1. Presented By: Raquel Whittlesey-Harris 12/04/02
Elements of Security
Presented By:
Raquel Whittlesey-Harris
12/04/02

2. Contents Introduction Definitions Security Against an Adversary
Example
Theorems
References
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3. Introduction
Stabilization Theory was introduced by Edsger Dijkstra in 1974
Two building blocks of stabilization theory was introduced in 1991, by Anish Arora and Mohamed Gouda
The two elements can adequately explain fault-tolerant computing
Closure
Convergence
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4. Introduction The third element, protection, is introduced here
The three adequately explain system security
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5. Definitions Computing System, Consist of
Nonempty set of variables with values from predefined domains, and
Nonempty set of actions that can be executed to update the values of the variables
<guard>  <statement>
<guard> is a Boolean expression over the system variables
<statement> is a sequence of assignment statements over the system variables
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6. Definitions State of system S Computation of system S
Triple (p,c,p’) where,
States of S, p (tail state) and p’ (head state)
c – guard is true at state p
Computation of system S
An infinite sequence of transitions of S where the following hold,
Order – head state of each transition is the same as the tail state of the next transition in the sequence
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7. Definitions The starting state of the computation,
Fairness – if a sequence has a transition where action c of system S is enabled at p, then c is executed or the sequence has a later transition where c is executed or where c is not enabled at p
The starting state of the computation,
Is the tail state, p, of the 1st transition in a computation
A computation is said to reach state p if a transition has a state, p, as the tail or head state of that transition
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8. Definitions State Predicate of System S
A function that has a Boolean value at each state of S
P is a state predicate of system S
A state is called a P-state iff the value of P is true at that state
P is closed in S iff for each transition (p,c,p’) of S, if p is a P-state, then p’ is a P-state
P and Q are two state predicates of system S
P implies Q, P  Q, iff for every state p of S, if p is a P-state, then p is a Q-state
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9. Definitions
V is a subset of variables of a system S and P and Q are state predicates of S
S is called V-safe from P to Q iff the following conditions hold,
Closure: P and Q are closed in S and Q  P in S
Convergence: Every computation of S that starts at a P-state reaches a Q-state; every computation that starts at an illegitimate state eventually reaches a legitimate state
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10. Definitions
Protection: No variable in V is written in any transition (p,c,p’) of S where p is a P-state but no a Q-state; no transition of S that starts at an illegitimate stat can affect the critical variables in V
P identifies all reachable states of S that can be reached under any interleaving of system execution and adversary interference
Q identifies all legitimate states of system S that can be reached under system execution only
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11. Security Against an Adversary
An Adversary, D, of system is a set of actions of the form
<guard>  <statement>
Transition of D is a triple (p,d,p’) where,
p and p’ are states of S,
d is an action of D,
guard of D is true at p
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12. Security Against an Adversary
Execution of d when S is in p yields S in p’
P is a state predicate of S and D is an adversary of S
P is closed in D iff for each transition (p,d,p’) of D, if p is a P-state, then p’ is a P-state
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13. Security Against an Adversary
V is a subset of variables of S and P and Q are state predicates of S
System P is called V-secure from P to Q against D iff the following hold,
Safety: S is V-safe from P to Q
Adversary Closure: P is closed in D
Adversary will maintain the system within the reachable states
Adversary cannot corrupt the critical variables in V
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14. Security Against an Adversary
Adversary Protection: No variable in V is written in any transition (p,d,p’) of D where p is a P-state
If S is V-secure from P to Q against D, then all computations, C, that start at a Q-state and consist of an infinite # of S transitions and a finite # of D transitions, satisfies,
Computation C has an infinite suffix whose transitions are all S transitions and whose states are all Q-states
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15. Security Against an Adversary
Every transition in C that updates the variables in V is an S transition whose tail and head states are Q-states
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16. Example Secure Data xfer System S
Sender process sends a continuous stream of data items to a receiver process via three shared variables that are written by the sender and read by the receiver
Shared var seq, data, chk : integer
seq – contains the sequence number of the current data item
data – contains the current data item
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17. Example Sender has the following variables
chk – contains an integrity check for the current values of seq and data
chk = H.(ss|seq|data); a secure hash function applied to the concatenation of a secret value ss and the current values of seq and data
ss is known to only the sender and receiver
Sender has the following variables
local var sent : array [integer] of integer,
x : integer
sent is an infinite array containing all of the data items to be sent by the sender
x is an index of sent
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18. Example Sender contains action,
true  seq := x; data := sent[seq]
chk := H.(ss|seq|data); x := x+1
Receiver contains the following local variables,
local var rcvd : array[integer] of integer,
y, z : integer
rcvd is an infinite array containing all data items received by the receiver
z is an index of array rcvd
y contains the sequence # of the last data item received by the receiver
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19. Example Receiver has one action, Set V of critical variables is,
true  if seq > y H.(ss|seq|data) = chk 
y,rcvd[z], z:= seq, data, z+1
[] seq  y H.(ss|seq|data) chk 
skip fi
Set V of critical variables is,
V = {rcvd, z}
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20. Example P and Q are state predicates,
P defines the set of reachable states
Q defines the set of legitimate states
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21. Example
S is V-safe from P to Q can be shown by showing that the 3 conditions hold
P & Q are closed in S & Q  P in S; closure
Sender action is continuously enabled & any execution of this action starting from a P-state leads S to a Q-state; convergence
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22. Example Adversary D has one action,
Neither variable in V is updated in any S transition that starts at a (P  not Q)-state; protection
Adversary D has one action,
Q  seq := any value in the range 0..x-1; data :=
sent[seq];
chk := H.(ss|seq|data); x:= x+1
D attacks S only when S is at a Q-state
By replaying old messages
Closure and protection hold
S is V-secure from P to Q against D
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23. Theorems Base Theorem Union Theorem
If P is closed in S,
Then S is V-secure from P to P against E
Union Theorem
if S is V-secure from P to Q against D and
S is V’-secure from P to Q against D,
then S is (VV’)-secure from P to Q against D
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24. Theorems Adversary Union Theorem
If S is V-secure from P to Q against D and
S is V-secure from P to Q against D’
then S is V-Secure from P to Q against (DD’)
Junctivity Theorem
If S is V-secure from Q to P against D and
S is V-secure from Q’ to P’ against D,
then S is V-secure from Q  Q’ to P  P’ against D, and
S is V-secure from Q  Q’ to P  P’ against D
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25. Theorems Transitivity Theorem
If S is V-secure from P to Q against D and
S is V-secure from Q to R against D,
Then S is V-secure from P to R against D
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26. Theorems Weakening Theorem If S is V-secure from P to Q against D,
V’ is a subset of V,
P’ is closed and P’  P in S,
Q’ is closed and Q  Q’ and Q’  P’ in S, and
D’ is a subset of D and P’ is closed in D’,
Then S is V’-secure from P’ to Q’ against D’
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27. Theorems Proof of the weakening theorem
From the antecedent of the weakening theorem, the
following 5 assertions hold
S is V-secure from P to Q against D
V’ is a subset of V
P’ is closed and P’  P in S
Q’ is closed and Q  Q’ & Q’  P’ in S
D’ is a subset of D and P’ is closed in D’
From (1), the following five assertions hold
S, P, & Q satisfy the closure condition
S, P, & Q satisfy the convergence condition
S, P, Q, & V satisfy the protection condition
D & P satisfy the adversary closure condition
D, P, & V satisfy the adversary protection condition
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28. Theorems From assertions (2) through (9), the following are concluded
From (3), (4), & (6), S, P’, & Q’ satisfy the closure condition
From (3), (4), and (6), S, P’, & Q’ satisfy the convergence condition
From (2), (3),(4), and (8), S, P’, Q’, & V’ satisfy the protection condition
From (5), D’ & P’ satisfy the adversary closure condition
From (2), (3), (5), & (10), D’, P’, & V’ satisfy the adversary protection condition
From assertions (11) through (15), it is concluded that S is V’-secure from P’ to Q’ against D’
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29. References
M. G. Gouda, Elements of Security: Closure, convergence, and protection, Information Processing Letters 77 (2001)
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