Elliptic Curve Cryptography in Practice
Joppe W. Bos1 , J. Alex Halderman2 , Nadia Heninger3 , Jonathan Moore,
Michael Naehrig1 , and Eric Wustrow2
1

Microsoft Research
University of Michigan
3
University of Pennsylvania
2

Abstract. In this paper we perform a review of elliptic curve cryptography (ECC) as it is used in practice today in order to reveal unique mistakes and vulnerabilities that arise in implementations of ECC. We study
four popular protocols that make use of this type of public-key cryptography: Bitcoin, secure shell (SSH), transport layer security (TLS), and
the Austrian e-ID card. We are pleased to observe that about 1 in 10 systems support ECC across the TLS and SSH protocols. However, we findthat despite the high stakes of money, access and resources protected by
ECC, implementations suffer from vulnerabilities similar to those that
plague previous cryptographic systems.

1

Introduction

Elliptic curve cryptography (ECC) [32, 37] is increasingly used in practice toinstantiate public-key cryptography protocols, for example implementing digital
signatures and key agreement. More than 25 years after their introduction to
cryptography, the practical benefits of using elliptic curves are well-understood:
they offer smaller key sizes [34] and more efficient implementations [6] at thesame security level as other widely deployed schemes such as RSA [44]. In thispaper, we provide two contributions:• First, we study the current state of existing elliptic curve deployments inseveral different applications. Certicom released the first document providing
standards for elliptic curve cryptography in 2000, and NIST standardized
ECDSA in 2006. What does the deployment of these algorithms look like in2013? In order to study this question, we collect cryptographic data from anumber of different real-world deployments of elliptic curve cryptography:
Bitcoin [38], secure shell (SSH) [47], transport layer security (TLS) [9], and
the Austrian Citizen Card [29].
• Next, we perform a number of “sanity checks” on the data we collected, inparticular on the public keys, key exchange data, and digital signatures, inorder to detect implementation problems that might signal the presence of
cryptographic vulnerabilities.
The security of deployed asymmetric cryptographic schemes relies on the believed hardness of number theoretic problems such as integer factorization andthe computation of discrete logarithms in finite fields or in groups of points
on an elliptic curve. However, most real-world cryptographic vulnerabilities do
not stem from a weakness in the underlying hardness assumption, but rather
from implementation issues such as side-channel attacks, software bugs or design flaws (cf. [26]). One such example are so-called cache attacks [40] (see [13]for an application to the asymmetric setting) that exploit the memory accesspattern in cryptographic schemes using data dependent table lookups. Anotherclass of problems is related to implementations which do not provide sufficient
randomness and subsequently generate insecure cryptographic keys. Recent examples of implementations suffering from a lack of randomness are the Debian
OpenSSL vulnerability [51], the discovery of widespread weak RSA and DSA
keys used for TLS, SSH, and PGP as documented in [33, 28] and recent resultsin [4] that show how to break a number of RSA keys obtained from Taiwan’s
national Citizen Digital Certificate database.
In order to survey the implementation landscape for elliptic curve cryptography, we collected several large cryptographic datasets:• The first (and largest) dataset is obtained from the Bitcoin block chain. Bitcoin is an electronic crypto-currency, and elliptic curve cryptography is central to its operation: Bitcoin addresses are directly derived from elliptic-curve
public keys, and transactions are authenticated using digital signatures. The
public keys and signatures are published as part of the publicly available
and auditable block chain to prevent double-spending.
• The second largest dataset we collected is drawn from an Internet-wide
scan of HTTPS servers. Elliptic-curve cipher suites that offer forward secrecy by establishing a session key using elliptic-curve Diffie-Hellman key
exchange [19] were introduced in 2006 and are growing in popularity for TLS.
This dataset includes the Diffie-Hellman server key exchange messages, aswell as public keys and signatures from servers using ECDSA.
• We also performed an Internet-wide scan of SSH servers. Elliptic-curve cipher
suites for SSH were introduced in 2009, and are also growing more common
as software support increases. This dataset includes elliptic curve DiffieHellman server key exchange messages, elliptic-curve public host keys, and
ECDSA signatures.
• Finally, we collected certificate information, including public keys from thepublicly available lightweight directory access protocol (LDAP) databasefor the Austrian Citizen Card. The Austrian e-ID contains public keys forencryption and digital signatures, and as of 2009, ECDSA signatures areoffered.
Our main results can be categorized as follows.
Deployment. Elliptic curve cryptography is far from being supported as astandard option in most cryptographic deployments. Despite three NIST curves
having been standardized at the 128-bit security level or higher, the smallest
curve size, secp256r1, is by far the most commonly used. Many servers seem toprefer the curves defined over smaller fields.

Weak keys. We observed significant numbers of non-related users sharing public(and hence private) keys in the wild in both TLS and SSH. Some of these cases
were due to virtual machine deployments that apparently duplicated keys acrossdistinct instances; others we were able to attribute to default or low-entropy
keys generated by embedded devices, such as a network firewall product.
Vulnerable signatures. ECDSA, like DSA, has the property that poor randomness used during signature generation can compromise the long-term signing
key. We found several cases of poor signature randomness used in Bitcoin, which
can allow (and has allowed) attackers to steal money from these clients. Thereappear to be diverse causes for the poor randomness, including test values foruncommonly used implementations, and most prominently an Android Java bugthat was discovered earlier this year (see [35] for a discussion of this bug in Android and related Java implementations).

2

Preliminaries

This section briefly discusses the standardized elliptic curves that are mainly
used in practice. It also fixes notation for elliptic curve public-key pairs andintroduces the basic concepts for key establishment and digital signatures in theelliptic curve setting.
Elliptic Curves Used in Practice. First, we briefly recap standardized elliptic
curves that are used most commonly in real-world applications. All these curves
are given in their short Weierstrass form E : y 2 = x3 + ax + b and are definedover a finite field Fp , where p > 3 is prime and a, b ∈ Fp . Given such a curve
E, the cryptographic group that is employed in protocols is a large prime-ordersubgroup of the group E(Fp ) of Fp -rational points on E. The group of rational
points consists of all solutions (x, y) ∈ F2p to the curve equation together witha point at infinity, the neutral element. The number of Fp -rational points isdenoted by #E(Fp ) and the prime order of the subgroup by n. A fixed generatorof the cyclic subgroup is usually called the base point and denoted by G ∈ E(Fp ).
In the FIPS 186-4 standard [50], NIST recommends five elliptic curves for use
in the elliptic curve digital signature algorithm targeting five different security
levels. Each curve is defined over a prime field defined by a generalized Mersenne
prime. Such primes allow fast reduction based on the work by Solinas [46]. All
curves have the same coefficient a = −3, supposedly chosen for efficiency reasons, and their group orders are all prime, meaning that n = #E(Fp ). The fiverecommended primes are
p192 = 2192 − 264 − 1,
p224 = 2224 − 296 + 1,
256
224
192
96
p256 = 2 − 2 + 2 + 2 − 1, p384 = 2384 − 2128 − 296 + 232 − 1,
p521 = 2521 − 1.
In the standard, these curves are named P-192. P-224, P-256, P-384, and P-521,
but in practice they also appear as nistp192, nistp224 etc. These along withother curves are also recommended by Certicom in the standards for efficient
cryptography SEC2 [15], in which the curves are named secp192r1, secp224r1,

secp256r1, secp384r1, secp521r1. But sometimes, other names are used, forexample P-192 and P-256 are named prime192v1 and prime256v1 in OpenSSL.
For 256-bit primes, in addition to the NIST curve defined over Fp256 , SEC2
also proposes a curve named secp256k1 defined over Fp where p = 2256 − 232 −
977. This curve is used in Bitcoin. It has a 256-bit prime order. Interestingly, thischoice deviates from those made in FIPS 186-4 in that the curve coefficients are
a = 0 and b = 7. This means that secp256k1 has j-invariant 0 and thus possesses
a very special structure. A curve with j-invariant 0 has efficiently computableendomorphisms that can be used to speed up implementations, for example using
the GLV decomposition for scalar multiplication [25]. Since for secp256k1 p ≡ 1(mod 6), there exists a primitive 6th root of unity ζ ∈ Fp and a corresponding
curve automorphism ψ : E → E, (x, y) 7→ (ζx, −y). This map allows the fastcomputation of certain multiples of any point P ∈ E(Fp ), namely ψ(P ) = λP
for an integer λ with λ6 ≡ 1 (mod n). But efficient endomorphisms not only
speed up scalar multiplication, they also speed up Pollard’s rho algorithm [41]for computing discrete logarithms [23]. The automorphism group of E has order
6 and is generated by the map ψ above. In contrast, an elliptic curve with jinvariant different from 0 and 1728 only has an automorphism group of order 2,such√ that the speed-up in Pollard’s rho algorithm is a constant factor of up to
2 over such a curve.
Another consequence of the larger automorphism group is the existence ofsix twists (including the curve itself and the standard quadratic twist). An implementation using x-coordinate only arithmetic (such as the formulas in [11])must pay attention to the curve’s twist security (see [2, 3]). This means that its
quadratic twist needs to have a large enough prime divisor for the discrete logarithm problem on the twist to be hard enough. This prevents an invalid-curve
attack in which an attacker obtains multiples with secret scalars of a point on thequadratic twist, e.g. via fault injection [24]. The quadratic twist of secp256k1
has a 220-bit prime factor and thus can be considered twist secure (e.g. as in[5]). A non-laddering implementation (using both x- and y-coordinates) can becompromised by an invalid-curve attack if the implementation does not check
whether the point satisfies the correct curve equation [7]. This could lead to amore serious attack on secp256k14 since an attacker might obtain scalar multiples with secret scalars of a point on any curve over Fp with coefficient a = 0,i.e. on any of secp256k1’s twists. The largest prime divisors of the remaining
four twists’ group orders are of size 133, 188, 135, and 161 bits, respectively,
but there are several other smaller prime factors that offer more choices for aninvalid-curve attack.
Elliptic Curve Public-Key Pairs. Given a set of domain parameters that
include a choice of base field prime p, an elliptic curve E/Fp , and a base point
G of order n on E, an elliptic curve key pair (d, Q) consists of a private key d,4This invalid curve attack on secp256k1 using fault injection has been mentioned
before, for example by Paulo S.L.M. Barreto (@pbarreto): ”In other words: given 13
faults and a good PC, one can break secp256k1 (and Bitcoin) in 1 minute.”, October
21, 2013, 10:20 PM, Tweet.

which is a randomly selected non-zero integer modulo the group order n, and apublic key Q = dG, the d-multiple of the base point G. Thus the point Q is arandomly selected point in the group generated by G.
Elliptic Curve Key Exchange. There are several different standardized key
exchange protocols (see [48, 16]) extending the basic elliptic curve Diffie-Hellman
protocol, which works as follows. To agree on a shared key, Alice and Bob individually generate key pairs (da , Qa ) and (db , Qb ). They then exchange the publickeys Qa and Qb , such that each can compute the point P = da Qb = db Qa using
their respective private keys. The shared secret key is derived from P by a key
derivation function, generally being applied to its x-coordinate.
Elliptic Curve Digital Signatures. The Elliptic Curve Digital Signature Algorithm (ECDSA) was standardized in FIPS 186-4 [50]. The signer generates a
key pair (d, Q) consisting of a private signing key d and a public verification key
Q = dG. To sign a message m, the signer first chooses a per-message random
integer k such that 1 ≤ k ≤ n − 1, computes the point (x1 , y1 ) = kG, transforms
x1 to an integer and computes r = x1 mod n. The message m is hashed to a
bitstring of length no more than the bit length of n, which is then transformed
to an integer e. The signature of m is the pair (r, s) of integers modulo n, wheres = k −1 (e + dr) mod n. Note that r and s need to be different from 0, and k
must not be revealed and must be a per-message secret, which means that it
must not be used for more than one message.
It is important that the per-message secret k is not revealed, since otherwise
the secret signing key d can be computed by d ≡ r−1 (ks − e) (mod n) becauser and s are given in the signature and e can be computed from the signed
message. Even if only several consecutive bits of the per-message secrets for acertain number of signatures are known, it is possible to compute the private key(see [30]). Also, if the same value for k is used to sign two different messages m1
and m2 using the same signing key d and producing signatures (r, s1 ) and (r, s2 ),then k can be easily computed as k ≡ (s2 − s1 )−1 (e1 − e2 ) (mod n), which thenallows recovery of the secret key.
One solution to prevent the generation of predictable or repeated nonces is togenerate the nonce deterministically from the private key and the message [42].

3Applications of Elliptic Curves

In this section, we survey deployments of elliptic curve cryptography in the realworld and provide statistics on usage.
Bitcoin. The cryptocurrency Bitcoin is a distributed peer-to-peer digital currency which allows “online payments to be sent directly from one party to another without going through a financial institution” [38]. The (public) Bitcoin
block chain is a journal of all the transactions ever executed. Each block inthis journal contains the SHA-256 [49] hash of the previous block, hereby chaining the blocks together starting from the so-called genesis block. In Bitcoin, an
ECDSA private key typically serves as a user’s account. Transferring ownershipof bitcoins from user A to user B is realized by attaching a digital signature(using user A’s private key) of the hash of the previous transaction and information about the public key of user B at the end of a new transaction. The
signature can be verified with the help of user A’s public key from the previoustransaction. Other issues, such as avoiding double-spending, are discussed in theoriginal document [38].
The cryptographic signatures used in Bitcoin are ECDSA signatures anduse the curve secp256k1 (see Section 2). Given an ECDSA (possibly compressed) public-key K, a Bitcoin address is generated using the cryptographic
hash functions SHA-256 and RIPEMD-160 [21]. The public key is hashed twice:
HASH160 = RIPEMD-160(SHA-256(K)). The Bitcoin address is computed directly from this HASH160 value (where k denotes concatenation) as
base58(0x00 k HASH160 k bSHA-256(SHA-256(0x00 k HASH160))/2224 c),where base58 is a binary-to-text encoding scheme.
By participating in the Bitcoin peer-to-peer network, we downloaded the Bitcoin block chain up to block number 252 450 (all transactions up to mid-August2013) in the Berkeley DB [39] format. We extracted 22 159 078 transactions inplain text: this resulted in a single 26 GB file. In our dataset we have 46 254 121
valid public keys containing an elliptic curve point on the curve, and 15 291 112of these points are unique. There are 6 608 556 unique points represented in compressed (x-coordinate only) format and 8 682 692 unique points in uncompressed
format (we found 136 points which occur in both compressed and uncompressed
public keys). Since it is hard to tell if address reuse is due to the same user
reusing their key in Bitcoin (see e.g. [43, 36] regarding privacy and anonymity in
Bitcoin), there is no simple way to check if these duplicate public keys belong
to the same or different owners.
Currently (January 2014) there are over 12.2 million bitcoins in circulation
with an estimated value of over 10 billion USD. Bitcoin has been analyzed before
in different settings (e.g. [1, 45]), but we perform, as far as we are aware, the first
asymmetric cryptographic “sanity” check; see Section 4.1.
Secure Shell (SSH). Elliptic curve cryptography can be used in three positions in the SSH protocol. In SSH-2, session keys are negotiated using a DiffieHellman key exchange. RFC 5656 [47] specifies the ephemeral Elliptic Curve
Diffie-Hellman key exchange method used in SSH, following SEC1 [16]. Each
server has a host key that allows the server to authenticate itself to the client.
The server sends its host key to the client during the key exchange, and theuser verifies that the key fingerprint matches their saved value. The server thenauthenticates itself by signing a transcript of the key exchange. This host key
may be an ECDSA public key [47]. Finally, clients can use ECDSA public keys
for client authentication.
We surveyed the state of elliptic curve deployment on the server side for SSH
by scanning the complete public IPv4 space in October 2013 for SSH host keys,
server Diffie-Hellman values, and signature values. We also collected the list of
key exchange and authentication cipher suites offered by each server. We used

ZMap [22], a fast Internet-wide port scanner, to scan for hosts with port 22
open, and attempted an SSH protocol handshake with the addresses accepting
connections on port 22.
In order to focus on elliptic curve values, our client offered only elliptic curve
cipher suites. This resulted in us discovering several implementations that provided unexpected responses to our non-standards-compliant SSH handshake:
servers that provided RSA or prime-order DSA public keys, or servers that provided empty keys.
Of the 12 114 534 hosts where we successfully collected a set of cipher suites,
1 249 273 (10.3%) supported an ECDSA cipher suite for the host key. Of these,
1 247 741 (99.9%) supported ecdsa-sha2-nistp256, 74 supported ecdsa-sha2-nistp384,
and 1458 (0.1%) supported ecdsa-sha2-nistp521. 1 674 700 hosts (13.8%) supported some form of ECDH key exchange. Of these, 1 672 458 (99.8%) supported
the suites ecdh-sha2-nistp256, ecdh-sha2- nistp384, ecdh-sha2-nistp521
in order of increasing security, and 25 supported them in the opposite order. We
successfully collected 1 245 051 P-256, 73 P-384, and 1436 P-521 public keys. In
addition, 458 689 servers responded with a DSA public key, 29 648 respondedwith an RSA public key, and 7 935 responded with an empty host key, despite
our client only claiming ECDSA support. The hosts responsible for these responses included several kinds of routers and embedded devices, including thosefrom Huawei and Mikrotik.
Transport Layer Security (TLS). In TLS, elliptic curves can arise in several
locations in the protocol. RFC 4492 [9] specifies elliptic curve cipher suites for
TLS. All of the cipher suites specified in this RFC use the elliptic curve DiffieHellman (ECDH) key exchange. The ECDH keys may either be long-term (inwhich case they are reused for different key exchanges) or ephemeral (in whichcase they are regenerated for each key exchange). TLS certificates also contain apublic key that the server uses to authenticate itself; with ECDH key exchanges,
this public key may be either ECDSA or RSA.
ECC support was added to TLS [9] through an additional set of cipher suites
and three extensions in the client and server hello messages. The cipher suites
indicate support for a particular selection of key exchange, identity verification, encryption, and message authenticity algorithms. For example, the cipher
suite TLS ECDHE RSA WITH AES 128 CBC SHA uses ephemeral ECDH for a key exchange, signed with an RSA key for identity verification, and uses AES-128 [18]in CBC mode for encryption and the SHA-1 hash function in an HMAC for message authentication. In addition, if a cipher suite that involves ECC is desired,the client must include a set of supported elliptic curves in a TLS extension inits client hello message.
Unlike in SSH, a TLS server does not send its full preference of cipher suites
or curves that it supports. Rather, the client sends its list of supported cipher
suites and elliptic curves, and the server either replies with a single cipher suitefrom that list or closes the connection if it does not support any cipher suites incommon with the client. If the suite requires ECC, the server similarly includes
only a single curve type along with the key or signature. This makes learningwhich curves a server supports more difficult; a client must use multiple TLS
connections to offer a varying set of curves in order to learn a server’s support
and ordered preference.
In October 2013, we used ZMap [22] to scan the IPv4 address space on port
443, and used an event-driven program to send a specially crafted client hellomessage to each host with the port open. We offered 38 ECDH and ECDHE
cipher suites and 28 different elliptic curves. Of the 30.2 million hosts with port
443 open, 2.2 million (7.2%) supported some form of ECDH and provided an
ECC public key, along with information about which curve it uses. We thenconnected to these hosts again, excluding their known-supported curve typefrom our client hello’s curve list. We repeated this process until we had an
empty curve list, the server disconnected with an error, or the server presented
a curve that was not offered to them (a violation of the protocol). This process
allowed us to learn each server’s ordered preference of support across our 28
curves. We found the most commonly supported curve type across the 2.2 million
ECC-supporting hosts was secp256r1, supported by 98% of hosts. The curves
secp384r1 and secp521r1 were supported by 80% and 17% respectively, withthe remaining curves supported by fewer than 3% of hosts each.
We also found that the majority of hosts appear to prefer smaller curve sizes
over larger ones. Of the 1.7 million hosts that supported more than one curve,
98.9% support a strictly increasing curve-size preference. For example, 354 767
hosts had a preference of “secp256r1, secp384r1, secp521r1”, while only 190
hosts had the opposite order of “secp521r1, secp384r1, secp256r1”. Thissuggests that most hosts prefer lower computation and bandwidth costs overincreased security.
Austrian e-ID. Physical smart cards are increasingly being deployed for user
authentication. These smart cards contain cryptographic hardware modules that
perform the cryptographic computations; most often, these cards contain private
keys for encryption and signatures. Elliptic curve cryptography is an attractive
option for these types of deployments because of the decreased key size andcomputational complexity relative to RSA or large prime-order groups.
Austria’s national e-ID cards contain either an RSA or ECDSA public key,and can be used to provide legally binding digital signatures. We collected
828 911 Citizen Card certificates from the LDAP database ldap.a-trust.atin January 2013. Each certificate contained a public key and an RSA signaturefrom the certificate authority. 477 985 (58%) certificates contained an elliptic
curve public key, and 477 785 parsed correctly using OpenSSL. Of these, 253 047
used curve P-192, and 224 738 used curve P-256.

4

Cryptographic Sanity Check

There is long history of practical problems in cryptography related to insufficient randomness. The most notorious example in recent history is the Debian
OpenSSL vulnerability [51]: a 2006 change in the code prevented any entropy
from being incorporated into the OpenSSL entropy pool, so that the state ofthe pool was dependent only on the process ID and architecture of the hostmachine. A fixed number of cryptographic keys, nonces, or other random valuesof a given size could ever be generated by these implementations. The problem
was discovered in 2008.
In 2012 two different teams of researchers showed independently that a significant number of RSA keys (not considering the keys affected due to the Debian
OpenSSL bug) are insecure due to insufficient randomness [33, 28]. The latter paper also examined prime-order DSA SSH host keys and signatures, and found asignificant number of SSH host keys could be compromised due to poor randomness during signature generation. Most of the vulnerable keys were attributed topoor entropy available at first boot on resource-limited embedded and headlessdevices such as routers. In 2013, another paper showed that a number of RSA
keys obtained from Taiwan’s national Citizen Digital Certificate database could
be factored [4] due to a malfunctioning hardware random number generator on
cryptographic smart cards. In order to verify if similar vulnerabilities occur inthe setting of elliptic curve cryptography, we gathered as much elliptic curve
data as we could find and performed a number of cryptographic sanity checks:
Key Generation. An elliptic curve public key is a point Q = dG which is amultiple of the generator G for 1 ≤ d < n. Poor randomness might manifest itself
as repeated values of d, and thus repeated public keys observed in the wild. In
contrast to RSA, where poor random number generators and bugs have resulted
in distinct RSA moduli that can be factored using the greatest common divisor
algorithm when they share exactly one prime factor in common, an elliptic curve
public key appears to have no analogous property. We are unaware of any similarmathematical properties of the public keys alone that might result in complete
compromise of the private keys, and they are unlikely to exist because discrete
logarithms have strong hardcore properties [10, 31]. We checked for these problems by looking for collisions of elliptic curve points provided in public keys. In
practice, however, it is not uncommon to encounter the same public key multiple
times: individuals can use the same key for multiple transactions in Bitcoin or
the same key pair can be used to protect different servers owned by the same
entity.
Repeated Per-Message Signature Secrets. ECDSA signatures are randomized: each signature consists of two values (r, s): the value r is derived from anephemeral public key kG generated using a random per-message secret k, anda signature value s that depends on k. It is essential for the security of ECDSA
that signers use unpredictable and distinct values for k for every signature, sincepredictable or repeated values allow an adversary to efficiently compute the longterm private key from one or two signature values, as explained in Section 2. In
a widely known security failure, the Sony PlayStation 3 video game console useda constant value for signatures generated using their ECDSA code signing key,
allowing hackers to compute the secret code signing key [14].
We checked for these problems by parsing each signature and checking forcolliding values of the ephemeral public key.

4.1

BitcoinUnexpected, Illegal, and Known Weak Values. We checked for public keys
corresponding to the point at infinity, points that do not lie on the curve, and“public keys” that possibly do not have corresponding private keys. In addition,
we generated a large list of elliptic curve points for which we know the private
key. This is realized by multiplying the generator of the curve, as specified in
the standard, by various integers s from different sets in the hope that poor
entropy might have generated these scalars. We computed the elliptic curve
scalar multiplication sG for these different values of the scalar s and stored the xcoordinate of this resulting point in a database (by restricting to the x-coordinate
we represent both points ±sG). We checked these self-generated points in thisdatabase against all the elliptic curve points extracted from the ECDSA publickeys and signatures to verify if we find collisions: if so, we can compute theprivate key. We considered three different sets in the setting of the secp256k1
curve (as used in Bitcoin) and the NIST P-256 curve. The first set contains
small integers i: where 100 ≤ i ≤ 106 . The second set contains 256-bitscalarsof low Hamming weight: we used integers of Hamming-weight one ( 256
= 256
1


256
scalars), two ( 256=32640
scalars),
and
three
(
=
2
763
520
scalars).
The
2
3
third set contains the Debian OpenSSL vulnerable keys. We generated the set of
scalars produced by the broken Debian OpenSSL implementation run on a 64bit little-endian byte order architecture implementation. For the Bitcoin curve
we extended the first set by also considering the scalars iλ such that the scalarmultiplication corresponds to iλP = ψ(iP ) (see Section 2).
We found that two values from the set of small integers have been used inpractice: the Bitcoin addresses corresponding to the secret key 1 and 2. For thesecret key 1 both the addresses derived from the compressed and decompressed
public point have been used while for the secret scalar 2 only the address to thedecompressed point has been used. One value from the Hamming-weight one set
appeared in practice, the address corresponding to the decompressed public key
268 G. All these three addresses currently have a zero balance.
Repeated Per-Message Secrets. We extracted 47 093 121 elliptic curve points
from the signatures and verified that they are correct: i.e. the points are on
the curve secp256k1 (see Section 2). We also looked for duplicated noncesin the signature and found that 158 unique public keys had used the same
signature nonces r value in more than one signature, making it possible tocompute these users’ private keys. We find that the total remaining balanceacross all 158 accounts is small: only 0.00031217 BTC, which is smaller than
the transaction fee needed to claim them. However, we find that one address,
1HKywxiL4JziqXrzLKhmB6a74ma6kxbSDj, appears to have stolen bitcoins from10 of these addresses. This account made 11 transactions between March and October 2013. Each transaction contained inputs from addresses that duplicatedsignature nonces, and appear in our list. These transactions have netted thisaccount over 59 bitcoins (approximately $48,000 USD).
To understand the root causes of the repeated signature nonces, we made agraph of transactions, starting with the vulnerable addresses and adding edges

Fig. 1: Visualization of transactions between Bitcoin addresses that duplicatedsignature nonces (red), and addresses one (yellow) and two (blue) hops away inthe transaction graph. The unique pattern across graphs suggests that multiple
distinct implementations or usage patterns may be to blame for the generationof repeated nonces that expose users’ private keys.

to other addresses indicating if they had sent or received bitcoins to one another. Next, we created edges from those second layer addresses, terminating
the graph 2 degrees from the original vulnerable keys. This resulted in five distinct connected components, with the largest connected component containing
1649 addresses. Figure 1 shows the second and third largest connected graphs.
The unique patterns of these two graphs suggest that there are several sets ofunique users or implementations at play creating these types of failure.
We were able to identify three keys belonging to Bitcoincard [8], an embedded device that acts as a standalone Bitcoin client. We also identified several
Blockchain.info accounts that duplicated nonces due to a bug in a Javascriptclient’s random number generator not being seeded correctly [27]. These funds
were then subsequently transferred to the same address mentioned above. In
some cases, nonce repetition may be intentional: there exists a timestamping
scheme for Bitcoin that purposely leaks the private key of a transaction by deliberately using the same random nonce [17]. If this scheme is implemented andtested, then this might explain very small transactions signed with duplicatednonces.
Unspendable Bitcoins. It is possible to transfer bitcoins to an account for
which (most likely) no corresponding cryptographic key-pair exists. These bitcoins remain stuck at these accounts forever and are essentially removed from circulation. This might result in deflation: increasing the value of the other (spendable) bitcoins. We investigate a lower bound on the number of “unspendable”
bitcoins. Since the HASH160 values and the Bitcoin addresses (which are directly
derived from this HASH160 value) are an integral part of the Bitcoin block chain(i.e. the transaction history), people have used “interesting” invalid values for
the ECDSA public-key or used the HASH160 value to embed a message. Suchtransactions to addresses without corresponding cryptographic key-pair are pos-

HASH160

Bitcoin addressbalance in BTC

0000000000000000000000000000000000000000

1111111111111111111114oLvT20000000000000000000000000000000000000001

11111111111111111111BZbvjr

0000000000000000000000000000000000000002

11111111111111111111HeBAGj

0000000000000000000000000000000000000003

11111111111111111111QekFQw

0000000000000000000000000000000000000004

11111111111111111111UpYBrS

0000000000000000000000000000000000000005

11111111111111111111g4hiWR

0000000000000000000000000000000000000006

11111111111111111111jGyPM80000000000000000000000000000000000000007

11111111111111111111o9FmEC

0000000000000000000000000000000000000008

11111111111111111111ufYVpS

2.94896715
0.01000000
0.00000001
0.00000001
0.00000001
0.00000001
0.00000001
0.00000001
0.00000001
0.00012000
0.01000005
1.32340175

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 1GZQKjsC97yasxRj1wtYf5rC61AxpR1zmr
ffffffffffffffffffffffffffffffffffffffff 1QLbz7JHiBTspS962RLKV8GndWFwi5j6Qr

151 miscellaneous ASCII HASH160 values
publickey

valid
encoding

point on
curve

∅

5
!
5
!

5
!
5
5

00
0 128
... 0
040 126... 0

Bitcoin addressbalance in BTC

1HT7xU2Ngenf7D4yocz2SAcnNLW7rK8d4E

68.80080003
2.08000002
0.00010000
0.01000000

1FYMZEHnszCHKTBdFZ2DLrUuk3dGwYKQxh
13VmALKHkCdSN1JULkP6RqW3LcbpWvgryV
16QaFeudRUt8NYy2yzjm3BMvG4xBbAsBFM

Table 1: A summary of the interesting HASH160 and public key values usedin the Bitcoin block chain with the corresponding Bitcoin address and balance.
Most likely, these addresses have no valid private key, leaving the account balances unspendable. The dots in the notation 0 128. . . 0 represent 128 zeros (the key
has 130 zeros in all). We find these addresses hold a total of 75 unspendable
BTC.

sible since the actual ECDSA keys are only required when the money in theseaccounts is spent. Given a Bitcoin address, or HASH160 value, it is infeasible
to compute the corresponding cryptographic key-pair (since this requires computing preimages of the hash function used). In this section we assume thatthe interesting (or strange) values we encounter do not correspond to a valid
cryptographic key-pair. Of course, it is possible (but unlikely) that these weregenerated in a valid manner.
Interesting HASH160 Values. Since no cryptographic key is required togenerate a Bitcoin address, just a HASH160 value, our first idea was to checkfor addresses which have a HASH160 value which is a small integer i, where0 ≤ i < 100. We found that the first nine values all exist and have a non-zerobalance. This motivated us to search for repeated patterns when the HASH160
is displayed in hexadecimal. All of these 16 possibilities exist and three of them
have a non-zero balance; see Table 1.
People have sometimes used HASH160 values to embed an ASCII encoded
string into one or multiple HASH160 values within a transaction. ASCII encodes 128 specific characters (97 printable and 33 non-printable). The probability
that an ECDSA public key results in a hexadecimal written HASH160 contain-ing ASCII characters only is 2−20 (where we assume the cryptographic hashfunctions used outputs uniform random data). Our dataset contains 53 019 716
HASH160 values (16 526 211 unique). Hence, we expect to find approximately
16 valid Bitcoin addresses with a HASH160 value containing ASCII characters
only. In our dataset we found 248 ASCII-only HASH160 values (180 unique).
Out of these, 20 unique addresses have spent their money; i.e. they correspondto a valid Bitcoin address. This is in line with our estimate of 16. Out of theother 160 unique addresses 137 have a non-zero balance. When inspecting thesevalues it is clear that people have inserted various messages in the Bitcoin transaction history (the messages range from a happy birthday message to a tribute).
Typically only a small number of bitcoins are used in these transactions. See
Table 1 for the details.
Interesting ECDSA Public Keys. Following the same reasoning as in the
HASH160 setting, one could use “interesting” values for the public key itself.
Before we outline our search for such values, let us recall the format of ECDSA
public keys as specified in [16] where we assume the keys are represented in theirhexadecimal value (this is the setting used in Bitcoin). A point P = (x, y) canbe represented as follows where p = 2256 − 232 − 977 is the prime used in Bitcoin.
• If P is the point at infinity, then it is represented by the single byte 00.
• An uncompressed point starts with the byte 04 followed by the 256-bit x- and256-bit y-coordinate of the point (04 k x k y). Hence 2dlog2 (p)/8e + 1 = 65
bytes are used to represent a point.
• A point is compressed by first computing a parity bit b of the y-coordinate
as b = (y mod 2) + 2 and converting this to a byte value (b ∈ {02, 03}). The
dlog2 (p)/8e + 1 = 33-byte compressed point is written as b k x.
Similar to the HASH160 search, we started by looking for points that encode
a small integer value. We generated all the Bitcoin addresses corresponding to
the public keys with values the first 256 integers i (0 ≤ i < 256) and various
values for the parity bit. We used a single byte containing i, a 33-byte valueb k 0 .60
. . 0 k i, b ∈ {00, 02, 03}, and a 65-byte value b k 0 124. . . 0 k i, forb ∈ {00, 04}. We found three addresses with a non-zero balance: the single byte
00, and the 65-byte b k 0 124. . . 0 k i for i = 00 and b ∈ {00, 04}. This first point is
the point at infinity, which is a correctly encoded and valid point on the curve.
Note, however, that this value is explicitly prohibited as a public key [16] since it
can only occur for the private key d = 0 which is not allowed. The 65-byte values
both seem to try and encode the point at infinity: in the case where b = 00 theencoding is invalid while in the case b = 04 the encoding is valid but the point(x, y) = (0, 0) is not on the curve.
When looking for other values, we also tried the empty public key (∅). Thisaddress contains a significant amount of bitcoins (over 68 BTC). We suspect
money has been transferred to this account due to software bugs. These results
are included in Table 1. In total we found that at least 75 BTC (over 61,000 USD)
has been transferred to accounts which have (most likely) no valid corresponding
ECDSA private key. Note that this is strictly a lower bound on the number ofunspendable bitcoins, as we do not claim that this list is complete.

4.2

Secure Shell (SSH)

Duplicate public keys. An August 2013 SSH scan collected 1 353 151 valid elliptic curve public keys, of which 854 949 (63%) are unique. There were 1 246 560
valid elliptic curve public keys in the October 2013 scan data, of which 848 218(68%) are unique. We clustered the data by public key. Many of the most commonly repeated keys are from cloud hosting providers. For these types of hosts,
repeated host keys could be due either to shared SSH infrastructure that is accessible via multiple IP addresses, in which case the repeated keys would not
be a vulnerability, or they could be due to mistakes during virtual machine deployment that initialize multiple VMs for different customers from a snapshotthat already contains an SSH host key pair. It appears that both cases are represented in our dataset. Digital Ocean released a security advisory in July 2013 [20]recommending that customers should regenerate SSH host keys due to repeatedkeys deployed on VM snapshots; we found 5 614 hosts that had served the publickey whose fingerprint appears in Digital Ocean’s setup guide.
We were also able to identify several types of network devices that appeared
to be responsible for repeated host keys, either due to default keys present in
the hardware or poor entropy on boot. These include the Juniper Web Device
Manager, the Juni FemtoAP, and ZTE Wireless Controller. We were able to attribute the repeated keys to these implementations because these devices served
login pages over HTTP or HTTPS which identified the manufacturer and brand.
We were unable to easily give an explanation for most of the repeated keys, as(unlike in the results reported in [28]) many of the clusters of repeated keys appeared to have almost nothing in common: different SSH versions and operating
systems, different ports open, different results using nmap host identification,different content served over HTTP and HTTPS, and IP blocks belonging tomany different hosting providers or home/small commercial Internet providers.
We can speculate that some of these may be VM images, but in many cases wehave no explanation whatsoever. We can rule out Debian weak keys as an explanation for these hosts, because the Debian bug was reported and fixed in 2008,
while OpenSSH (which is almost universally given in the client version strings
for the elliptic curve results) introduced support for elliptic curve cryptography
in 2011. We checked for repeated signature nonces and did not find any. We also
checked for overlap with the set of TLS keys we collected and did not find any.
4.3

Transport Layer Security (TLS)

Duplicate public keys. Although we collected a total of over 5.4 million publickeys from ECDH and ECDHE key exchanges, only 5.2 million of these wereunique. As observed in [12], OpenSSL’s default behavior is to use ephemeralstatic ECDH (the key pair is ephemeral for each application instance and notnecessarily per handshake instance) which might explain some of the observed
duplicate keys. We found 120 900 distinct keys that were presented by morethan one IP address, with the most common duplicated key presented by over
2 000 hosts. Many of these duplicated keys appear to be served from a single

or small set of subnets, and appear to serve similarly configured web pages forvarious URLs, suggesting that these are part of a single shared hosting. We
also discovered one instance of a default key being used on a device sold todifferent consumers. We found about 1 831 Netasq devices that present the same
secp256r1 public key for their ECDHE key exchange. Each device must also
have the same private key, allowing an attacker who buys or compromises one
device to passively decrypt traffic to other devices.
Duplicate server randomness. We also were surprised to find that several
hosts duplicated the 32-byte random nonce used in the server hello message.
We found 20 distinct nonces that were used more than once, 19 of which werere-used by more than one IP address. The most repeated server random wasrepeated 1 541 times and was simply an ASCII string of 32 “f” characters. These
devices all appear to be a UPS power monitor, which appears to outsource its
SSL implementation to a company called Ingrasys according to the certificate
presented. However, we were unable to successfully establish any TLS sessions
with these devices, either using a browser or OpenSSL.
For servers that happen to always duplicate a server random, it is clear
there is an implementation problem to be fixed. However, for servers that only
occasionally produce the same server random, it is indeed more troubling. Moreinvestigation is required to find the root cause of these collisions and determine
if the problem extends to cryptographic keys.
4.4

Austrian e-ID

We did not find any abnormalities with the ECDSA keys in this dataset. Of the477 985 elliptic curve public keys that we extracted from the Austrian Citizen
Card certificate database, 24 126 keys appear multiple times. However, in all
but 5961 of these cases, the certificate subjects were equal. Of the nonequal
subjects, all but 70 had identical “CN” fields. All of these remaining certificateswith identical public keys issued to nonequal names appeared to be due to either
minor character encoding or punctuation differences or name changes.

5

Conclusions

We explore the deployment of elliptic curve cryptography (ECC) in practiceby investigating its usage in Bitcoin, SSH, TLS, and the Austrian citizen card.
More than a decade after its first standardization, we find that this instantiationof public-key cryptography is gaining in popularity. Although ECC is still far
from the dominant choice for cryptography, the landscape shows considerable
deployment in 2013.
Our cryptographic sanity checks on these datasets confirmed that, as expected, ECC is not immune to insufficient entropy and software bugs. We foundmany instances of repeated public SSH and TLS keys, some of which seem to correspond to different owners. For the Bitcoin data set, there are many signatures

sharing ephemeral nonces, allowing attackers to compute the corresponding private keys and steal coins. We hope that our work will encourage researchers anddevelopers alike to remain diligent in discovering and tracking down these typesof implementation problems, ultimately improving the security of the cryptographic protocols and libraries we depend on.
Acknowledgments. We thank Jaap W. Bos for valuable discussions about thefinancial market, Andy Modell for support in TLS scanning, Sarah Meiklejohn
for sharing her knowledge about Bitcoin, and Felipe Voloch for pointing out
the existence of the private keys 1 and 2 in Bitcoin. We thank the Microsoft
Security Vulnerability Research team for their help with responsibly disclosing
the vulnerabilities we found to affected companies.

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