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U n u s u a l Side Channel Countermeasure Ideas

U n u s u a l Side Channel Countermeasure Ideas (that lend themselves to some form of provability). The Solutions’ Galaxy. The Solutions’ Galaxy. Hamster Wheel Keys Eric Brier, David Naccache, Nigel Smart. A common practice. ID, i.

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U n u s u a l Side Channel Countermeasure Ideas

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  1. Unusual Side Channel Countermeasure Ideas (that lend themselves to some form of provability)

  2. The Solutions’ Galaxy

  3. The Solutions’ Galaxy

  4. Hamster Wheel Keys Eric Brier, David Naccache, Nigel Smart

  5. A common practice ID, i In the past several authors proposed to prevent side channel attacks by having a key evolve in time. The typical setting is the following: ki move ki to RAM k write ki+1=H(ki) in NVM erase ki from NVM k0=f(ID,k) ki=H(H(…H(k0)…) secure communication using ki

  6. The time consuming part ID, i ki move ki to RAM k write ki+1=H(ki) in NVM erase ki from NVM k0=f(ID,k) ki=H(H(…H(k0)…) secure communication using ki

  7. Implemented Solutions Repeated application of H Hashing trees, even patented.

  8. Issues Repeated application of H: The system slows down with time. Hashing trees: Clumsy bookkeeping and sensitive to card tearing. Most importantly: we want to quantify leakage, i.e. model leakage depending on the H we use.

  9. The Alternative H(k) = a kb mod p Why? Because the terminal has an easy shortcut: Hi(k) = au kv mod p where u=(bi-1)/(b-1) mod(p) and v=bi mod (p)

  10. Quick Implementation Hi(k) = au kv mod p where u=(bi-1)/(b-1) mod(p) and v=bi mod (p) Precompute C=k a1/(b-1) Precompute D=1/a Hi(k) = DCv mod p

  11. Variants H(k) = {akb mod 3, akb mod 5,…, akb mod pi} Advantage: Word operations instead of long-integer arithmetic. Note that different a and b values could be used for different coordinates. However, as will be seen later, this is less secure wrt side channel leakage as each coordinate can be an independent target to side channel analysis.

  12. Before We Proceed We do not claim the invention of these PRNGs! The main contribution of this work is : • Stress that one can capitalize on the shortcut offered by their arithmetic properties to very simply implement key-evolving smart-card based protocols. • Analyze the resilience of these generators to leakage of a piece of the key.

  13. Realistic Assumptions a, k and p can be arbitrary and secret. No penalty. b would typically be of moderate size because of the burden of exponentiation on the card’s size. Hence, we should reasonably assume that b is public.

  14. Leakage Model At each iteration some bits of axb mod p and x leak. Question: Under which assumptions can we infer k? Advantage of looking at the problem from this angle: we have algebraic tools to analyze multivariate modular equations. The variables in question are the chunks of axb that the side channel does not provide at each session.

  15. H(x) = x2 mod n This is the BBS generator. If less than log log n bits leak at each step then this is secure under the factoring assumption even when n is known to the attacker. If n is known to the attacker and each operation leaks “more” bits of x, then x can be inferred. Analysis of “more” in two slides. If leakage is in between or n unknown: open problem.

  16. H(x) = x2 mod p See Gomez, Gutierrez and Ibeas for known p. If ¾ of x leak than x is revealed. (Same performance as brutal linearization). But we can do better. Consider the equation (A+x)2=B+y mod p Here A and B is what leaks via side channel. Denote this equation E Gomez, Gutierrez and Ibeas “Cryptanalysis of the quadratic generator“

  17. Consider all the equations of the type xi Ej which are verified modulo nj with i+2j2d This gives a constraint of the order of n to the power of the sum of the j for i+2j2d, which equals d(d+1)(2d+1)/6 The degrees of freedom on all linerarized variables is of the order n to the power the sum of the (i+j) for i+2jd, which gives d(d+1)^2 We hence get a size ratio < the quotient of these two sizes, which simplfies into (2d+1)/6/(d+1). This quantity tends to 1/3 when d.

  18. If H(x)=xe mod n, the constraints are sont i+ejed. We get the same contraintes with more freedom i.e. d(d+1)(6+e+e^2+2de+2de^2). The factorized ratio is then 2(2d+1)/(6+e+e^2+2de+2de^2) This tends 2/(e^2+e) when d tends to infinity. As e increases the attacker’s handicap increases very quickly.

  19. H(x) = x+P on an ECC See Gutierrez and Ibeas for known p. If 5/6 of x leak than x is revealed. For unknown P or unknown ECC: open problem. Same techniques should normally apply but we did not check in detail. Gutierrez and Ibeas, « inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits » H(x) = 2x on an ECC

  20. Practical Recommendations H(k) = a kb mod n Use unknown a, unknown composite n and b=8. a and k should be of the size of n. Use only ¼ of the bits of H(k) as key material. Use one bit out of four in H(k) as key material.

  21. Quick Implementation Let C=f(1,ID,MasterKey)=k a1/7 Let D=f(2,ID,MasterKey)=1/a Solve and personalize k and a in the card The terminal uses the shortcut formula: Hi(k) = DCv mod nwhere v=8i mod (n)

  22. A Possible Implementation ID, i ki move ki to RAM k write ki+1=aki8 in NVM erase ki from NVM C=f(1,ID,k) D=f(2,ID,k) v=8i mod(n) ki=DCv mod n secure communication using ki

  23. An Ideal Power Attack Countermeasure (in 3 slides) Jean-Max Dutertre, Amir Pasha Mirbaha,David Naccache, Assia Tria

  24. Idea Power the µP from a photovoltaic panel facing a powerful LED. Vcc IO VccµP CLK RST + - Vss

  25. Constructing the Device We are currently ordering a photovoltaic panel about the size of a smart card and an OLED panel about the same size. Step 1: Place both panels face-to-face, have the OLED glow to its maximal capacity and check that the derived power allows to power the µP. LED PV Panel

  26. Constructing the Device Step 2: Characterize the energy transfer-rate as function of resistor value. Step 3: Construct a generic power attack isolation board. Vcc IO VccµP CLK RST + - Vss

  27. For More on PV Physics http://en.wikipedia.org/wiki/Solar_cell

  28. Can’t Do Less David Naccache, Christof Paar, Florian Praden

  29. Investors deal with two questions How to get funds? Logistics How to spend funds rationally? Tactics & Strategy Here we address the second.

  30. The Subleq Machine Subleq is a Turing-complete machine having only one instruction. subleq a b c  *(b)=*(b)-*(a)  if the result is negative or zero, go to c else execute the next instruction.

  31. The Subleq Machine Since subleq has only three arguments and since there is no confusion of instructions possible (there is only one!), a subleq code can be regarded as a sequence of triples. a1 b1 c1 a2 b2 c2 a3 b3 c3 :

  32. …interleaved with data Since data can be embedded in the code, the sequence of triples can be interleaved with data. For instance: a1 b1 c1 data1 data2 a2 b2 c2 data3 a3 b3 c3 :

  33. How does it work? *b = *b-*a; if (*b0) program_counter = c; else program_counter = program_counter+3;

  34. Genealogy Subleq is an OISC (“One Instruction Set Computer) which comes from the Minsky machine concept. The Minsky machine is a register machine with only two instructions: “increment” and “decrement-and-branch”.

  35. Allowing for comfort Memory is loaded with instructions and data altogether (no distinction). Hence the code can potentially self-modify and consider that any cell is a, b or c. We can pre-store constants (like 0,1 etc) e.g. we devote a cell called Z to contain zero, N to contain -1

  36. What does this do? subleq Z Z c

  37. JMP c subleq Z Z c

  38. What does this do? subleq a a $+1

  39. CLR a subleq a a $+1

  40. What does this do? CLR b subleq a Z $+1 subleq Z b $+1 CLR Z

  41. MOV a b subleq b b $+1 *b=0 subleq a Z $+1 Z=-*a subleq Z b $+1 *b=0-(-*a)=*a subleq Z Z $+1Z=0

  42. What does this do? subleq a Z $+1 subleq b Z $+1     CLR c subleq Z c $+1 CLR Z

  43. ADD a b c subleq a Z $+1 Z=0-*a subleq b Z $+1     Z=-*a-*b subleq c c $+1 *c=*c-*c=0 subleq Z c $+1 *c=0+*a+*b sublez Z Z $+1 Z=0

  44. What does this do? CLR t CLR s     subleq a t $+1 subleq b s $+1 subleq s t $+1 CLR c CLR s subleq t s $+1 subleq s c $+1

  45. SUB a b c subleq t t $+1 *t=0 subleq s s $+1    *s=0 subleq a t $+1 *t=-*a subleq b s $+1 s=-*b subleq s t $+1 t=-*a+*b subleq c c $+1 *c=0 subleq s s $+1 *s=0 subleq t s $+1 *s=0-(-*a+*b)=*a-*b subleq s c $+1 *c=0-(*a-*b)=*b-*a not really optimal code just to illustrate

  46. What does this do? CLR t subleq a t $+1     CLR s subleq t s $+1 subleq b s c

  47. BLE a b c subleq t t $+1 t=0 subleq a t $+1     *t=-*a subleq s s $+1 *s=0 subleq t s $+1 *s=*a subleq b s c *s=*a-*b if *a-*b0 goto c

  48. What does this do? CLR t subleq a t $+1     CLR s subleq b s $+1 subleq s t $+1 subleq N t c

  49. BHI a b c subleq t t $+1 *t=0 subleq a t $+1     *t=-*a subleq s s $+1 *s=0 subleq b s $+1 *s=-*b subleq s t $+1 *t=-*a+*b subleq N t c *t=-*a+*b-(-1) if *b-*a+10 goto c

  50. What have we got so far? JMP a goto a MOV a b *b=*a SUB a b c *c=*b-*a ADD a b c *c=*b+*a BHI a b c if *b-*a+10 goto c if *b<*b+1*a goto c if *b<*a goto c if *a>*b goto c BLE a b c if *a-*b0 goto c if *a*b goto c CLR a *a=0

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