A Survey on Elliptic Curve Cryptography

International Journal of Research in advance Engineering,
Volume -1 Isuue-2, Feb-2015, Available at www.knowledgecuddle.com
7
A Survey on Elliptic Curve Cryptography
Kinjal Mehta
Assistant Professor (EC)
L D College Of Engineering
Ahmedabad
Dharmesh Patel
M.E.(C.E.), student-
B.V.M.Engineering College
V V Nagar
_____________________________________________________________________________________
ABSTRACT :Elliptic curve cryptography (ECC) is the most efficient public key encryption scheme based on
elliptic curve concepts that can be used to create faster, smaller, and efficient cryptographic keys. ECC
generates keys through the properties of the elliptic curve equation instead of the conventional method of
key generation. This scheme can be used with public key encryption methods, such as RSA, Diffie-Hellman key
exchange and Digital Signature. Review of the four protocols which applies ECC namely Bitcoin, secure
shell (SSH), transport layer security (TLS), and the Austrian e-ID Card describes the high security by using
elliptic curve cryptography.
Keywords: Elliptic Curve Cryptography (ECC), Public Key Infrastructure (PKI), Finite field, Digital Signature,
Elliptic Curve Digital Signature Algorithm (ECDSA)
________________________________________________________________________________________________
I. INTRODUCTION
Rapid development on secure communication in particular is in demand for any kind of communication
network .The main component of secure communications software stack includes key exchange and signatures
which is required for public key algorithms like RSA,DSA and elliptic curve cryptography[1][2] . The discrete
logarithm problem on elliptic curve groups is believed to be more difficult than the corresponding problem in
the underlying finite field [3][4][5].Elliptic Curve Cryptography provides level of security with a 164-bit key that
RSA require a 1,024-bit key to achieve, Because ECC helps to establish equivalent security with lower computing
power and battery resource usage. The ECC covers all primitives of public key cryptography like digital signature ,key
exchange, key transport ,key management .Presently ECC has been commercially adopted by many standardize
organization such as NIST ,ISO ,and ANSI [1] .ECC covers the discipline of mathematics and computer science and
engineering . It can widely use for electronic commerce, secure communication, etc. The security of the Elliptic
Curve Cryptography depends on the difficulty of finding the value of k, given kP where k is a large number and P is a
random point on the elliptic curve[6][7]. This is the Elliptic Curve Discrete Logarithmic Problem. The elliptic curve
parameters for cryptographic schemes should be carefully chosen in order to resist all known attacks of Elliptic
Curve Discrete Logarithmic Problem (ECDLP)[7][8]. Additional to the ECC, this paper presents the collection of the
keys which are implemented in the Bitcoin, Secure Shell (SSH), Transport Layer Security (TLS), and Australian
E-card. 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. Elliptic-curve cipher suites that offer forward secrecy by establishing a session
key using elliptic-curve Diffie-Hellman key exchange [20b] were introduced in 2006 and are growing in popularity for
TLS. This dataset includes the Diffe-Hellman server key exchange messages, as well as public keys and signatures from
servers using ECDSA. 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 Diffe-Hellman server key exchange messages,
elliptic-curve public host keys, and ECDSA signatures. The Austrian e-ID contains public keys for encryption and
digital signatures, and as of 2009, ECDSA signatures are offered.
II. ECC PRELIMINARIES
The elliptic curve cryptosystem [9, 10] was discovered by Koblitz [11] and then Miller [12] in 1985 to design
public key cryptosystem and presently, it becomes an integral part of the modern cryptography. Let E/Fp denotes an
elliptic curve E over a prime finite field Fp, which can be defined by baxxy  32
, where, a, b are real numbers
and the discriminate 0274 23
 baD , which ensures that elliptic curve does not contain repeated factors.

8
Figure 1
Properties:
1. Negative: The negative of a point P = (xP,yP) is its reflection about the x-axis: the point -P is (xP,-yP). For each point
P on an elliptic curve, the point -P is also on the curve.
2. Addition: Suppose that P and Q are two distinct points on an elliptic curve, and P ≠ -Q. To add the points P and Q, a
line is drawn from these two points will intersect the elliptic curve in exactly one more point, call -R. The point -R is
reflected in the x-axis to the point R. The law for addition in an elliptic curve group is P + Q = R.
3. Point at Infinity: The line through P and -P is a vertical line which does not intersect the elliptic curve at a third point.
It is for this reason that the elliptic curve group includes the point at infinity O. By definition, P + (-P) = O. all elliptic
curves have an additive identity.
4. Doubling point P: To add a point P to itself, a tangent line to the curve is drawn at the point P. If yP is not 0, then the
tangent line intersects the elliptic curve at exactly one other point, -R. -R is reflected in the x-axis to R
Figure 2
Key Generation:
ECC is the public key cryptography therefore we have to generate both public key and private key. The sender
will be encrypting the message with receiver‟s public key and the receiver will decrypt its private key.

9
Select a number„d‟ within the range of „n‟.
The public key is calculated by Q = d * P
„Q‟ is the public key and„d‟ is the private key. d = The random number that we have selected within the range of (1 to n-1
). P is the point on the curve.
 Encryption
In ECC, the entire message to be sent is represented on the Elliptic curve. Let „m‟ be the message that we are
sending. We have to represent this message on the curve. These have in-depth implementation details. All the advance
research on ECC is done by a company called certicom. Consider „m‟ has the point „M‟ on the curve „E‟. Randomly
select „k‟ from [1 - (n-1)].Two cipher texts will be generated let it be C1 and C2. C1 = k*P and C2 = M + k*Q , where
C1 and C2 will be sent.
 Decryption
We have to get back the message „m‟ that was send to us, M = C2 – d * C1, M is the original message that we had
sent.
III. APPLICATIONS OF ECC
1. Bitcoin
The crypto currency 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" [13].The (public) Bitcoin block
chain is a journal of all the transactions ever executed. Each block in this journal contains the SHA-256 [14] hash of the
previous block, hereby chaining the blocks together starting from the so-called genesis block. In Bitcoin, User‟s
account‟s unique key is defined by the private key generated by an ECDSA. Thus by Transferring bitcoins, we
are transferring the ownership of bitcoins. When the bitcoins are transferred from user A to user B, digital signature is
attached with the help of user A‟s Private key. At the receiving side, ownership of bitcoins are gained by the hash of the
previous transaction and 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 previous transaction. Other issues, such as avoiding double-spending, are
discussed in the original document [13].The cryptographic signatures used in Bitcoin are ECDSA signatures. Given
an ECDSA (possibly compressed) public-key K, a Bitcoin address is generated using the cryptographic hash functions
SHA-256 and RIPEMD-160 [15].
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 ||
HASH160 || [SHA-256(SHA-256(0x00 || HASH160)) =2 ^224]), where base58 is a binary-to-text encoding scheme5.
Currently (October 2013) there are over 11.5 million bitcoins in circulation with an estimated value of over 2 billion
USD. Since it is hard to tell if address reuse is due to the same user reusing their key in Bitcoin (see e.g. [9, 8] 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.
2. Secure Shell (SSH)
Secure Shell (SSH) is a cryptographic network protocol for secure data Communication, remote
command line-login, remote command execution, and other secure network services between two networked computers.
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

10
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 snapshot that
already contains an SSH host key pair. 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. 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 [16]) 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 to many
different hosting providers or home/small commercial Internet providers. We can speculate that some of these may be
VM images, but in many cases we have 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.
3. Transport Layer Security (TLS)
Transport Layer Security (TLS) and its predecessor, Secure Sockets Layer (SSL), are cryptographic protocols
which are designed to provide communication security over the Internet .From a total of over 5.4 million public
keys from ECDH and ECDHE key exchanges, only 5.2 million of these were unique. As observed in [17], OpenSSL's
default behavior is to use ephemeral-static ECDH which might explain some of the observed duplicate keys. We found
120,900 distinct keys that were presented by more than 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 for various URLs, suggesting that these are part of a single shared hosting.
Sometimes one instance of a default key being used on a device sold to different consumers. Several researches
discovered 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 were re-used by more than one IP address. The most
repeated server random was repeated 1 541 times and was simply an ASCII string of 32 f” characters. These devices all
appear to be a UPS power monitor. 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. More investigation is required to find the root cause of these collisions and
determine if the problem extends to cryptographic keys.
4. Austrian e-ID
Of the 477,985 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 non equal subjects, all but 70 had identical CN" fields. All of these remaining certificates with identical public
keys issued to nonequal names appeared to be due to either minor character encoding or punctuation differences or name
changes. Hence, there appears to be no abnormalities with the ECDSA keys in this dataset.
IV.CONCLUSION
Elliptic Curve Cryptography offers the highest strength-per-key-bit of any known public-key system of first
generation techniques like RSA, Diffie-Hellman. ECC offers the same level of security with smaller key sizes,
computational power is high. There are several major organizations that develop standards like International Standards
Organization (ISO), American National Standards Institute (ANSI), Institute of Electrical and Electronics
Engineers (IEEE), Federal Information Processing Standards (FIPS). We explore the deployment of elliptic curve
cryptography (ECC) in practice by investigating its usage in Bitcoin, SSH, TLS, and the Austrian citizen card.
More than a decade after the first ECC standardization we find that this instantiation of public key cryptography is
gaining in popularity.

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[14] U.S. Department of Commerce/National Institute of Standards and Technology. Secure Hash Standard (SHS).
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