1. Fault-Secure Parity Prediction Arithmetic Operators MICHAEL NICOLAIDIS RICARDO O. DUARTE TIMA Laboratory SALVADOR MANICH JOAN FIGUERAS Polytechnic University of Catalonia

2. Outline What is the problem ? Efficient Self-Checking Arithimetic Analysis of fault secureness Single-cell fan-out networks Take some examples Multiple-cell fan-out networks Implementations and experiments Conclusions

3. What’s The Problem ? DRAWBACK: Parity prediction may not achieve fault- secureness, because single faults propagate o- -utput errors of random multiplicity. Arithmetic codes can ensure falut secureness for most arithmetic operators. Problem  1.Arithmetic code checkers are complex circuit 2.Arithmetic code are not compatible with parity checking 3.Must implement whole system with A~ code

4. Efficient self-checking arithimetic Parity prediction schemes carries a lower overhead and a lower performance penalty than arithmetic coding schemes for small and medium-size operands. And we are interest in efficient parity prediction implementations. Method : Include arithemetic operators

5. Analysis of fault secureness Definition: In a self-checking design, a circuit is fault secure if its output code detects all errors generated by each modeled fault. Restriction: Cannot achieve fault secureness because even-multiplicity errors escape detection by parity code. -  Solution : 1. 先分析 parity prediction cellular arithmetic operators 來看不符合的原因. 2. 一一排解. continuous

6. Analysis of fault secureness Basic components of cellular arithmetic operators : full- and half- adder cells. 5 大主題 : Lemma1. 將不符合 fault-secure 的 parity predictor 電路給 它獨立出來. Lemma2. 在不符合 fault-secure 的 parity predictor 電路中, 將經由運算得到的 carries, 拿來推算其可能產生的 parity. continuous

7. Analysis of fault secureness Lemma3. 証明如果上圖的 sum(S) 和 carry(C) 使用同一個電 路則 arithmetic operator 將無法 fault secure. continuous

8. Analysis of fault secureness Lemma4. 証明如果一個 cell 的 inputs 只能生成 sum,carry, 跟 redundant carry 任兩個, 則 operator 也無法 fault secure. continuous

9. Analysis of fault secureness Lemma5. 証明如果說 C,Cp 和 S 在 logic 有 fault 而產生 error 給 C 跟 Cp, 則儘管 error 會傳給下一個 S, 但其仍可符合 fault secureness

10. Single-cell fan-out networks Definition: any network in which each cell output enters exactly one input of exactly one cell is a single-cell fan-out networks. Theorem1 : Parity prediction in a single-cell fan-out network using full- and half- adder cell designed as in Fig.3 achieves fault secureness. continuous

11. Single-cell fan-out networks Theorem2 : The full-adder cell of Fig.4 prserves fault secureness.

12. Examples Ripple-carry adders.

13. Examples Parity prediction multipliers.

14. Examples Improved parity prediction multiplier.

15. Multiple-cell fan-out networks A more complex implementation is required if some signals enter an even number of cell inputs. Fig.9b is one solution. Fig.9c is another solution with a double-rail checker.

16. Examples Nonrestoring-array divider. The array’s basic cell is the controlled add/subtract cell.

17. Self-Checking nonrestoring-array divider

18. Divider points Each carry signal enters the row’s rightmost cell twice. The divider generates two independent sets of outputs(remainder and quotient). The leftmost cell’s carry and sum outputs have opposite values,and either can provide the quotient outputs. The parity of sum outputs is PS=PCp+PN+PD+1  for even number of rows,it becomes PS=PCp+PN+1.

19. Implementations and experiments

22. Conclusions Hardware cost of these fault-secure structures is significant. For small nad medium size operators,these solution are also more efficient than operators based on arithmetic codes which require complex checkers and code translators for compatibility.