### Table 2: Overview of pseudo-random generator constructions.

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

"... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table 2 for derandomizing the process. The symbol A... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table2 for derandomizing the process. The symbol A... In PAGE 3: ...Table2... In PAGE 8: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 8: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table 2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ...More Applications and New Derandomizations We will now apply the general framework of Section 4 to various fundamental constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, however, that weaker assumptions can be taken (the weakest being that the polynomial- time hierarchy does not collapse) in order to achieve weaker, but still subexponential, deterministic simulations.... ..."

### Table 2: Overview of pseudo-random generator constructions.

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

1999

"... In PAGE 2: ... We formally define the notion of a success pred- icate in Section 4. If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table 2 for derandomizing the process. The symbol A in Table 2 represents an arbitrary class of oracles.... In PAGE 2: ... We formally define the notion of a success pred- icate in Section 4. If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table2 for derandomizing the process. The symbol A in Table 2 represents an arbitrary class of oracles.... In PAGE 2: ... If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table 2 for derandomizing the process. The symbol A in Table2 represents an arbitrary class of oracles. To illustrate the power of our generalization, we apply it to the following randomized processes from different areas of theoretical computer science.... In PAGE 5: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of ran- dom bits, and suppose that B can efficiently check (F; ). Then the hardness conditions on the left-hand side of Table2 provide pseudo-random generators G with complexity and seed lengths s as specified on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x;Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will first analyze the complexity of an oracle B capable of effi- ciently checking the associated success predicate and then con- struct a pseudo-random generator secure against B based on a func- tion with presumed hardness against B.... In PAGE 5: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of ran- dom bits, and suppose that B can efficiently check (F; ). Then the hardness conditions on the left-hand side of Table 2 provide pseudo-random generators G with complexity and seed lengths s as specified on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x;Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will first analyze the complexity of an oracle B capable of effi- ciently checking the associated success predicate and then con- struct a pseudo-random generator secure against B based on a func- tion with presumed hardness against B.... In PAGE 5: ... 5 More Applications We will now apply the general framework of Section 4 to various other constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, however, that weaker assumptions can be taken (e.... ..."

Cited by 72

### Table 4. Characteristics of pseudo-random pattern generators

"... In PAGE 4: ... 5. Estimated overhead As shown in Table4 , the GLFSRs differ in hardware costs and in randomness, dependent on the selected char- acteristic polynomial which was calculated from the cho- sen ( ; m) pair and the generating polynomials. Gener- ally, it can be stated that the GLFSR implementation needs more area than LFSRs and approximately the same area as LHCAs (after optimization, i.... In PAGE 4: ...HCAs (after optimization, i.e., removing redundant XOR gates). Table4 shows that the LFSRs, i.... In PAGE 4: ... The table shows also, that choosing a suitable GLFSR leads to a much better SCC and less XOR gates than any LHCA. In Table4 , num- ber 5 is the GLFSR which fulfills the requirements of low area and good SCC in the best way. Regarding test time, GLFSRs have the advantage of higher fault coverage for the same test length.... ..."

### Table 1: Pseudo-random generators comparison Reference seed size circuit size

2006

### Table 4: Instances arising from pseudo-randomly distributed generators 24

1994

"... In PAGE 23: ... If we are not in the \nice case quot;, both methods have to make a similar number of comparisons, and therefore, if the \bad cases quot; outweigh, the di erence of e ciency of the two methods gets smaller. Table4 shows the same running time data for generator sets being uniformly pseudo- randomly distributed in [0; 1]2. The rst column contains the number of generators.... ..."

Cited by 2

### Table 1: Types of Pseudo-Random Number Generators in PRNGlib irng Generator typ

"... In PAGE 8: ...ower triangle and to 1 on the diagonal (see Section 3.3.3). For the LF and GSR generators, the initialized mantissa l should be close to l, the mantissa of double precision oating point numbers. The generator types are numbered by the parameter irng (see Table1 ). All informa- tion about the actual status and parameter settings is stored in the integer and real work arrays iw(1:liw) and rw(1:lrw) (see Tables 2 and 3).... In PAGE 16: ... Currently PRNGlib provides the interface to the four random number generator types LFA, LFS, GSR, and MLC (cf. Table1 ). For future extension of PRNGlib by a generator typ the six routines INItyp, UNItyp, SKPtyp, PERtyp, CHKtyp, INFtyp must be provided (Package 5).... In PAGE 50: ...A Error messages Table1 1: Returned error values in PRNGlib value description subroutines 0 no error -1 error in distribution RANDIS, : : : 1 wrong number of processors npe PARBLK 2 wrong n or lstart PARBLK 10 open error RANIN, RANOUT 20 integer read/write error RANIN, RANOUT 30 double precision read/write error RANIN, RANOUT 100 idis is out of choices RANDIS 200 irng is out of choices RANCHK, : : : 300 lvec is out of choices RANUNI 1001 liw 10 RANCHK, CHKLF 1002 lrw iw(4) = r 1003 l = iw(2) not multiple of 2. 1004 pointer iw(3) lt; 0 or iw(3) iw(4) = r 1005 s = iw(5) 0 or iw(5) gt; iw(4) = r 1006 r = iw(4) gt; 9689 3001 liw 10 + 2iw(4) RANCHK, CHKGSR 3002 lrw 6 3003 = l iw(2) not multiple of 4.... ..."

### Table 1: Frequency Component Con- structed using Pseudo-Random Num- ber Generator

"... In PAGE 8: ... Table1 0: Monthly Annuity Payments Based on a $100,000 House and an Annuitant Age 65 at Purchase Monthly Annuity Pay- ments Contract Interest Rate 11.... ..."

### Table 2: Overview of pseudo-random generator constructions. To illustrate the power of our generalization, we apply our technique to the following fundamental constructions from di erent areas of theoretical computer science. 2

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

"... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table 2 for derandomizing the process. The symbol A... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table2 for derandomizing the process. The symbol A... In PAGE 3: ...Table2... In PAGE 9: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table 2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ... 5 More Applications and New Derandomizations We will now apply the general framework of Section 4 to various fundamental constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, how- ever, that weaker assumptions can be taken (the weakest being that the polynomial-time hierarchy does not collapse) in order to achieve weaker, but still subexponential, deterministic simulations.... ..."