Basic Cryptography Digital Signature
Transcript of Basic Cryptography Digital Signature
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Elements of applied cryptography
Digital Signatures
! Digital Signatures with appendix
! Digital signatures with message recovery
! Digital signatures based on RSA
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The signing process
M Mh
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m m* s
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Mh S
Boolean
(m*,s) VAtrue
false
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The signing process
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The signing process
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BREAKING A SIGNATURE
1. Total break < 4(3'0.405*/.*4:&'*,$*;$=29,'*,-'*./)#'01.*private key
2. Selective forgery < adversary controls the messages
whose signature is forged
3. Existential forgery < adversary has no control on the
messages whose signature is forged
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BASIC ATTACKS
! KEY-ONLY ATTACKS < 4(3'0.405*"#$%.*$#&5*,-'*./)#'01.*
public key
! MESSAGE ATTACKS
a. known-message attack < adversary has signatures for a set of messages which are known by the adversary butnot chosen by him
b. chosen-message attack < in this case messages are
chosen by the adversary
c. adaptive chosen-message attack < in this casemessages are adaptively chosen by the adversary
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Introductory comments
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1. Generate two large, distinct primes p , q (100 200
decimal digits)
2. Compute n = p q and = (p -1) (q -1)
3. Select a random number 1 < e < such that gcd(e, ) = 1
4. Compute the unique integer 1 < d < such that
ed 1 mod
5. (d , n ) is the private key
6. (e , n ) is the public key
At the end of key generation, p and q must be destroyed
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Signature generation. In order to sign a message m, A does
the following
1. Compute m* = R(m) an integer in [0, n <1]
2. Compute s = m*d mod n
3. 61.*./)#4,90'*+$0*=*/.*.
Signature verification>*?#*$0('0*,$*3'0/+5*61.*./)#4,90'*.*4#(*
recover message m, B does the following
1. @:,4/#*61.*49,-'#,/;*29:&/;*"'5*A e , n )
2. Compute m* = se mod n
3. Verify that m* is in MR; if not reject the signature
4. Recover m = R-1(m*)
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! If s is a signature for a message m, then s = m*d mod n
where m* = R(m).
! Since ed = 1 (mod ), se = m*ed = m* (mod n). Finally,
R-1(m*) = R-1(R(m)) = m.
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! Redundancy function
! A suitable redundancy function is necessary in order to avoid
existential forgery
! IOS/IEC 9796 (1991) defines a mapping that takes a k-bit
integer and maps it into a 2k-bits integer
! The RSA digital signature scheme with appendix
! MD5 (128 bit)
! PKCS#1 specifies a redundancy function mapping 128-bit
integer to a k-bit integer, where k is the modulus size (k 512,
k = 768, 1024)
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! Performance characteristics
! Let p = q = k then
! signature generation requires O(k3
) bit operations! signature verification, in the case of small public exponent,
requires O(k2) bit operations
! Suggested value for e in practice are 3 and 216+1. Of course, p and q
must be chosen so that gcd(e, (p < 1)(q < 1)) = 1.
! The RSA signature scheme is ideally suited to situations where
signature verification is the predominant operation being
performed.
! Example. A trusted third party creates a public-key certificate for an entityA. This requires only one signature generation, and this signature may be
verified many times by various other entities
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! Parameter selection
! bitsize of the modulus: miminum 768; at least 1024 for signatures of
longer lifetime or critical for overall security of a large network (i.e.,
the private key of a certification authority)
! No weaknesses have been reported when the public exponent e ischosen to be a small number such as 3 or 216+1.
! It is not recommended to restrict the size of the private exponent d in
order to improve the efficiency of signature generation
! Bandwidth efficiency ! By definition, BWE = log2 ( MS ) / log2 ( MR )
! For (RSA, ISO/IEC 9796), BWE = 0.5, that is, with a 1024-bits
modulus can be signed 512-bits messages
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! System wide parameters
! Each entity must have a distinct RSA modulus; it is insecure to
use a system-wide modulus! The public exponent e can be a system-wide parameter, and is
in many applications. In this case, the low exponent attack must
be considered
! Short vs. long messages
! Suppose n is a 2k-bit RSA modulus which is used to sign k-bit
messages (i.e., BWE is 0.5)
! Suppose entity A wishes to sign a kt-bit message m
! For t = 1 RSA with message recovery is more efficient;
! For t > 1, RSA with appendix is more efficient
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! Signature
! select k Zp <1* randomly
! r = g k mod p , s = (h (m ) !xr )k !1 mod (p <1)
! The pair (r , s ) is the digital signature for m ! Verification
! Verify that 1 r p <1; if not reject the signature
! Compute v 1 = y r r s mod p
! Compute h (m ) and v 2 = g h (m ) mod p
! Accept the signature only if v 1 = v 2.
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Proof
! If the digital signature (r , s ) has been produced by Alice
then s = (h (m ) !xr )k !1 mod (p <1).
! Multiplying both sides by k gives ks = (h (m ) !xr ) mod (p <
1). Rearranging yields h (m ) ks +xr mod (p <1).
! This implies that g h (m ) g ar +ks (g x )r r s mod p
! Thus v 1 = v 2 as required.
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