Last weekend I was visiting with my grandmother and she said to me and my brothers "Can anybody explain crypto? I keep hearing about crypto on the news and I don't know what that is?"
We tried to briefly explain, but I don't think we did a good job. So I decided to lay out a simple groundwork to understand crypto that could be understood by anybody, even my grandmother.
The first thing to understand is that when the guys on the news talk about "crypto" they are probably talking about "cryptocurrencies", like Bitcoin, which could also be called cryptographic-currencies.
Let's start at the beginning: if you have a message written as letters, you can rewrite that as a big number. Here, let me demonstrate: lets's use a simple system where each letter corresponds to a two digit number, a is 01 on up to z is 26, make 00 a space and 27 a period and we can write a sentence. So to write "abc" we could use the number 010203, and 101112 would be "jkl". Or the number 160529051800091900071805012027 is the message "peter is great." Actual cryptography will use ASCII or a similar system so that you have the whole alphabet, upper and lower case letters, a wide variety of punctuation, and numerals; but the underlying idea is the same - any message can be written as a really big number.
The next thing to understand is the idea of one-way or "trapdoor" functions. Let's take prime factoring as an example: what are the prime factors of 527 ? You might start by noticing it is odd, so not 2; then you start dividing each prime number going up - 3 does not work (if it is a multiple of 3 then the sum of the digits will also be); it's not a multiple of 5 (does not end in a 5 or 0); I don't know a trick for 7 but that does not divide evenly either; some people make it to 11 and then quit. But if I say what is 17 x 31 you might even be able to do it in your head: 10(17 x 3) + (17 x 1) --> 51_ + 17 -> 527. So you see that going one way (finding the prime factorization) takes much more work than going the other way (multiplying two primes). You can use a computer to make it easier, up to a point. If you have a "small" number the computer can factor it quickly, but as the number gets bigger the factorization takes longer and longer, so if you have a big enough number then not even the world's largest supercomputer can crack that prime factorization. (4096 bits should be enough for everybody)
People can then use such a one-way function to create what is called asymmetric cryptography. The idea here is that each person creates a pair of keys with a "public key" portion and a "private key" portion. A message is stored as a large number, a one-way function is used on it using the private key, and then anybody can check using the public key with the one-way function to prove that the message was made by that person. (Alternately, a message created using the public key can only be read by the person holding the private key, so this is also useful for secure communication).
As an example of a digital signature, the RSA system uses prime factorization, as mentioned above, to keep the private key secure. In RSA, a private key is made by taking two large primes (2048 bits long) and publishing their product (N) as part of the public key, along with an unrelated number (e). Using the two primes, the key generator also calculates e's modular inverse (d), which is a unique number, and stores that as the private key. Since you need the two primes to calculate d, and the number N is so large that it is impossible to factor, you can give other people the public key (e, N) and still the private key (d, N) will stay a secret. A message m (remember, the message is converted from letters to a really big number) is then signed by taking the modular exponentiation c = m^d mod N, and anybody can check that you signed it because they can easily calculate m = c^e mod N (this is true because e and d are modular inverses).
Once you have an asymmetric cryptographic system like RSA, or elliptic curve cryptography (ECC) which is more complicated but the basic idea is the same, then you can create a cryptocurrency. This is as simple as each person having a key-pair, and people can sign messages, or transactions, like "move $1 from {Peter's key} to {John's key}" - signed by {Peter's key}. Then everybody can check to see that was, in fact, signed by Peter. And if Peter had $1, then it is subtracted from his account and added to John's.
In a centralized system, with one company keeping a ledger with all the accounts, that will be sufficient. But if you are running a world-wide, peer-to-peer system and you receive such a transaction, how do you know Peter did not just sign a transaction giving all his money to Rachel instead and give that transaction to everybody else? You could say whichever message is received first is valid, but it is hard to get people spread around the world to agree on things like the order of messages because somebody else could have seen the messages in a different order.
The innovation of Bitcoin was to introduce the idea of a "blockchain" to serve as a secure, trustable ledger for transactions of digital money. Anybody can create transactions to move their own money within the system, called bitcoins, and these are shared with all users. A block is created by collecting valid transactions together and also lists the previous block. Thus a chain of these blocks is created, and balances are updated based on the transactions that are included in the blocks. So if Peter, who has 1 bitcoin in his account, creates one transaction that says "move 1 to John", and another that says "move 1 to Rachel", the person who creates the block will only include the one they heard first, and everybody will update the accounts based on the transaction that ends up in the blockchain; the other transaction will then be rejected by everybody.
In systems like Bitcoin, the people who publish these blocks to the blockchain are sometimes called "miners" because of the particular way in which Bitcoin introduces new money into the system: Each block is created with a certain amount of new bitcoin (started as 50 per block, cuts in half every 4 years, now at 12.5), and people making transactions include a "fee" to get their transaction included, these all go to the one person who makes the block (so people doing the work to check that transactions are valid and making the blocks are rewarded with a supply of new money, like people who work in mines are rewarded with a supply of new gold).
Naturally this incentivizes each person to have their own block included in the blockchain so they get the "miner reward", and if two different blocks are created at the same time which gets included? This is solved by the idea of "difficulty": each block is identified by a "hash function", another one-way function, which converts the contents into a number. The function is chosen to give an essentially random distribution. The difficulty is then calculated as a function of the number of leading zeros in the number. So 1234 would have a difficulty of 0, 0234 has a difficulty of 1 (probability of 1 in 10), and 0056 has a difficulty of 10 (1 in 100, ten times as hard as previous). Anyway, the next block has to meet a minimum difficulty score, which is adjusted periodically so that a new block is found roughly every ten minutes. If there are two competing blocks, the one included is always the one that has the greatest difficulty score. So the miners will build slightly different versions of a block and calculate the hash function until they find one with the right score.
The hash function is designed to be computationally difficult for computers. But a stronger computer will calculate it faster, and so in the beginning of bitcoin anybody could have their computer working on hashing blocks and expect to find a valid one every once in a while, a computer that was twice as fast would just get twice as many hits over a long period of time. Within a couple years of bitcoin starting, though, people had discovered that graphics cards could be programmed to do the hash calculation much faster (by orders of magnitude) than a normal computer CPU. So for a while people would buy high end graphics cards and stack them together. Within a few more years, though, specialty circuits were made which could do this calculation faster by a couple more orders of magnitude. Because of the way that the difficulty requirement is periodically redefined, these application specific circuits still generate about one block every ten minutes, while the chance that a normal computer will find a valid block is essentially 0, and all mining is controlled by a few companies in China that have built their own custom bitcoin-mining supercomputers.
The rule is that the only valid cryptocurrency blockchain is the one with the highest difficulty score, and for that there is nothing close to Bitcoin, which has been running since 2009. However, all those blocks add up, so to store the bitcoin blockchain requires several hundred gigabytes of memory. There are 1 TB disk drives available (1024 GB), so anybody can build a computer that is capable of holding all this data, and then they can run the check themself to show that any Bitcoin transaction is valid or not. This peer-to-peer structure makes Bitcoin more resilient than other types of digital currency which have a central point of failure. Because of this resiliency, the fact that Bitcoins can easily be sent anywhere around the globe instantly, and the fact that there is a defined limit to the total number of bitcoin (unlike US dollars, which can be printed whenever the US needs more money, causing inflation), Bitcoin can be used as a secure store of value or as a way to securely transfer funds globally, which is why the exchange rate has consistently increased over time (current exchange rate is about 3600 US dollars per bitcoin).
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Showing posts with label Cryptography. Show all posts
Showing posts with label Cryptography. Show all posts
January 16, 2019
May 19, 2018
lamport.py
This is a script I wrote in Python to use the ideas expressed by Lamport and Stanislav.
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
#!/usr/bin/python
#
# (C) 2016 Peter Lambert.
# You do not have, nor can you ever acquire the right to use, copy or distribute this software;
# should you use this software for any purpose, or copy and distribute it to anyone or in any manner,
# you are breaking the laws of whatever soi-disant jurisdiction, and you promise to continue doing
# so for the indefinite future.
#
import hashlib
import os
import sys
#
# Comment out the line below with the hash you do not want to use
# sha = hashlib.sha256
sha = hashlib.sha512
def binmessagehash(filename):
m = open(filename, 'r')
messagehash = sha(m.read()).hexdigest()
m.close()
return bin(int(messagehash, 16))[2:].zfill(len(messagehash) * 4)
def decode(pubkeyfile, sigfile):
kf = open(pubkeyfile, 'r')
pubkey = [line.strip().split() for line in kf.readlines()]
kf.close()
sf = open(sigfile, 'r')
sig_keys = [line.strip() for line in sf.readlines()]
sf.close()
binmessage = ''
for k in range(len(sig_keys)):
key_hash = sha(sig_keys[k].decode('hex')).hexdigest()
if key_hash in pubkey[k]:
binmessage += str(pubkey[k].index(key_hash))
else:
return 'Bad signature, hash not found for %s' % sig_keys[k]
hexmessage = '%x' % (int(binmessage, 2))
return hexmessage
def encode(privkeyfile, messagefile):
pkf = open(privkeyfile, 'r')
key = [line.strip().split() for line in pkf.readlines()]
pkf.close()
bmess = binmessagehash(messagefile)
for k in range(len(bmess)):
print key[k][int(bmess[k])]
def generate_key(payloadbits, strengthbits):
if '-s' in sys.argv:
rs = randstr
else:
rs = os.urandom
key = [[rs(int(strengthbits) / 8) for n in [0, 1]] for m in range(int(payloadbits))]
for k in key:
print ' '.join(kp.encode('hex') for kp in k)
def output_help():
print '''Usage:
./lamport.py -g PAYLOADBITS STRENGTHBITS > PRIVKEYFILE
generates a key
./lamport.py -g -s PAYLOADBITS STRENGTHBITS > PRIVKEYFILE
generates a key with the slower, more secure /dev/random
./lamport.py -p PRIVKEYFILE > PUBKEYFILE
creates a pubkey from a privkey file
./lamport.py -e PRIVKEYFILE MESSAGE > ENCODED.TXT
creates a digital signature
./lamport.py -d PUBKEYFILE ENCODED.TXT
decode the digital signature
./lamport.py -v PUBKEYFILE ENCODED.TXT MESSAGE
check that encoded is the correct signature for message
'''
def priv_to_pubkey(privkeyfile):
pkf = open(privkeyfile, 'r')
for line in pkf.readlines():
print ' '.join(sha(k.decode('hex')).hexdigest() for k in line.strip().split())
pkf.close()
def randstr(num_bytes):
stream = open('/dev/random', 'rb')
res = stream.read(num_bytes)
stream.close()
return res
def verify(pubkeyfile, sigfile, messagefile):
decoded = decode(pubkeyfile, sigfile)
print decoded
m = open(messagefile, 'r')
messagehash = sha(m.read()).hexdigest()
m.close()
if decoded == messagehash:
print 'Good signature'
else:
print 'Bad signature'
if __name__ == "__main__":
if '-g' in sys.argv and sys.argv[-2].isdigit() and sys.argv[-1].isdigit():
generate_key(sys.argv[-2], sys.argv[-1])
elif '-p' in sys.argv:
priv_to_pubkey(sys.argv[-1])
elif '-e' in sys.argv:
encode(sys.argv[-2], sys.argv[-1])
elif '-d' in sys.argv:
print decode(sys.argv[-2], sys.argv[-1])
elif '-v' in sys.argv:
verify(sys.argv[-3], sys.argv[-2], sys.argv[-1])
else:
output_help()
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May 18, 2018
Constructing Digital Signatures from One Way Functions
This is something I pulled from a pdf a while ago. It was referenced on Loper-OS, and so I thought it would be worthwhile to de-pdf-enate it.
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Hash: SHA1
Op. 52
Constructing Digital Signatures from One Way Function
Leslie Lamport
Computer Science Laboratory
SRI International
18 October 1979
CSL 98
333 Ravenswood Ave. Menlo Park, California 94025
(415) 326-6200 Cable: SRI INTL MPK TWX: 910-373-1246
1. Introduction
A digital signature created by sender P for a document m is data
item o_p(m) having the property that upon receiving m and o_p(m) one can
determine (and if necessary prove in court of law) that P generated the
document m .
A one way function is function that is easy to compute, but whose
inverse is difficult to compute [1]. More precisely one way function T is
a function from set of data objects to a set of values having the following
two properties:
1. Given any value v , it is computationally infeasible to find a
data object d such that T(d) = v .
2. Given any data object d , it is computationally infeasible to find
a different data object d' such that T(d') = T(d) .
If the set of data objects is larger than the set of values, then such a
function is sometimes called one way hashing function.
We will describe a method for constructing digital signatures from such a
one way function T . Our method is an improvement of a method devised by
Rabin [2]. Like Rabin's, it requires the sender P to deposit a piece of
data o in some trusted public repository for each document he wishes to
sign. This repository must have the following properties:
- o can be read by anyone who wants to verify P's signature.
- It can be proven in court of law that P was the creater of o .
Once o has been placed in the repository, P can use it to generate a
signature for any single document he wishes to send.
Rabin's method has the following drawbacks not present in ours.
1. The document must be sent to single recipient Q , who then
requests additional information from P to validate the signature.
P cannot divulge any additional validating information without
compromising information that must remain private to prevent
someone else from generating new document m' with valid
signature o_p(m') .
2. For a court of law to determine if the signature is valid, it is
necessary for P to give the court additional private information.
This has the following implications.
- P -- or a trusted representative of P must be available
to the court.
- P must maintain private information whose accidental
disclosure would enable someone else to forge his signature on
a document.
With our method, P generates a signature that is verifiable by anyone,
with no further action on P's part. After generating the signature, P can
destroy the private information that would enable someone else to forge his
signature. The advantages of our method over Rabin's are illustrated by the
following considerations when the signed document m is a check from P
payable to Q .
1. It is easy for Q to endorse the check payable to third party
R by sending him the signed message "make m payable to R ".
However, with Rabin's scheme, R cannot determine if the check m
was really signed by P , so he must worry about forgery by Q as
well as whether or not P can cover the check. With our method,
there is no way for Q to forge the check, so the endorsed check
is as good as check payable directly to R signed by P .
(However, some additional mechanism must be introduced to prevent
Q from cashing the original check after he has signed it over to
R .)
2. If P dies without leaving the executors of his estate the
information he used to generate his signatures, then Rabin's method
cannot prevent Q from undetectably altering the check m -- for
example, by changing the amount of money payable. Such posthumous
forgery is impossible with our method.
3. With Rabin's method, to be able to successfully challenge any
attempt by Q to modify the check before cashing it, P must
maintain the private information he used to generate his signature.
If anyone (not just Q ) stole that information, that person could
forge a check from P payable to him. Our method allows P to
destroy this private information after signing the check.
2. The Algorithm
We assume a set M of possible documents, set K of possible keys [footnote 1],
and set V of possible values. Let S denote the set of all subsets of
{1, ... , 40} containing exactly 20 elements. (The numbers 40 and 20 are
arbitrary, and could be replaced by 2n and n. We are using these numbers
because they were used by Rabin, and we wish to make it easy for the reader to
compare our method with his.)
We assume the following two functions.
1. function F : K -> V with the following two properties:
a. Given any value v in V , it is computationally infeasible
to find a key k in K such that F(k) = v .
b. For any small set of values v_1, ... , v_m , it is easy to
find a key such that F(k) is not equal to any of the
v_i .
2. A function G : M -> S with the property that given any document
m in M , it is computationally infeasible to find document
m' /= m such that G(m') = G(m)
For the function F , we can use any one way function T whose domain is
the set of keys. The second property of F follows easily from the second
property of the one way function T . We will discuss later how the function
G can be constructed from an ordinary one way function.
For convenience, we assume that P wants to generate only a single
signed document. Later, we indicate how he can sign many different documents.
The sender P first chooses 40 keys k_i such that all the values F(k_i) are
distinct. (Our second assumption about F makes this easy to do.) He puts
in public repository the data item o = (F(k_1), ... , F(k_40)) . Note that
P does not divulge the keys k_i , which by our first assumption about F
cannot be computed from o .
To generate a signature for a document m , P first computes G(m) to
obtain a set [i_1, ... , i_20] of integers. The signature consists of the 20
keys k_i_1, ... , k_i_20 . More precisely, we have o_p(m) = (k_i_1, ... , k_i_20) ,
where the i_j are defined by the following two requirements:
(i) G(m) = {i_1, ... , i_20}
(ii) i_1 < ... < i_20
After generating the signature, P can destroy all record of the 20 keys k_s
with s not in G(m) .
To verify that 20-tuple (h_1, ... , h_20) is valid signature o_p(m)
for the document m, one first computes G(m) to find the i_j and then uses
o to check that for all j , h_j is the i_j^th key. More precisely, the
signature is valid if and only if for each j with 1 _< j _< 20 :
F(h_j) o_i_j , where o_i denotes the i^th component of o , and the i_j are
defined by the above two requirements.
To demonstrate that this method correctly implements digital signatures,
we prove that it has the following properties.
1. If P does not reveal any of the keys k_i , then no-one else can
generate valid signature o_p(m) for any document m .
2. If P does not reveal any of the keys k_j except the ones that
are contained in the signature o_p(m) , tnen no-one else can
generate valid signature o_p(m') for any document m' /= m .
The first property is obvious, since the signature o_p(m) must contain
20 keys k_i such that F(k_i) = o_i , and our first assumption about F states
that it is computationally infeasible to find the keys k_i just knowing the
values F(k_i) .
To prove the second property, note that since no-one else can obtain any
of the keys k_i , we must have o_p(m') = o_p(m) . Moreover, since the o_i are
all distinct, for the validation test to be passed by o_p(m') we must also
have G(m') = G(m) . However, our assumption about G states that it is
computationally infeasible to find such document m' . This proves the
correctness of our method.
For P to send many different documents, he must use a different o for
each one. This means that there must be sequence of 40-tuples o_1, o_2, ...
and the document must indicate which o_i is used to generate that document's
signature. The details are simple, and will be omitted.
3. Constructing the Function G
One way functions have been proposed whose domain is the set of documents
and whose range is a set of integers of the form {0, ... , 2^n - 1} for any
reasonably large value of n . (It is necessary for n to be large enough to
make exhaustive searching over the range of T computationally infeasible.)
Such functions are described in [1] and [2]. The obvious way to construct the
required function G is to let T be such a one way function, and define
G(m) to equal R(T(m)) , where R : {0, ... , 2^n - 1} -> S .
It is easy to construct function R having the required range and
domain. For example, one can compute R(s) inductively as follows:
1. Divide s by 40 to obtain quotient q and a remainder r
2. Use r to choose an element x from {1, ... , 40} (This is
easy to do, since 0 _< r _< 40 .)
3. Use q to choose 19 elements from the set {1, ... , 40} - {x} as
follows:
a. Divide q by 39 to obtain quotient ...
It requires careful analysis of the properties of the one way function T
to be sure that the resulting function G has the required property. We
suspect that for most one way functions T , this method would work. However,
we cannot prove this.
The reason constructing G in this manner might not work is that the
function R from {0, ... , 2^n} into S is a many to one mapping, and the
resulting "collapsing" of the domain might defeat the one way nature of T .
However, it is easy to show that if the function R is one to one, then
property (ii) of T implies that G has the required property. To construct
G we need only find an easily computable one to one function R from
{0, ... , 2^n - 1} into S , for a reasonably large value of n .
We can simplify our task by observing that the function G need not be
defined on the entire set of documents. It suffices that for any document
m , it is easy to modify m in a harmless way to get new document that is
in the domain of G. For example, one might include meaningless number as
part of the document, and choose different values of that number until he
obtains a document that is in the domain of G . This is an acceptable
procedure if (i) it is easy to determine whether a document is in the domain,
and (ii) the expected number of choices one must make before finding a
document in the domain is small.
With this in mind, we let n = 40 and define R(s) as follows: if the
binary representation of s contains exactly 20 ones, then R(s) = {i : the
i^th bit of s equals one} , otherwise R(s) is undefined. Approximately
13% of all 40 bit numbers contain exactly 20 ones. Hence, if the one way
function T is sufficiently randomizing, there is a 0.13 probability that any
given document will be in the domain of G . This means that randomly
choosing documents (or modifications to a document), the expected number of
choices before finding one in the domain of G is approximately 8. Moreover,
after 17p choices, the probability of not having found document in the
domain of G is about 1/10^p. (If we use 60 keys instead of 40, the expected
number of choices to find document in the domain becomes about 10, and 22p
choices are needed to reduce the probability of not finding one to 1/10^p.)
If the one way function T is easy to compute, then these numbers
indicate that the expected amount of effort to compute G is reasonable.
However, it does seem undesirable to have to try so many documents before
finding one in the domain of G . We hope that someone can find more
elegant method for constructing the function G , perhaps by finding a one to
one function R which is defined on a larger subset of {0, ... , 2^n} .
Note; We have thus far insisted that G(m) be a subset of
{1, ... , 40} consisting of exactly 20 elements. It is clear that the
generation and verification procedure can be applied if G(m) is any proper
subset. An examination of our correctness proof shows that if we allow G(m)
to have any number of elements less than 40, then our method would still have
the same correctness properties if G satisfies the following property:
- For any document m , it is computationally infeasible to find a
different document m' such that G(m') is a subset of G(m) .
By taking the range of G to be the collection of 20 element subsets, we
insure that G(m') cannot be proper subset of G(m) . However, it may be
possible to construct a function G satisfying this requirement without
constraining the range of G in this way.
REFERENCES
[1] Diffie, W. and Hellman, M. "New Directions in Cryptography".
_IEEE Trans. on Information Theory IT-22_ (November 1976),
644-654.
[2] Rabin, M. "Digitalized Signatures", in _Foundations of
Secure Computing_, Academic Press (1978), 155-168.
FOOTNOTES
[1] The elements of K are not keys in the usual cryptographic sense, but are
arbitrary data items. We call them keys because they play the same role as
the keys in Rabin's algorithm.
- ---
Editor's note:
Some characters in the original paper were changed to members of the
Roman alphabet.
/= takes the place of the "not equal" symbol.
_< takes the place of the "less than or equal" symbol.
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