Basic Cryptography Digital Signature

8/3/2019 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

Sh

S A

m m* s

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Mh S

Boolean

(m*,s) VAtrue

false



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M

MS

S

R S A

m m* s

MR

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The signing process

M

MS

S

R S A

m m* s

MR



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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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Basic Cryptography Digital Signature

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