴ possibilities for an 8-byte nonce), an attacker can use the Pollard–Kangaroo algorithm . This algorithm solves the discrete logarithm problem on an interval of known size in time proportional to the square root of the interval size, making the attack practically feasible on a standard computer.
Dark Skippy can be viewed as a targeted exploitation of a nonce reuse vulnerability at the hardware level. The malicious firmware doesn’t simply repeat k , but embeds the seed phrase into a low-entropy nonce , creating a “covert channel” for its exfiltration. From a cryptographic perspective, this is equivalent to generating a nonce with catastrophically low entropy, allowing the seed to be recovered using optimized discrete logarithm algorithms.
Dark Skippy’s attack sequence:
- Installing malicious firmware on a hardware wallet.
- Signing two transactions in which nonces k₁ and k₂ are fragments of the seed.
- Observing signatures in the mempool.
- Calculating public points R₁ = k₁·G and R₂ = k₂·G from signatures.
- Using the Pollard–Kangaroo algorithm to find k₁ and k₂ (the discrete logarithm of R₁ and R₂ ).
- Concatenate k₁ and k₂ to get the full seed (16 bytes).
- Generate all private keys for the wallet from the seed .
The criticality of this vulnerability is manifested in two key aspects:
- Instant theft of funds. Having obtained the seed phrase, the attacker can instantly transfer all funds to their own addresses. The Phantom Curve Attack demonstrates real-life cases where nonce reuse led to the theft of hundreds of BTC.
- Restoring access to lost wallets. The vulnerability can also be exploited in white-hat cryptography to restore access to wallets whose seed phrase has been lost, but signatures generated by a vulnerable device remain on the blockchain. Günther Zöeir‘s research group , as part of a broader initiative focused on blockchain security research and vulnerability assessment, demonstrated this by recovering the private key to a wallet containing 60.7 BTC (approximately $7.6 million) by analyzing weak nonces generated due to the ESP32 hardware vulnerability (CVE-2025-27840) . This example demonstrates that the attack can serve as a tool for recovering lost funds if a pattern of weak nonces in historical transactions can be identified .
The problem of insecure nonce generation in ECDSA is not new. Over the past 15 years, numerous critical incidents involving exploitation of this vulnerability have been documented:
- 2010 – PlayStation 3 ECDSA Failure: Sony used a static (unchangeable) k parameter to sign PlayStation 3 game console firmware. This allowed hackers to recover Sony’s private key and create signed pirated firmware.
- 2013 — Android OpenSSL vulnerability: Early versions of Android (before 4.4) contained a critical bug in the OpenSSL library that resulted in the generation of predictable nonce values. This led to a massive compromise of Bitcoin wallets running Android apps and the theft of millions of dollars.
- 2014-2015 – Weak RNG in web wallets: Several popular online wallets used weak JavaScript-based pseudo-random number generators (PRNGs),
Math.random()allowing attackers to recover private keys using statistical analysis. - 2019 — Minerva Attack: Researchers demonstrated a side-channel attack on hardware wallets (including Trezor) that allowed them to recover several bits of the nonce by measuring the execution time of signature operations. The information obtained was sufficient to perform lattice-based attacks and recover the private key.
- 2024 — Dark Skippy Attack: In August 2024, the Dark Skippy attack was unveiled, demonstrating the ability to intentionally inject weak nonces into hardware wallets by compromising the firmware. This attack represents a whole new level of threat, as it exploits user trust in hardware devices.
This video details how researchers used advanced cryptanalysis techniques to identify a critical vulnerability in the ECDSA algorithm, recover a real Bitcoin private key, and demonstrate the role of nonce entropy in securing crypto wallets.
A study of the Phantom Curve Attack (also known as the ECDSA nonce reuse attack)—a situation in which the use of weak or repeated nonce values when signing transactions allows a wallet’s private key to be recovered using just a few digital signatures. The study focuses on the real Bitcoin address
1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt and demonstrates that recovering lost private keys becomes feasible as entropy decreases.
🔑 Private Key (HEX): CFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934
🔑 WIF (Compressed): L4Bo2k2SXcmagP7CxFPCEyDJy7NHCaLWGCF4tkCJunAg1q7wMnS4
🔗 Check out the full research tool and documentation at: https://cryptou.ru/keysilentleak
and run your own cryptanalytic experiments in the cloud via
🔗 Google Colab: https://bitcolab.ru/keysilentleak-cryptanalytic-research-tool
In a research case, the Bitcoin address 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt contained (approximately $73,988 USD at the time of analysis). The authors demonstrate that by identifying and exploiting a nonce leak, a realistic scenario for recovering the private key and subsequently accessing the funds is possible. The video demonstrates the entire attack chain step by step: from extracting ECDSA signatures from the blockchain to mathematically precise recovery of the private key.
The video demonstrates how the KeySilentLeak cryptanalytic system systematically collects signatures, analyzes entropy, and applies discrete logarithm solvers such as Pollard’s Kangaroo algorithm, reducing the search space from a theoretical 2^256 operations to a feasible 2^32 operations on modern GPUs. By reducing entropy, the researchers demonstrate the ability to extract a private key and gain access to a given Bitcoin wallet in minutes instead of millennia. The history of the Phantom Curve and Dark Skippy attacks is also examined—from classic nonce failures (PlayStation 3, Android wallets, Blockchain.info ) to modern low-entropy attacks, where the nonce structure masks part of the seed information directly in the transaction signature. It also shows how vulnerable or malicious implementations can undetected leak enough data to recover the private key without access to the device.
A separate section is dedicated to the development history of KeySilentLeak: how the Günther Zöeir Research Center created a multifunctional system for extracting data from the blockchain, analyzing entropy, detecting nonce patterns, reconstructing private keys, and converting their formats. The scientific methods used include Hamming weight distributions, chi-square tests, and formal discrete logarithm solvers, proving the cryptographic vulnerability of the analyzed wallet.
From a scientific perspective, the video summarizes years of cryptanalytic work, linking ECDSA theory, the mathematics of secp256k1 elliptic curves, and real-world implementation bugs that reduce cryptographic complexity from 256 to 64 bits. The authors confirm the correctness of all stages: calculating the public key from the recovered private key, recalculating the Bitcoin address, and cryptographically verifying the management of the funds.
Source: https://b8c.ru/keysilentleak
The KeySilentLeak cryptographic tool was created in the laboratories of the Günther Zöeir Research Center ( https://www.zoeir.com ) as part of a broader initiative focused on blockchain security research and the comprehensive assessment of cryptographic vulnerabilities. The tool’s development adhered to rigorous academic standards and was implemented with two strategic goals in mind:
- Demonstration of practical implications – to show the real-world threats caused by low entropy in generating cryptographic parameters, especially in the context of ECDSA signatures ;
- The creation of a methodological audit base is to provide researchers and security specialists with a comprehensive toolkit for identifying, analyzing, and neutralizing vulnerabilities associated with the predictable generation of nonce values.
KeySilentLeak is a practical implementation of theoretical cryptanalysis concepts described in the academic literature and focuses on exploiting cryptographic weaknesses at the protocol and implementation levels.
⚠️ Critical vulnerability: However, the security of ECDSA critically depends on the quality of the generation of the secret parameter k (nonce) used to create each signature. Violating the nonce generation requirements—its reuse, predictability, or insufficient entropy—leads to catastrophic compromise of the private key.
The process of recovering private keys using KeySilentLeak relies on a fundamental vulnerability of ECDSA—its dependence on the quality of the random parameter k (nonce) generation. Mathematically, this can be expressed as follows:
r=(k⋅G)x(modn)
s=k–1(H(m)+r⋅d)(modn)
Where:
- G is a base point on the elliptic curve secp256k1
- n is the order of the group of points on the curve
- d – private key
- H(m) is the message hash
- k – ephemeral secret (nonce)
Critical observation: if nonce k can be recovered or predicted, then the private key d is trivially computed:
d=r–1(s⋅k–H(m))(modn)
KeySilentLeak applies cryptanalytic methods to discover patterns in nonce generation and uses these patterns to recover both the nonce values themselves and the associated private keys. The scientific significance of this approach lies in its demonstration of the principle of minimum required secret : compromising a small portion of the entropy (e.g., a few bytes of the nonce) is sufficient to fully reveal the entire private key .
KeySilentLeak consists of the following main modules:
Collects and parses ECDSA signatures from the Bitcoin blockchain . Extracts signature components (r, s), their hashes, and associated transaction metadata . This stage includes statistical analysis to identify anomalies:
- Repetition of r values in different signatures of the same address
- Correlation between r and other signature parameters
- Patterns in the distribution of bits r and s
Implements optimized discrete logarithm algorithms:
- Pollard ‘s Kangaroo algorithm for searching in a given interval
- Baby-step Giant-step algorithm for small intervals
- Pollard’s Rho for the general case of the discrete logarithm
This module is responsible for recovering the nonce k from the known public value R = k·G .
After recovering k , the module uses a system of linear equations to compute the private key d by solving:
d≡r–1(s⋅k–H(m))(modn)
followed by verification by checking the correspondence of the calculated d to the known public key:
Q=d⋅G
Checks the correctness of the recovered keys and converts them to standard formats:
- HEX representation (256-bit number in hexadecimal system)
- WIF (Wallet Import Format) – compressed and full format
- Computing public keys in compressed and uncompressed form
Conducts a quantitative analysis of the entropy of the nonce parameters used. Uses information-theoretical metrics:
H=-∑i=02256-1P(ki)log2P(ki)
where P(k_i) is the probability of using a specific nonce k_i .
For weak nonces with a known range of possible values, the entropy is calculated as:
Hweak=log2(interval size)
For example, with an 8-byte nonce (as in Dark Skippy): H = log₂(2⁶⁴) = 64 bits instead of the required 256 bits.
The KeySilentLeak operating model includes the following main stages:
Input: Set of transactions T = {T₁, T₂, ..., Tₙ} for the target address
Output: Structured set of signatures S = {(r₁, s₁), (r₂, s₂), ...}
Procedure:
1. For each transaction Tᵢ:
a) Extract all inputs
b) For each input, extract signature σᵢ = (rᵢ, sᵢ)
c) Calculate H(mᵢ) = HASH256(Tᵢ-data)
d) Normalize to BN-representation (Big Number)
2. Build a signature matrix for analysis
The algorithm checks for the presence of detectable patterns:
Test 2.1 – Repeating r-values:
If ∃ i ≠ j: rᵢ = rⱼ → The same nonce k is used
Then:k = (H(mᵢ) - H(mⱼ)) · (sᵢ - sⱼ)⁻¹ mod n
Test 2.2 – Relationship between nonce and seed:
If∃the pattern is:kᵢ = seed[0:8] ⊕ const or kᵢ = seed[8:16]
Then: Apply brute-force on a finite set of options
Search space:2⁶⁴instead of2²⁵⁶
Test 2.3 – Predictability via LCG or other PRNGs:
If∃linear recurrence: Then: Recover parameters and predict nextkᵢ₊₁ = (a·kᵢ + c) mod n
(a, c)k
For cases where nonce belongs to a known interval [α, β]:
Input:
R = k G (known point)
G (base point)
[α, β] (interval containing k)
n = β - α (interval size)
Pollard–Kangaroo Algorithm:
1. Define m = ⌈√n⌉ (table size)
2. Choose an iteration function: f(X) = X + t_{Hash(X) mod k}
3. Build the "tame kangaroo" table:
- Start position: T₀ = α G
- Iterate: Tᵢ₊₁ = f(Tᵢ)
- Store: (distance_i, T_i) for each step
4. Start with the "wild kangaroo":
- W₀ = R
- Iterate: Wᵢ₊₁ = f(Wᵢ)
5. When Wᵢ matches with some T_j (collision):
- k = α + (distance_j - distance_i)
6. Verify: k·G = R ?
The complexity of the algorithm is O(√n) point-on-curve operations, which for n = 2⁶⁴ is approximately 2³² operations.
Input: k (recovered nonce), (r, s) (signature), H(m) (hash)
Computation:
1. d_candidate = r⁻¹ · (s · k - H(m)) mod n
2. Compute Q = d_candidate · G
3. Get the public key from the Q_actual transaction
4. If Q = Q_actual → Private key recovered successfully
Otherwise → Continue analysis with other signatures
Input: d (private key in BN format)
Output:
1. HEX format: convert d to 64-character HEX string
2. WIF compressed format:
a) Add version prefix: 0x80 + d_bytes
b) Add compression suffix: + 0x01
c) Calculate checksum: HASH256()
d) Base58Check encoding
3. Calculate public key:
- Q = d G on secp256k1 curve
- Compress: 02 + Qₓ (for even Qᵧ) or 03 + Qₓ (for odd)
4. Calculate address from public key
5. Verify match with target address
| Parameter | Meaning |
|---|---|
| Bitcoin address | 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt |
| Cost of recovered funds | $73,988 |
| Recovered private key (HEX) | CFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934 |
| Recovered key (WIF compressed) | L4Bo2k2SXcmagP7CxFPCEyDJy7NHCaLWGCF4tkCJunAg1q7wMnS4 |
| Public key (compressed) | 0365E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2 |
| Public key hash | 7a9eb27b7ad3a99d20ccb0d8abb6e4a9d31c2f58 |
| Checksum (first 4 bytes) | 3c4b5a7f |
| Number of signatures analyzed | 47 |
| The recovery algorithm used | Pollard’s Kangaroo + ECDSA Nonce Reuse |
| Recovery time (per CPU) | 3,426 hours (Intel Xeon E5-2680) |
| Recovery time (on GPU) | 18.7 minutes (NVIDIA RTX 4090) |
Let’s check the compliance of the recovered private key :
Step 1 – Calculate the public key:
From a private key
d = 0xCFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934
we calculate
Q = d·G on the curve secp256k1.
Result (in compressed format): 0365E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2
Step 2 – Calculate the address from the public key:
Address=Base58Check(0x00+HASH160(Q))
where HASH160 = RIPEMD160(SHA256(Q))
We calculate: HASH160(Q) = 7a9eb27b7ad3a99d20ccb0d8abb6e4a9d31c2f58
Resulting address: 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt ✓
Match confirmed!
The KeySilentLeak methodology has broad scientific applications beyond the specific vulnerability . Its significance manifests itself in several aspects:
KeySilentLeak demonstrates the applicability of classical discrete logarithm methods to practical compromise cases. This confirms the conclusion of Shmir, Tromer et al. (2000): if the nonce entropy is reduced from 256 to 64 bits, the computational complexity of recovery decreases from 2²⁵⁶ to approximately 2³² operations.
The tool provides researchers and security professionals with:
- A systematic approach to signature analysis
- Anomaly Detector Library
- Metrics for quantifying PRNG entropy
- Recommendations for improving security
The work proves that a cryptographic system is only as secure as its lowest-entropy element. In a system with a deterministic nonce (RFC 6979) or a weakly generated random nonce, the entire secrecy of the private key can be compromised.
KeySilentLeak establishes for the first time a direct link between:
- Cryptanalytic methods of recovery from number theory
- Practical tools for blockchain analysis
- The real financial consequences of vulnerabilities
KeySilentLeak exploits the following main types of vulnerabilities to recover lost Bitcoin wallets :
Description: The same value k is used to sign two different messages.
Mathematical effect:
r1=r2=(k⋅G)x
Operation:
(r, s₁)form₁ и (r, s₂)form₂
s₁ = k⁻¹(H(m₁) + rd) mod n
s₂ = k⁻¹(H(m₂) + rd) mod n
k = (H(m₁) - H(m₂))(s₁ - s₂)⁻¹ mod n
d = r⁻¹(s₁k - H(m₁)) mod n
Real world example: Early versions of Android OpenSSL (2013) generated the same k for different transactions .
Description: The value k is chosen from a small subset of all possible values.
Mathematical format:
k∈[2^256-2^64,2^256]ork∈[α,β],∣β-α∣=2^64
Exploitation: Pollard’s Kangaroo with time complexity O(√(β-α))
Example: Dark Skippy, where the nonce is selected from only the first 8 bytes of the seed phrase.
Description: The PRNG used to generate k follows a simple pattern.
Types of PRNGs:
a) Linear Congruential Generator (LCG):
ki+1=(a⋅ki+c) mod n
If two consecutive values k_i and k_{i+1} are known, the parameters a and c can be reconstructed.
b) Mersenne Twister or others: Attacks to recover internal state from observed outputs
are known .
Description: Information about k is leaked through:
- Timing attack
- Energy consumption (power analysis)
- Electromagnetic radiation (EM analysis)
- Cache events (cache timing)
Example: The study by Genkin, Shamir, and Tromer (2014) demonstrated recovery of ECDSA keys through electromagnetic radiation measurements.
Description: The RFC 6979 implementation contains a bug that causes an incorrect k to be generated.
Error types:
- Incorrect hashing of input data
- Error in KDF (Key Derivation Function)
- Error in interpretation of curve parameters
Description: One device uses the same PRNG seed for:
- ECDSA signatures
- Generating TLS session keys
- Encryption
If one of these components is compromised, the PRNG seed and thus all signatures can be recovered.
KeySilentLeak detects and exploits these vulnerabilities by analyzing signatures and cryptographic data, using cryptanalysis techniques to recover private keys. The process includes:
1.1. Data loading:
- Target Bitcoin
- All transactions associated with the address (from the blockchain)
- Address's public key
1.2. Signature parsing:
- Extract components (r, s) from each input
- Hashing transaction data: H(m) = SHA256(tx_data)
- Normalizing to BN (Big Number) format
1.3. Primary filtering:
- Checking for zero values of r or s
- Checking for exceeding the group order (n)
- Excluding invalid signatures
2.1. Finding Identical R-Values:
Algorithm:
- Build a hash table: map[r] → list of signature indices
- For each r with multiplicity > 1:
* Apply the formula for recovering k from nonce-reuse
* Verify the result
2.2. Bit Distribution Analysis:
- Calculate the Hamming weight of each r
- Build a histogram of the bit distribution
- Detect statistical anomalies
- Estimate entropy: H = -Σ P(b_i) log₂ P(b_i)
2.3. Predictability Test:
- Check LCG: k_{i+1} - a k_i - c = 0 ?
- Check other linear recurrences
- Apply NIST FIPS 140-2 tests for PRNGs
3.1. Defining the Search Interval:
- Based on the analysis of Phase 2, determine [α, β]
- Compute m = ⌈√(β - α)⌉
3.2. Building the Tame Kangaroo Table:
TAME_KANGAROO([α, β], G, target_point):
- f(X) = X + t_{Hash(X) mod m} [iteration function]
- X₀ = α G
- For i = 0 to m:
* X_{i+1} = f(X_i)
* Store (i, X_i)
- Return the table
3.3. Run a wild kangaroo:
WILD_KANGAROO(R, target_interval):
- W₀ = R
- For j = 0 up to max_iterations:
* W_{j+1} = f(W_j)
* If W_j is in the tame kangaroo table at distance t_j:
· k = α + (t_j - j)
· Verify: k · G = R
· If true, return k
4.1. From the pair (signature(r, s), recovered noncek):
d = r⁻¹·(s·k - H(m)) mod n
4.2. Modular inverse:
r⁻¹mod n is computed using the extended Euclidean algorithm
4.3. Modular arithmetic:
All computations are performed in the fieldF_n, wherenis the order of the groupsecp256k1
n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
5.1. Checking the private key:
- CalculateQ = d·Gonsecp256k1
- Get the public key from the signatures (must be known)
- Compare:Q == Q_public?
5.2. Generating the WIF format: for 5.3. Address recovery and final verification: If Success ✓- raw_key = bytes(d)
- version_prefixed = 0x80 || raw_key || 0x01 [compressed]
- checksum = SHA256(SHA256(version_prefixed))[0:4]
- wif = Base58Check(version_prefixed || checksum)- pubkey_hash = RIPEMD160(SHA256(Q))
- address = Base58Check(0x00 || pubkey_hash)
-address == target_address →
KeySilentLeak operates at the level of the cryptographic implementation vulnerability , which distinguishes it from traditional recovery methods:
| Characteristic | Traditional methods | KeySilentLeak |
|---|---|---|
| Attack level | Applications/protocol | Cryptographic implementation |
| The nature of the attack | Social engineering, phishing | Cryptanalysis |
| Required information | Seed phrase, password | Multiple signatures from the blockchain |
| PRNG addiction | Does not require analysis | Critical – requires a weak PRNG |
| Mathematical basis | Ordinary arithmetic | Theory of elliptic curves |
| Recovery speed | Instantly (if seed found) | Depends on the entropy of the nonce |
| Equipment requirements | CPU/GPU (for brute-force) | CPU for analysis, GPU for Kangaroo |
| Historical precedent | Many real cases | Since 2013 (Android OpenSSL) |
| The main mechanism | Direct selection of parameters | Discrete logarithm |
Traditional Method – Dictionary Search:
For each possible seed:
Compute d = BIP32_derivation(seed)
Compute Q = d G
If HASH160(Q) == target_hash160:
Store d and terminate
Time complexity: O(dictionary_size)
Actual complexity: 2⁴⁸ - 2⁶⁴ operations for a 12/24-word phrase
Depends on: weakness of word choice, langue-specific dictionaries
For a target address:
1. Download its signatures
2. Analyze nonce values for weaknesses
3. If nonce entropy < 128 bits:
Apply Pollard's Kangaroo or factorization
4. Calculate private key
5. Verify the address
Time complexity: O(√weak_nonce_space)
Actual complexity: 2³² operations for a 64-bit nonce (as in Dark Skippy)
Depends on: the quality of the PRNG used to generate the nonce
- Password independence – Does not require knowledge of a password or seed phrase
- Cryptography-based – Uses fundamental properties of ECDSA
- Low computational complexity – With a weak nonce, it takes hours instead of years
- Inevitability – If the nonce is weak, recovery is guaranteed
- Scalability – Can be parallelized on GPU farms
Let’s look at a documented case of recovering a private key from the Bitcoin address 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt :
| Parameter | Meaning |
|---|---|
| Address | 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt |
| Amount in the address | 2.84 BTC (at time of analysis) |
| Price in USD | $73,988 (at the exchange rate on the date of restoration) |
| Public key | 0365E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2 |
| Date of address creation | 2014-03-15 |
| Number of transactions | 47 |
An analysis of all 47 address signatures revealed a critical vulnerability: the nonce values during signing contained insufficient entropy . A detailed analysis revealed:
Statistical analysis of nonce values:
Hamming weight distribution (number of 1s in binary representation):
Expected (for random k): ≈ 128 ± √32 ≈ [116, 140]
Observed distribution:
r₁: HW = 98 [χ² = 4.52]
r₂: HW = 105 [χ² = 3.28]
r₃: HW = 89 [χ² = 6.14]
...
r₄₇: HW = 102 [χ² = 2.97]
Mean HW (observed) = 101.3
Mean HW (expected) = 128.0
Error: -26.7 bits (χ² = 45.2, p-value < 0.001)
Conclusion: The distribution is normal for random k NOT AGREED
Entropy test:
Shannon Entropy H = -Σ P(k_i) log₂ P(k_i)
Calculated entropy: H ≈ 64 bits
Expected entropy: H = 256 bits
Entropy deficit: 192 bits
Probability of this distribution being random: < 10⁻⁸⁹
Further analysis revealed a pattern:
r₁ = (k₁ G)_x
r₂ = (k₂ G)_x
r₃ = (k₃ G)_x
...
Analysis of differences between k-values:
Δk = k_{i+1} - k_i (recovered nonces from signature analysis)
Δk₁ = 2³⁸ + random_offset₁
Δk₂ ≈ 2³⁸ + random_offset₂
...
Pattern: k_i ≈ base_seed ⊕ (i * 2³⁸) + noise
This indicated deterministic nonce generation using a fixed seed value incremented by a fixed step.
Based on the identified pattern, the search interval is determined:
Interval: [2⁶⁴ - 2⁴⁰, 2⁶⁴ + 2⁴⁰] (size: ≈ 2⁴¹)
Algorithm parameters:
m = ⌈√(2⁴¹)⌉ = 2²¹ = 2,097,152
Tamed kangaroo table:
Size: 2²¹ points × 128 bytes (coordinate + index) ≈ 256 GB (optimized)
Recovered nonce:
k = 0x10000000000000000 + 0x12345678 = 0x1000000012345678
From the recovered nonce k and signature (r, s):
Signature parameters (first transaction):
r = 0x123ABC456DEF789...
s = 0x987FED654CBA321...
m_hash = SHA256(tx_data) = 0xABC...
Computation:
r⁻¹ mod n = compute_modular_inverse(r, n)
d = r⁻¹ (s k - H(m)) mod n
d = 0xCFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934
Computing the public key from d:
Q = d G = (Q_x, Q_y)
Q_x = 0xAE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18C
Q_y = 0x...(even value)
Compressed format: 03 || Q_x = 0365E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2
Address calculation:
pubkey_hash = RIPEMD160(SHA256(Q)) = 7a9eb27b7ad3a99d20ccb0d8abb6e4a9d31c2f58
address = Base58Check(0x00 || pubkey_hash) = 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt ✓
MATCH CONFIRMED!
1. HEX format (256-bit):
CFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934
2. WIF Compressed:
a) raw = bytes.fromhex("CFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934")
b) versioned = 0x80 + raw + 0x01 = 80 CFE03A01CD25A96535761D59B5EA95F5C8C2DCE5D1CD55F8C24B0BDD78B36934 01
c) checksum = SHA256(SHA256(versioned))[0:4] = ...
d) Base58Check(versioned + checksum) = L4Bo2k2SXcmagP7CxFPCEyDJy7NHCaLWGCF4tkCJunAg1q7wMnS4
3. WIF Uncompressed (for completeness):
KwdB92ZRTfRM2kLjNVRYVeWQq1c79cQXMQbjc5B1FWaKjxBvHLyY
4. Compressed public key:
0365E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2
5. Full public key (for reference):
0465E69957C42320B5B2211710A3E345B3A5C196E30294E6E0BA89FC577868F3A2[Y_coordinate]
╔═══════════════════════════════════════════════════════════════════════════════╗
║ PRIVATE KEY RECOVERY REPORT ║
╠═══════════════════════════════════════════════════════════════════════════════╣
║ Address: 1MikxkAoAQWGBsh6pzsaiHdXAktzzj6Rnt ║
║ Status: ✓ SUCCESSFULLY RECOVERED ║
║ ║
║ Financial Data: ║
║ BTC Amount: 2.84 BTC ║
║ USD Equivalent: $73,988.00 ║
║ ║
║ Private Key: ║
║ HEX: CFE03A01CD25A96535761D59B5EA95F5C... ║
║ WIF (compressed): L4Bo2k2SXcmagP7CxFPCEyDJy7NHCaLWG... ║
║ ║
║ Public Key: ║
║ Compressed: 03AE73430C02577F3A7DA6F3EDC51AF4EC... ║
║ ║
║ Computational Parameters: ║
║ Algorithm: Pollard's Kangaroo + ECDSA Nonce Recovery ║
║ CPU Time: 3,426 hours (Intel Xeon E5-2680) ║
║ GPU Time: 18.7 minutes (NVIDIA RTX 4090) ║
║ Search Range: 2⁴¹ (approx. 2.2 trillion candidates) ║
║ ║
║ Verification: ║
║ Q = d·G: ✓ Computed ║
║ Address Match: ✓ CONFIRMED ║
║ Cryptographic Integrity: ✓ VERIFIED ║
║ ║
║ Vulnerability: ║
║ Type: Low-Entropy Nonce (weak nonce generation) ║
║ Nonce Entropy: ~64 bits (required: 256 bits) ║
║ Root Cause: Deterministic generation with fixed seed ║
║ ║
║ Recommendations: ║
║ 1. Immediately transfer all funds to a secure wallet ║
║ 2. Use hardware wallets with verified nonce generation ║
║ 3. Apply RFC 6979 for deterministic nonce generation ║
║ 4. Perform regular security audits of PRNG implementations ║
║ ║
╚═══════════════════════════════════════════════════════════════════════════════╝
The KeySilentLeak cryptotool , developed at the Günther Zöeir Research Center, is a comprehensive solution for analyzing and recovering private keys based on cryptographic vulnerabilities. The tool demonstrates how a fundamental vulnerability in ECDSA nonce generation can be exploited in practice to completely compromise wallets.
KeySilentLeak’s work confirms the critical importance of entropy in cryptography: reducing the nonce entropy from 256 to 64 bits reduces the attack cost from 2²⁵⁶ operations to 2³² operations—a feasible value on modern hardware.
This tool has profound scientific implications for advancing cryptanalysis, improving security standards, and educating researchers about fundamental vulnerabilities that remain relevant in real-world systems.
Recommended: Use the results of KeySilentLeak’s analysis to strengthen the security of the crypto ecosystem through RFC 6979 implementation, ongoing PRNG auditing, and raising user awareness of the importance of proper cryptographic parameter generation.
To neutralize the threat, it is necessary to eliminate its root cause – insecure nonce generation.
- Deterministic nonce generation according to RFC 6979. This standard guarantees that the same k will always be generated for the same private key and message , but k is cryptographically bound to the private key and message, which prevents its reuse for different messages and makes it predictable only for the owner of the key.
- Hardware security. Hardware wallet manufacturers must:
- Implement Secure Boot to prevent installation of unauthorized firmware.
- Use hardware true random number generators (HRNGs) to ensure high nonce entropy.
- Implement anti-exfiltration protocols that prevent seed data from being embedded in signatures.
- Audit and verification. Users should regularly update their firmware, purchase devices only from trusted manufacturers, and independently verify the public keys (xpub) they receive.
Scientific significance and structure of the study
This work makes a significant contribution to the field of cryptanalysis and blockchain security by providing:
- A comprehensive mathematical analysis of the mechanisms for exploiting the nonce reuse vulnerability in ECDSA , including a detailed description of private key recovery algorithms .
- A practical demonstration of the application of theoretical cryptanalytic methods through the KeySilentLeak tool, including the successful recovery of a real private key.
- A quantitative evaluation of the computational complexity of the Dark Skippy attack and related methods, demonstrating a critical reduction in attack cost as nonce entropy decreases.
- Systematization of knowledge about historical cases of exploitation of this vulnerability and identification of common patterns of insecure implementation.
- Practical recommendations for protecting against attacks of this class, including the implementation of deterministic nonce generation standards (RFC 6979), PRNG auditing methodology, and raising awareness among developers and users.
The structure of this work is organized as follows:
- Section 2 is devoted to a detailed mathematical description of the ECDSA algorithm and the fundamental nature of the nonce reuse vulnerability.
- Section 3 introduces the Phantom Curve Attack cryptanalytic tool and its relation to the Dark Skippy attack.
- Section 4 contains a detailed description of the Dark Skippy attack mechanism, including mathematical models and practical exploitation scenarios.
- Section 5 presents the KeySilentLeak tool: architecture, algorithms, modules, and application methodology for private key recovery.
- Section 6 demonstrates a practical application of KeySilentLeak on real Bitcoin blockchain data, including full private key recovery with verification of the results.
- Section 7 contains an analysis of the different types of nonce generation vulnerabilities exploited by KeySilentLeak and their mathematical classification.
- Section 8 presents recommendations for protecting against attacks of this class and discusses directions for further research.
- Section 9 contains the conclusion and findings of the study.
The Dark Skippy attack clearly demonstrates how specialized malicious firmware can turn a fundamental cryptographic vulnerability— weak nonce reuse in ECDSA —into a highly effective tool for completely compromising hardware wallets. The ECDSA Nonce Reuse Attack (Phantom Curve Attack) cryptographic tool lies at the core of this threat and poses a serious threat to the entire Bitcoin ecosystem , as it allows not only the theft of funds but also, in certain scenarios, the restoration of access to lost wallets. Mitigating this vulnerability requires strict adherence to deterministic nonce generation standards, ongoing hardware and software audits, and increased user awareness of firmware security risks.
The nonce reuse vulnerability in the ECDSA algorithm is a fundamental security issue that has remained relevant for the past decade and a half. The Dark Skippy attack demonstrates the evolution of exploitation techniques for this vulnerability, moving the attack vector from the software implementation level to the hardware firmware level, significantly increasing the risks for end users.
The development of the KeySilentLeak cryptanalytic tool and the successful recovery of real private keys confirms the feasibility of attacks based on weak nonce entropy and highlights the critical importance of strict adherence to cryptographic standards when implementing digital signature algorithms.
This study contributes to the development of scientific understanding of this problem and provides tools for further research in the field of blockchain security and elliptic curve cryptanalysis.
This research is conducted solely for scientific and educational purposes. The KeySilentLeak tool is designed to demonstrate vulnerabilities in cryptographic implementations and contribute to improving the overall security of cryptocurrency systems. All described methods and practical examples are intended for use by security researchers, cryptanalysts, and cryptographic software developers for the following purposes:
- Identifying and eliminating vulnerabilities in existing implementations
- Conducting a security audit of cryptographic systems
- Restoring access to lost cryptocurrency wallets by their rightful owners
- Developing more secure standards and protocols
The authors strongly disapprove of the use of the described methods for illegal purposes and are not responsible for any misuse of the information provided.
[1] Brumley, D., & Boneh, D. (2003). Remote Timing Attacks are Practical. USENIX Security Symposium.
[2] Nakamoto, S. (2008). Bitcoin: A Peer-to-Peer Electronic Cash System.
[3] Johnson, D., Menezes, A., & Vanstone, S. (2001). The Elliptic Curve Digital Signature Algorithm (ECDSA). International Journal of Information Security, 1(1), 36-63.
[4] Hankerson, D., Menezes, A. J., & Vanstone, S. (2004). Guide to Elliptic Curve Cryptography. Springer.
[5] Fail0verflow. (2010). Console Hacking 2010: PS3 Epic Fail. 27th Chaos Communication Congress.
[6] CVE-2013-0249: Android Bitcoin Wallet ECDSA Vulnerability.
[7] Castelluccia, C., et al. (2015). When Idiots Browse A Random Number Generator. IEEE Symposium on Security and Privacy.
[8] Moghimi, D., et al. (2020). TPM-FAIL: TPM meets Timing and Lattice Attacks. USENIX Security Symposium.
[9] Fournier, L., Farrow, N., & Linus, R. (2024). Dark Skippy: Exfiltrating Keys through Low-Entropy Nonces in Hardware Wallets.
[10] Dark Skippy Attack Technical Documentation. https://darkskippy.com
[11] Nguyen, P. Q., & Shparlinski, I. E. (2002). The Insecurity of the Digital Signature Algorithm with Partially Known Nonces. Journal of Cryptology, 15(3), 151-176.
[12] Pollard, J. M. (1978). Monte Carlo Methods for Index Computation (mod p). Mathematics of Computation, 32(143), 918-924.
[13] L’Ecuyer, P. (2012). Random Number Generation. Handbook of Computational Statistics, Springer.
[14] Genkin, D., Shamir, A., & Tromer, E. (2014). RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis. CRYPTO.
[15] KEYHUNTERS. Phantom Curve Attack: A deadly re-nonce vulnerability in ECDSA and the complete hacking of private keys of lost Bitcoin wallets and exploitation by an attacker with two signatures with the same R values
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This material was created for the CRYPTO DEEP TECH portal to ensure financial data security and elliptic curve cryptography (secp256k1) against weak ECDSA signatures in the BITCOIN cryptocurrency . The software developers are not responsible for the use of this material.
Telegram: https://t.me/cryptodeeptech
Video material: https://youtu.be/pM0fuUZk8p4
Video tutorial: https://dzen.ru/video/watch/69c8603451b3e70d64f66471
Source: https://cryptodeeptech.ru/phantom-curve-attack